Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
lwm pl:ún nig Esn Bisidæ
sñaédeá£Bum<pßay
esovePAKNitviTüa ½ 1-dMeNa£RsaylMhatKNitviTüa eá£Bum< qñaM2000 ( sRmabeRtomRbLgcUlsaklviTüal&y nig GaharUbkrN_ ) 2-BiPBsVIútcMnYnBit ( sRmabfñak´TI11 nig sisßBUEkKNitviTüa ) 3- GnuKmn_RtIekaNmaRt ( sRmabfñak´TI11 nig sisßBUEkKNitviTüa )
4- dMeNa£RsayKMrU cMnYnkMupøic lImIt edrIev ( sRmabfñak´TI12) 5- segçbrUbmnþKNitviTüa ( sRmabfñak´TI11-12 ) 6- KMrUsikßaGnuKmn_ ( sRmabfñak´TI11-12 ) 7- kMEnlMhatKNitviTüafñakTI10km μviFIsikßafμI ( PaK1 qñaM2008 ) 8- 151 KNnalImIt ( sRmabfñakTI11-12 ) 9- KNitviTüaCMuvijBiPBelak PaK1- PaK2 nig PaK3 ( sRmabsisßBUEkKNitviTüafñakTI11 nig 12 ) 10- kMEnlMhatKNitviTüafñakTI11 kMritmUldæan km μviFIsikßafμI ( PaK1 qñaM2009 ) 11-sIVúténcMnYnBit sRmab´fñakTI11 kMritx<s´ nig kMritmUldæan
GñkcUlrYmRtYtBinitübec©keTs
elak lwm qun elak Esn Bisidæ elakRs I Tuy rINa elak Titü emg elak RBwm sunitü elak pl b‘unqay
GñkRtYtBinitüGkçraviruTæ elak lwm miKþsir
karIkMuBüÚT&r kBaØa lI KuNÑaka
GñkniBnæ nig eroberog elak lwm plþún nig elak Esn Bisidæ
GarmÖkfa esovePA GnuKmn_ Gics,:ÚNgEsül elakarIt nig RtIekaNmaRt sRmabfñak´TI11 kMritmUldæan nig kMritx<s´ EdlGñksikßakMBugkan´enAkñúgéden£ xMJúáTáneroberog niBnæ eLIgkñúgeKalbMngTukCaÉksarsRmabGñksikßaEdlman bMngcg´yldwgGMBIemeronsIVúténcMnYnBiten¼[kan´Etc,as´ nigmüageTotedIm,I CaCMnYydlGñksikßaEdlmanbMngeRtomRbLgsisßBUEkEpñkKNitviTüa nig eRtomRbLgykGaharUbkrN_nana . enAkñúgesovePAen£eyIgxMJúánxitxMRsavRCaveRCIserIsyklMhat´ya¨g sRmaMgbMputykmkeFIVdMeNa£Rsayyagek,a£k,ayEdlGac[elakGñk gayylqabcgcaMGMBIsil,£énkareda£RsayTaMgGs´en£ . b¨uEnþeTa£Caya¨gNak¾eday kgV£xat nig kMhusqþgedayGectnaRákd CaekItmanCaBMuxan TaMgbec©keTs nig GkçraviruTæ . GaRs&yehtuen£ eyIgxMJúCaGñkeroberogrg´caMedayrIkrayCanic©nUvmtiri£Kn´ EbbsSabnaBIsMNak´GñksikßakñúgRKbmCÄdæanedIm,ICYyEklMG esovePAen£ [ánkan´EtsuRkiRtPaBEfmeTot . CaTIbBa©b´en£eyIgxMJúGñkeroberogsUmeKarBCUnBrdlGñksikßaTaMgGs´ [mansuxPaBmaMmYn nig TTYlC&yCMn£RKbParkic© .
átdMbgéf¶TI 12 kk;da 2009 GñkniBnæ lwm plþún Tel : (017) 768 246
Éksareyag 1> esovePAKNitviTüafñak;TI11 kMritmUldæan nig kMritx<s;rbs;RksYg Gb;rM qñaM 2009 2> Gius,Ú:Ng;Esül nig elakarIt rbs;elak lwm yUesg qñaM1999 3>103 Trigonometry Problems ( Titu Andreescu and Zuming Feng )
4> 360 Problems for Mathematical Contests ( Titu Andreescu and Dorin Andrica )
5> Mathematical Olympiad Treasures ( Titu Andreescu and Boddan Enescu )
6> International Mathematical Olympiads 1959-1977 ( Samuel L. Greitzer )
eroberogeday lwm pl:ún nig Esn Bisidæ
- 1 -
eroberogeday lwm pl:ún nig Esn Bisidæ
- 2 -
EpñkRbFanlMhat 1111----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ 22 3243 22)( xxxxxf −−−−++++++++−−−− ++++==== k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf . . . . x> rktémø x> rktémø x> rktémø x> rktémø x edIm,I[ edIm,I[ edIm,I[ edIm,I[ 20)( ====xf . . . . 2222----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_
22
22)(
1
++++++++====
++++
x
x
xf
k> k> k> k> cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit m nig nig nig nig n ebI ebI ebI ebI 1====++++ nm enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa 3)()( ====++++ nfmf . . . . x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk ∑∑∑∑
====
++++====
p
kp p
kfS
1 1 . . . .
3333----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xx
xx
ee
eexf −−−−
−−−−
++++−−−−====)(
k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit a nig nig nig nig b cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³ (((( ))))
)()(1)()(
bfafbfaf
baf++++
++++====++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 3 -
4444----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ (((( )))) (((( ))))xxxf 3232)( −−−−++++++++====
k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 2)( ≥≥≥≥xf . . . . K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar 14)( ====xf . . . . 5555----edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> 02433.49 2 ====++++−−−− ++++xx x> x> x> x> 01282.94 2 ====++++−−−− ++++xx K> K> K> K> 026.53 32132 ====++++−−−− ++++++++++++ xxx X> X> X> X> (((( )))) (((( )))) 102323 ====−−−−++++++++ xx g> g> g> g> (((( )))) (((( )))) 143232 ====++++++++−−−−
xx 6666----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
x
xx
x
xx
x
xx
x
xx
6
6
5
5
4
4
3
32222 −−−−−−−−−−−−−−−−
====++++++++ 7777----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 882264
2)1(2)1(2)1( ++++++++====++++ −−−−−−−−−−−− xxx 8888----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ (((( )))) 222210114 6333 xxxxx −−−−====++++++++ −−−−−−−−−−−−
eroberogeday lwm pl:ún nig Esn Bisidæ
- 4 -
9999----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 04882)14(4 ====++++−−−−−−−−++++ xx xx 10101010----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 0
421
215
7443
2.34.9 ====−−−−++++
−−−−−−−− −−−−−−−− xxxx
11111111----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 324232
3331
3 −−−−++++−−−−−−−− ++++====++++ xxxxx 12121212----edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar ³³³³
====
====
4321
43
721
32
yx
yx
13131313----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
323.2 11
2 ====++++−−−−
−−−− xx
x 14141414----eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday 10 ====a nig TMnak;TMng nig TMnak;TMng nig TMnak;TMng nig TMnak;TMng kMeNIn kMeNIn kMeNIn kMeNIn (((( ))))nana
na ++++++++ ++++==== 1
21 24log Edl Edl Edl Edl ...;2;1;0====n . . . . k>eKtag k>eKtag k>eKtag k>eKtag na
nb 21++++==== . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 5 -
bgðajfa bgðajfa bgðajfa bgðajfa 21 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n ....
x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa n
nb 23==== . . . . K>TajrktY K>TajrktY K>TajrktY K>TajrktY na énsIVút énsIVút énsIVút énsIVút (((( ))))na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . 15151515----eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday 10 ====a nigTMnak;TMng nigTMnak;TMng nigTMnak;TMng nigTMnak;TMng kMeNIn kMeNIn kMeNIn kMeNIn (((( ))))nbnana
na ++++++++++++ ++++++++==== 121
31 3327log Edl Edl Edl Edl ...;2;1;0====n . . . . k>eKtag k>eKtag k>eKtag k>eKtag na
nb 31++++==== . . . . bgðajfa bgðajfa bgðajfa bgðajfa 3
1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n .... x>KNna x>KNna x>KNna x>KNna nb nig nig nig nig na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . 16161616----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_
++++−−−−====
xx
xf11
lg)(
k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit 11 <<<<<<<<−−−− a nig nig nig nig 11 <<<<<<<<−−−− b cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa (((( )))) (((( ))))
++++++++====++++abba
fbfaf1
. . . .
17171717----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit (((( )))) 0≥≥≥≥nnu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³ au ====0 nig nig nig nig (((( )))) nu
nn uu 2log1 ====++++ Edl Edl Edl Edl 2>>>>a . . . .
k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 6 -
x>eKtag x>eKtag x>eKtag x>eKtag nn uV 2log==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa 21 nn VV ====++++ . . . .
K>eKtag K>eKtag K>eKtag K>eKtag nn VW 2log==== . . . . bgðajfa bgðajfa bgðajfa bgðajfa (((( ))))nW CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt X>KNna X>KNna X>KNna X>KNna nn VW ; nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . 18181818----eK[ eK[ eK[ eK[ x nig nig nig nig y CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; 12.544 −−−−++++====++++ yxyx . . . . cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa
++++++++====++++3
22log21 2
yx
yx
19191919----edaHRsaysmIkarxedaHRsaysmIkarxedaHRsaysmIkarxedaHRsaysmIkarxageRkam ageRkam ageRkam ageRkam ³³³³ k> k> k> k> (((( )))) 0
4loglog 2
22 ====
++++ xx
x> x> x> x> (((( )))) 04
loglog3
5,02
2 ====
++++ x
x
K> K> K> K> 4log1log2 xx ++++==== X> X> X> X> (((( ))))(((( )))) (((( )))) 0212log312log 3
2
3 ====++++++++−−−−++++ xx g> g> g> g> 03437.87 2log12
2log ====++++−−−− ++++ xx 20202020----edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) (((( )))) 06log5log 2
22 ====−−−−++++−−−−++++ xxxx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 7 -
x> x> x> x> (((( )))) 0216log)5(2log 32
3 ====++++−−−−−−−−++++ xxxx
K> K> K> K> (((( )))) 044log3log21
2
21 ====−−−−++++++++++++
xxxx
21212121----k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa AaBa BA loglog ==== Edl Edl Edl Edl BAa ;; CabIcMnYnBitviC¢man nig xusBI CabIcMnYnBitviC¢man nig xusBI CabIcMnYnBitviC¢man nig xusBI CabIcMnYnBitviC¢man nig xusBI 1 . . . . x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar 6867 72log2log ====++++ xx 22222222----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar (((( )))) (((( )))) 2loglogloglog 2442 ====++++ xx 23232323----eK[sIVút eK[sIVút eK[sIVút eK[sIVút )21(log 2
2
n
nu ++++==== Edl Edl Edl Edl INn ∈∈∈∈ KNna KNna KNna KNna (((( )))) n
n
kkn uuuuuS ++++++++++++++++======== ∑∑∑∑
====....210
0 . . . .
24242424---- eK[sIVút eK[sIVút eK[sIVút eK[sIVút
−−−−==== 12
cos2ln nn
xu Edl Edl Edl Edl INn ∈∈∈∈
KNna KNna KNna KNna (((( )))) n
n
kkn uuuuuS ++++++++++++++++======== ∑∑∑∑
====....210
0 . . . .
25252525---- edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar
====
====
9001
6.5
4001
5.4
yx
yx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 8 -
26262626----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³ ³ ³ ³ 0logloglog30 3
42
5,03 ====−−−−++++ xxx xxx
27272727---- edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar ³³³³
====
====++++
64
26)loglog(5
yx
yx xy
28282828----eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½ 06)54(log5)54(log 3
632
6 ====++++−−−−−−−−−−−− −−−−−−−− xx xx ¿¿¿¿
29292929----eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½ )9(log
31
)9(log1 323
322)3( 3 xxxx xx
−−−−====−−−−++++ −−−−−−−−¿¿¿¿
30303030----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_
++++++++
++++++++====
1344
log.1
13log)(
2
222 xx
x
xxf
k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar
++++++++====
++++++++
1344
log1
13log
2
222 xx
x
x
x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam
++++++++
1
13log
22 x
x nig nig nig nig
++++++++1344
log2
2 xx viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 9 -
K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ )(xf elIcenøa¼ elIcenøa¼ elIcenøa¼ elIcenøa¼ ]3;0[ . . . . 31313131----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ IRxxxxf ∈∈∈∈++++++++==== ,)1ln()( 2 ccccUrRsaybBa¢ak´fa UrRsaybBa¢ak´fa UrRsaybBa¢ak´fa UrRsaybBa¢ak´fa )
11()
1(
bb
aa
fba
baf
++++++++
++++<<<<
++++++++++++
cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb 0,0 >>>>>>>> ba .... 32323232----eK[ eK[ eK[ eK[ 1≥≥≥≥a nig nig nig nig 1≥≥≥≥b . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )
2(log2loglog 222
baba
++++≤≤≤≤++++
33333333----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³ (((( ))))nn
nun ++++
++++==== 222 log.
11log
Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ++++++++++++++++==== ....321 CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n K>kMnt; K>kMnt; K>kMnt; K>kMnt; n edIm,I[ edIm,I[ edIm,I[ edIm,I[ 100====nS . . . . 34343434----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xxxf 2log2)( ++++==== Edl Edl Edl Edl 0>>>>x eKtag eKtag eKtag eKtag (((( )))) (((( ))))(((( )))))(;)(;)( 321 xfffuxffuxfu ============ nig nig nig nig (((( ))))(((( ))))(((( ))))....)(.... xffffu nn ==== . . . . k>eKyk k>eKyk k>eKyk k>eKyk nn uV 2log1++++==== . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa 2
1 nn VV ====++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 10 -
x>eRbIGnumanrYmKNitviTüacUrRsayfa x>eRbIGnumanrYmKNitviTüacUrRsayfa x>eRbIGnumanrYmKNitviTüacUrRsayfa x>eRbIGnumanrYmKNitviTüacUrRsayfa n
VVn2
1==== . . . . K>KNna K>KNna K>KNna K>KNna nV nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig x . . . . 35353535----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
(((( ))))2log )1( ++++==== ++++ nu nn Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrk>cUrk>cUrk>cUrbgðajfa bgðajfa bgðajfa bgðajfa (((( ))))nu CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ln....lnlnln 321 ++++++++++++++++==== CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n 36363636----eK[ eK[ eK[ eK[ xxxxf 2424 sin4coscos4sin)( ++++−−−−++++==== k> sRmYl k> sRmYl k> sRmYl k> sRmYl )(xf x>kMnt; x>kMnt; x>kMnt; x>kMnt; x edIm,I[ edIm,I[ edIm,I[ edIm,I[
21
)( ====xf
37373737----cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; INn∈∈∈∈ eK[ eK[ eK[ eK[ 12
sin12
cosππππππππ nn
nS ++++==== k> KNnatémø k> KNnatémø k> KNnatémø k> KNnatémø
12cos
ππππ nig nig nig nig 12
sinππππ
x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 0624 12 ====++++−−−− ++++++++ nnn SSS 38383838----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa
21
73
cos7
2cos
7cos ====++++−−−− ππππππππππππ
(IMO 1963)
eroberogeday lwm pl:ún nig Esn Bisidæ
- 11 -
39393939----KNna KNna KNna KNna )5
3sin41()
5sin41( 22 ππππππππ −−−−−−−−====A
40404040----eK[ eK[ eK[ eK[ 144
)( 2
3
−−−−++++====
xx
xxf . KNnatémø . KNnatémø . KNnatémø . KNnatémø )
7(cos
ππππf . . . .
41414141----eK[ eK[ eK[ eK[ )cos(sin1
)( xxk
xf kkk ++++====
Edl Edl Edl Edl ...;3;2;1====k cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa
121
)()( 64 ====−−−− xfxf . . . . 42424242----cUrbgðajfacMeBaHRKb;cMnYnKt;FmcUrbgðajfacMeBaHRKb;cMnYnKt;FmcUrbgðajfacMeBaHRKb;cMnYnKt;FmcUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati µCati µCati µCati n nig cMeBaHRKb; nig cMeBaHRKb; nig cMeBaHRKb; nig cMeBaHRKb; cMnYnBit cMnYnBit cMnYnBit cMnYnBit t
kx
2
ππππ≠≠≠≠ Edl Edl Edl Edl nt ....,,2,1,0==== nig nig nig nig k CacMnYnKt;eKmansmPaB CacMnYnKt;eKmansmPaB CacMnYnKt;eKmansmPaB CacMnYnKt;eKmansmPaB ³³³³
xxxxx
nn 2cotcot
2sin
1.....
4sin1
2sin1 −−−−====++++++++++++
(IMO 1966)
43434343----eK[ eK[ eK[ eK[ dcba ,,, CacMnYnenAkñúgcenøa CacMnYnenAkñúgcenøa CacMnYnenAkñúgcenøa CacMnYnenAkñúgcenøaH H H H ];0[ ππππ edaydwgfaedaydwgfaedaydwgfaedaydwgfa
++++====++++++++====++++
)cos2(cos4cos7cos
)sin2(sin4sin7sin
dcba
dcba
cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )cos(7)cos(2 cbda −−−−====−−−− . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 12 -
44444444----cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam ³³³³ )sin()sin()sin( xzzyyx −−−−++++−−−−++++−−−− 45454545----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag
2cba
p++++++++==== CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN .
kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa bc
cpbpA ))((2
sin−−−−−−−−==== rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng
BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH . xxxx----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa
81
2sin
2sin
2sin ≤≤≤≤CBA
46464646---- eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaNABC ehIyeKtag ehIyeKtag ehIyeKtag ehIyeKtag r nig nig nig nig R erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg; carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa
Rr
CBA ++++====++++++++ 1coscoscos 47474747----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag
2cba
p++++++++==== CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy r nig nig nig nig R
erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrprcabcab 422 ++++++++====++++++++ xxxx----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrrpcba 822 22222 −−−−−−−−====++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 13 -
48484848----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag
2cba
p++++++++==== CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy r
CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa ³³³³ k> k> k> k> r
Ccp
Bbp
Aap ====−−−−====−−−−====−−−−
2tan)(
2tan)(
2tan)(
x> x> x> x> p
rRCBA ++++====++++++++ 42
tan2
tan2
tan
K> K> K> K> rpCBA ====
2tan
2tan
2tan
X> X> X> X> rpCBA ====++++++++
2cot
2cot
2cot
49494949----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYyman mYyman mYyman mYymanRCug RCug RCug RCug cba ;; . . . . tag tag tag tag S CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 22
coscoscosR
abcCcBbAa ====++++++++
x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa SCcBbAa 4cotcotcot 222 ====++++++++ 50505050----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag r CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )(2cotcotcot RrCcBbAa ++++====++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 14 -
51515151----bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman
2tan
2tan
CBAbaba −−−−====
++++−−−−
52525252----bgðbgðbgðbgðajfakñúgRtIekaN ajfakñúgRtIekaN ajfakñúgRtIekaN ajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman )
2sin
2sin
2(sin4 222
222 CBAabc
cab
bca ++++++++≥≥≥≥++++++++
53535353----bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman 3333 16
81coscoscos
pc
C
b
B
a
A ≥≥≥≥++++++++
54545454----bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyEdl mYyEdl mYyEdl mYyEdl CBA ≠≠≠≠≠≠≠≠ eKmaneKmaneKmaneKman k> k> k> k> 0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa x> x> x> x> abc
BAACCBBAcACbCBa
3)sin()sin()sin(
)(sin)(sin)(sin 333333
====−−−−−−−−−−−−
−−−−++++−−−−++++−−−−
55555555----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
4sin.2
ππππnU
nn ==== Edl Edl Edl Edl *INn ∈∈∈∈
k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 4
sin4
cos4
)1(cos.2
ππππππππππππ nnn−−−−====
++++ x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa
4)1(
cos)2(4
cos)2( 1 ππππππππ ++++−−−−==== ++++ nnU nn
n K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk nn UUUUS ++++++++++++++++==== .........321 CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 15 -
56565656----eK[ eK[ eK[ eK[ x CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl 0217160 2 <<<<++++−−−− xx . . . . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 0
13sin <<<<
−−−−xππππ . . . .
57575757----eK[ eK[ eK[ eK[ 2
0ππππ<<<<<<<< x . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa ³³³³
21cos
11
sin1
1 ++++≥≥≥≥
++++
++++xx
55558888----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC . . . . 1 1 1 1----cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; [,0], ππππ∈∈∈∈yx cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³ )
2(sin2sinsin
yxyx
++++≤≤≤≤++++ . . . .
2 2 2 2----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa 2
33sinsinsin ≤≤≤≤++++++++ CBA . . . .
3 3 3 3----Tajbgðajfa Tajbgðajfa Tajbgðajfa Tajbgðajfa ³³³³ k/ k/ k/ k/
833
sin.sin.sin ≤≤≤≤CBA
x/ x/ x/ x/ 8
332
cos2
cos2
cos ≤≤≤≤CBA 59595959----eKeKeKeK[sIVúténcMnYnkMupøic [sIVúténcMnYnkMupøic [sIVúténcMnYnkMupøic [sIVúténcMnYnkMupøic )( nZ kMnt´eday kMnt´eday kMnt´eday kMnt´eday ³³³³
(((( ))))
∈∈∈∈++++====
++++====
++++ INnZZZ
iZ
nnn ;||212
31
1
0
eroberogeday lwm pl:ún nig Esn Bisidæ
- 16 -
( ( ( ( || nZ CamUDulén CamUDulén CamUDulén CamUDulén nZ ) . ) . ) . ) . snµtfa snµtfa snµtfa snµtfa INniZ nnnn ∈∈∈∈∀∀∀∀++++==== ,)sin.(cos θθθθθθθθρρρρ Edl Edl Edl Edl IRnnn ∈∈∈∈>>>> θθθθρρρρρρρρ ;,0 . . . . kkkk----rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag nθθθθ nignignignig 1++++nθθθθ ehIy ehIy ehIy ehIy nρρρρ nignignignig 1++++nρρρρ . . . . xxxx----rkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVút )( nθθθθ rYcKNnarYcKNnarYcKNnarYcKNna nθθθθ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . KKKK----cUrbgHajfacUrbgHajfacUrbgHajfacUrbgHajfa
2cos....
2cos
2coscos 121
00−−−−==== n
nθθθθθθθθθθθθθθθθρρρρρρρρ
rYbBa¢ak rYbBa¢ak rYbBa¢ak rYbBa¢ak nρρρρ GnuKmn_én GnuKmn_én GnuKmn_én GnuKmn_én n . . . . 60606060----eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit )( nU kMntelI kMntelI kMntelI kMntelI n eday½eday½eday½eday½
10 ====U nig nig nig nig aaUUINn nn sincos: 1 ++++====∈∈∈∈∀∀∀∀ ++++ Edl Edl Edl Edl
20
ππππ<<<<<<<< a . . . . k¿ tag k¿ tag k¿ tag k¿ tag
2cot
aUV nn −−−−==== . . . .
cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )( nV CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy . x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt )....(lim 10 n
nVVV ++++++++++++
+∞+∞+∞+∞→→→→ nig nig nig nig n
nU
+∞+∞+∞+∞→→→→lim
eroberogeday lwm pl:ún nig Esn Bisidæ
- 17 -
61616161----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt´eday ½kMnt´eday ½kMnt´eday ½kMnt´eday ½ 1;0 10 ======== UU nig nig nig nig nnn UaUUINn −−−−====∈∈∈∈∀∀∀∀ ++++++++ cos2: 12
Edl Edl Edl Edl IRa ∈∈∈∈ .... k¿ tag k¿ tag k¿ tag k¿ tag INnUaiaUZ nnn ∈∈∈∈∀∀∀∀−−−−−−−−==== ++++ ,)sin(cos1 . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa nn ZaiaZ )sin(cos1 ++++====++++ rYcTajrk rYcTajrk rYcTajrk rYcTajrk
nZ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ n nig nig nig nig a . . . . x¿ Tajrk x¿ Tajrk x¿ Tajrk x¿ Tajrk nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n rYcTajrk rYcTajrk rYcTajrk rYcTajrk n
aU
0lim
→→→→ . . . .
66662222----eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½ 223)
12cos()
4sin(4 ++++====++++++++ ππππππππ
xx 63636363----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;elI kMnt;elI kMnt;elI kMnt;elI IN eday eday eday eday ³³³³
22
0 ====U nig nig nig nig INnU
U nn ∈∈∈∈∀∀∀∀
−−−−−−−−====++++ ,
2
11 2
1
KNna KNna KNna KNna nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 18 -
64646464---- eK[ eK[ eK[ eK[
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
====++++++++
====++++
====
4
3
2
2cos2222
2cos222
2cos22
ππππ
ππππ
ππππ
BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg 65656565----ykykykyk a; b ;c CaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nig A,B,C CargVas;mMuTaMgbICargVas;mMuTaMgbICargVas;mMuTaMgbICargVas;mMuTaMgbI énRtIekaN énRtIekaN énRtIekaN énRtIekaN ABCmYyEdl mYyEdl mYyEdl mYyEdl S CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH.
Rsayfa Rsayfa Rsayfa Rsayfa S
cbaCotCCotBCotA
4
222 ++++++++====++++++++ 66666666---- eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cABbACaBC ============ ,, . cUrRsaybBa¢k;fa . cUrRsaybBa¢k;fa . cUrRsaybBa¢k;fa . cUrRsaybBa¢k;fa ³³³³
abccba
cC
bB
aA
2coscoscos 222 ++++++++====++++++++
67676767----edaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkar k> k> k> k> (((( )))) 0)sin2(logsinlog2 3
22
2 ====++++ xx
x> x> x> x> 02coslog3coslog 2
2
22 ====++++++++
xx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 19 -
68686868----edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar
(((( )))) (((( ))))(((( ))))
====
====yx
yx
sinlnsinln
2ln2ln
22
sin2sin2
Edl Edl Edl Edl 2
0ππππ<<<<<<<< x nig nig nig nig
20
ππππ<<<<<<<< y . . . . 69696969----edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar
++++====++++
====++++
2222
23
sinsin
sinsin yx
yx
70707070----eK[ eK[ eK[ eK[ 2
a0ππππ<<<<<<<< nig nig nig nig
2b0
ππππ<<<<<<<< . . . .
cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa 1bcosacos
bsinasin
2222
====
++++
lu¼RtaEt lu¼RtaEt lu¼RtaEt lu¼RtaEt ba ==== . . . . 71717171----eK[ eK[ eK[ eK[ ABC CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ AcosCsinBsin1CsinBsin 22 ++++====++++ . . . . bgHajfa bgHajfa bgHajfa bgHajfa ABC CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 20 -
72727272----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCúg mYymanRCúg mYymanRCúg mYymanRCúg c,b,a . . . . kMntRbePTénRtIekaN kMntRbePTénRtIekaN kMntRbePTénRtIekaN kMntRbePTénRtIekaN ABC ebIeKdwgfa ½ ebIeKdwgfa ½ ebIeKdwgfa ½ ebIeKdwgfa ½ )
c1
b1
a1
(21
cCcos
bBcos
aAcos ++++++++====++++++++
73737373----kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN ABC mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½
rR
4
2C
sin
1
2B
sin
1
2A
sin
1 ≥≥≥≥++++++++
Edl Edl Edl Edl r nig nig nig nig R CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . 74747474----k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa
aaa
a
2sin
1
sin2
1
2sin
2cos222 −−−−====
x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk ∑∑∑∑====
====n
kk
k
kn x
x
S0 2
2sin
2cos
.2
1 . . . .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 21 -
eroberogeday lwm pl:ún nig Esn Bisidæ
- 22 -
lMhat´TI1lMhat´TI1lMhat´TI1lMhat´TI1 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ 22 3243 22)( xxxxxf −−−−++++++++−−−− ++++==== k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf . . . . x> rktémø x> rktémø x> rktémø x> rktémø x edIm, edIm, edIm, edIm,I[ I[ I[ I[ 20)( ====xf . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> bgðajfaRKb;cMnYnBit k> bgðajfaRKb;cMnYnBit k> bgðajfaRKb;cMnYnBit k> bgðajfaRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf cMeBaHRKb;cMnYnBitviC¢man cMeBaHRKb;cMnYnBitviC¢man cMeBaHRKb;cMnYnBitviC¢man cMeBaHRKb;cMnYnBitviC¢man a nig nig nig nig b eKman eKman eKman eKman (((( )))) 0
2 ≥≥≥≥−−−− ba eK)an eK)an eK)an eK)an 02 ≥≥≥≥++++−−−− baba b¤ b¤ b¤ b¤ abba 2≥≥≥≥++++ edayyk edayyk edayyk edayyk 432
2 ++++−−−−==== xxa nig nig nig nig 2322 xxb −−−−++++==== eK)an eK)an eK)an eK)an 22 3243 2.22)( xxxxxf −−−−++++++++−−−−≥≥≥≥ 1622)( 6 ====≥≥≥≥xf dUcenHRKb;cMnYnBit dUcenHRKb;cMnYnBit dUcenHRKb;cMnYnBit dUcenHRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf . . . . x> rktémø x> rktémø x> rktémø x> rktémø x edIm,I[ edIm,I[ edIm,I[ edIm,I[ 20)( ====xf eK)an eK)an eK)an eK)an 2022
22 3243 ====++++ −−−−++++++++−−−− xxxx
(((( ))))i
xx
xx
xxxx
0202
642
02022
43
43
)43(643
2
2
22
====−−−−++++
====−−−−++++
++++−−−−++++−−−−
++++−−−−−−−−++++−−−−
eroberogeday lwm pl:ún nig Esn Bisidæ
- 23 -
tag tag tag tag 02 432>>>>==== ++++−−−− xxt smIkar smIkar smIkar smIkar (((( ))))i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³
02064 ====−−−−++++t
t b¤ b¤ b¤ b¤ 064202 ====++++−−−− tt
03664100' >>>>====−−−−====∆∆∆∆ eKTajb¤seKTajb¤seKTajb¤seKTajb¤s 16610;4610 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 4====t eK)an eK)an eK)an eK)an 42 432
====++++−−−− xx naM[ naM[ naM[ naM[ 2432 ====++++−−−− xx b¤ b¤ b¤ b¤ 0232 ====++++−−−− xx eday eday eday eday 0====++++++++ cba enaH enaH enaH enaH 11 ====x b¤ b¤ b¤ b¤ 22 ====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 16====t eK)an eK)an eK)an eK)an 162 432
====++++−−−− xx naM[ naM[ naM[ naM[ 4432 ====++++−−−− xx b¤ b¤ b¤ b¤ 0)3( ====−−−−xx eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 01 ====x b¤ b¤ b¤ b¤ 32 ====x . . . . dUcenHeK)an dUcenHeK)an dUcenHeK)an dUcenHeK)an }3,2,1,0{∈∈∈∈x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 24 -
lMhat´TI2lMhat´TI2lMhat´TI2lMhat´TI2
eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ 22
22)(
1
++++++++====
++++
x
x
xf
k> cMeBak> cMeBak> cMeBak> cMeBaHRKb;cMnYnBit HRKb;cMnYnBit HRKb;cMnYnBit HRKb;cMnYnBit m nig nig nig nig n ebI ebI ebI ebI 1====++++ nm enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa ³³³³ 3)()( ====++++ nfmf . . . . x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk ∑∑∑∑
====
++++====
p
kp p
kfS
1 1 . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> k> k> k> Rsayfa Rsayfa Rsayfa Rsayfa 3)()( ====++++ nfmf ebI ebI ebI ebI 1====++++ nm eKman eKman eKman eKman (((( ))))
22
22 1
++++++++====
++++
m
m
mf nig nig nig nig (((( ))))22
22 1
++++++++====
++++
n
n
nf
22
22
22
22)()(
11
++++++++++++
++++++++====++++
++++++++
n
n
m
m
nfmf
3
224
)224(3
224
2.32.312
2222
22222222
21
21
21
21
21
21
21
21
21
21
23
21
121
23
1
====
++++++++
++++++++====
++++++++
++++++++====
++++++++++++
++++++++++++++++++++++++++++====
++++++++
++++++++
++++++++
++++++++
++++++++++++
++++++++++++++++++++++++++++++++
nm
nm
nm
nm
nmnm
nmnmnmnm
dUcenH ebI dUcenH ebI dUcenH ebI dUcenH ebI 1====++++ nm enaH enaH enaH enaH 3)()( ====++++ nfmf . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 25 -
x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk ∑∑∑∑====
++++====
p
kp p
kfS
1 1
eKman eKman eKman eKman (((( ))))ip
pf
pf
pfSn
++++++++++++
++++++++
++++====
1......
12
11
b¤ b¤ b¤ b¤ (((( ))))iip
fpp
fp
pfSn
++++++++++++
++++−−−−++++
++++====
11
....11
1
bUksmIkar bUksmIkar bUksmIkar bUksmIkar (((( ))))i nig nig nig nig (((( ))))ii eK)an eK)an eK)an eK)an ³³³³ pSn 33....332 ====++++++++++++==== naM[ naM[ naM[ naM[
23p
Sn ====
BIeRBaH BIeRBaH BIeRBaH BIeRBaH 311
1 ====
++++++++
++++ pp
fp
f
31
11
312
13
311
12
====
++++++++
++++
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
====
++++−−−−++++
++++
====
++++−−−−++++
++++
pf
pp
f
pp
fp
f
pp
fp
f
dUcenH dUcenH dUcenH dUcenH 2
311
pp
kfS
p
kp ====
++++==== ∑∑∑∑
==== . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 26 -
lMlMlMlMhat´TI3hat´TI3hat´TI3hat´TI3
eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xx
xx
ee
eexf −−−−
−−−−
++++−−−−====)(
k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit a nig nig nig nig b cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³ (((( ))))
)()(1)()(
bfafbfaf
baf++++
++++====++++
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> Rsayfak> Rsayfak> Rsayfak> Rsayfa )(xf CaG CaG CaG CaGnuKmn_essnuKmn_essnuKmn_essnuKmn_ess eKman eKman eKman eKman xx
xx
ee
eexf −−−−
−−−−
++++−−−−====)(
cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit x eKman eKman eKman eKman 0>>>>++++ −−−− xx ee dUcenHGnuKmn_ dUcenHGnuKmn_ dUcenHGnuKmn_ dUcenHGnuKmn_ f manEdnkMnt; manEdnkMnt; manEdnkMnt; manEdnkMnt; IRD f ==== cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; fDx ∈∈∈∈ eK)an eK)an eK)an eK)an fDx ∈∈∈∈−−−− eK)aeK)aeK)aeK)an n n n (((( )))) 0)( ====
++++−−−−++++
++++−−−−====++++−−−− −−−−
−−−−
−−−−
−−−−
xx
xx
xx
xx
ee
ee
ee
eexfxf
eKTaj eKTaj eKTaj eKTaj )()( xfxf −−−−====−−−− dUcenH dUcenH dUcenH dUcenH )(xf CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 27 -
x> Rsayfa x> Rsayfa x> Rsayfa x> Rsayfa (((( )))))()(1)()(
bfafbfaf
baf++++
++++====++++
eKman eKman eKman eKman 1
1)( 2
2
++++−−−−====
++++−−−−==== −−−−
−−−−
x
x
xx
xx
e
e
ee
eexf
eK)an eK)an eK)an eK)an (((( ))))1
1
1
122
22
)(2
)(2
++++−−−−====
++++−−−−====++++ ++++
++++
++++
++++
ba
ba
ba
ba
e
e
e
ebaf
(((( ))))1
12
2
++++−−−−==== a
a
e
eaf nig nig nig nig
1
1)( 2
2
++++−−−−==== b
b
e
ebf
eKmeKmeKmeKman an an an 1
1
1
1)()( 2
2
2
2
++++−−−−++++
++++−−−−====++++ b
b
a
a
e
e
e
ebfaf
(((( ))))iee
ebfaf ba
ba
)1)(1(
)1(2)()( 22
22
++++++++−−−−====++++
++++
ehIy ehIy ehIy ehIy )1)(1(
)1)(1(1)()(1 22
22
++++++++−−−−−−−−++++====++++ ba
ba
ee
eebfaf
(((( ))))iiee
ebfaf ba
ba
)1)(1(
)1(2)()(1 22
22
++++++++++++====++++
++++
EcksmPaB EcksmPaB EcksmPaB EcksmPaB (((( ))))i nig nig nig nig (((( ))))ii GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an ³³³³ )(
1
1)()(1)()(
22
22
bafe
ebfafbfaf
ba
ba
++++====++++−−−−====
++++++++
++++
++++
dUcenH dUcenH dUcenH dUcenH (((( )))))()(1)()(
bfafbfaf
baf++++
++++====++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 28 -
lMhat´TI4lMhat´TI4lMhat´TI4lMhat´TI4 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ (((( )))) (((( ))))xx
xf 3232)( −−−−++++++++==== k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 2)( ≥≥≥≥xf . . . . K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar 14)( ====xf . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>Rsayfa k>Rsayfa k>Rsayfa k>Rsayfa )(xf CaGnuKmn_KU CaGnuKmn_KU CaGnuKmn_KU CaGnuKmn_KU
(((( )))) (((( ))))xxxf 3232)( −−−−++++++++====
EdnkMnt; EdnkMnt; EdnkMnt; EdnkMnt; IRD f ==== cMeBaH cMeBaH cMeBaH cMeBaH fDx ∈∈∈∈ enaH enaH enaH enaH fDx ∈∈∈∈−−−− eK)an eK)an eK)an eK)an (((( )))) (((( )))) (((( )))) xx
xf−−−−−−−− −−−−++++++++====−−−− 3232
eday eday eday eday 1)32)(32( ====−−−−++++
enaH enaH enaH enaH xx
xf−−−−−−−−
++++++++
−−−−====−−−−
321
321
)(
(((( )))) (((( )))) (((( ))))xfxfxx ====−−−−++++++++====−−−− 3232)(
dUcenH dUcenH dUcenH dUcenH )(xf CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 29 -
x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x bgðajfa bgðajfa bgðajfa bgðajfa 2)( ≥≥≥≥xf ³ ³ ³ ³ tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eK)an eK)an eK)an eK)an ³³³³ (((( )))) (((( )))) xxxx
)32()32(23232 −−−−++++≥≥≥≥−−−−++++++++ (((( )))) (((( )))) 23232 ≥≥≥≥−−−−++++++++ xx dUcenH dUcenH dUcenH dUcenH 2)( ≥≥≥≥xf cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; x . . . . K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar 14)( ====xf eK)an eK)an eK)an eK)an (((( )))) (((( )))) 143232 ====−−−−++++++++ xx tag tag tag tag (((( )))) 032 >>>>++++==== x
t enaH enaH enaH enaH (((( ))))t
x 132 ====−−−−
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr 141 ====++++t
t b¤ b¤ b¤ b¤ 01142 ====++++−−−− tt 2)34(48149' ========−−−−====∆∆∆∆
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s (((( ))))21 32347 ++++====++++====t
ehIy ehIy ehIy ehIy (((( )))) 222 32)32(347
−−−−++++====−−−−====−−−−====t ----cMeBaH cMeBaH cMeBaH cMeBaH (((( ))))2
1 32 ++++====t eK)an eK)an eK)an eK)an (((( )))) (((( ))))2
3232 ++++====++++ x naM[ naM[ naM[ naM[ 2====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH (((( )))) 2
1 32−−−−++++====t
eK)an eK)an eK)an eK)an (((( )))) (((( )))) 23232
−−−−++++====++++ x naM[ naM[ naM[ naM[ 2−−−−====x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 30 -
lMhat´TI5lMhat´TI5lMhat´TI5lMhat´TI5 edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> 02433.49 2 ====++++−−−− ++++xx x> x> x> x> 01282.94 2 ====++++−−−− ++++xx K> K> K> K> 026.53 32132 ====++++−−−− ++++++++++++ xxx X> X> X> X> (((( )))) (((( )))) 102323 ====−−−−++++++++ xx g> g> g> g> (((( )))) (((( )))) 143232 ====++++++++−−−−
xx dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> 02433.49 2 ====++++−−−− ++++xx (((( )))) (((( )))) 02433363 2 ====++++−−−− xx tag tag tag tag (((( )))) 03 >>>>==== xt 81243324';0243362 ====−−−−====∆∆∆∆====++++−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 27918;9918 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 91 ====t eK)an eK)an eK)an eK)an 93 ====x naM[ naM[ naM[ naM[ 2====x ----cMeBaH cMeBaH cMeBaH cMeBaH 272 ====t eK)an eK)an eK)an eK)an 273 ====x naM[ naM[ naM[ naM[ 3====x dUcenH dUcenH dUcenH dUcenH 3;2 21 ======== xx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 31 -
x> x> x> x> 01282.94 2 ====++++−−−− ++++xx (((( )))) (((( )))) 01282362 2 ====++++−−−− xx tag tag tag tag 02 >>>>==== xt 196128324';0128362 ====−−−−====∆∆∆∆====++++−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 321418;21418 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 21 ====t enaH enaH enaH enaH 22 ====x naM[ naM[ naM[ naM[ 1====x ----cMeBaH cMeBaH cMeBaH cMeBaH 322 ====t enaH enaH enaH enaH 322 ====x naM[ naM[ naM[ naM[ 5====x dUcenH dUcenH dUcenH dUcenH 5;1 21 ======== xx . . . . K> K> K> K> 026.53 32132 ====++++−−−− ++++++++++++ xxx (((( )))) (((( )))) (((( )))) 048630927 ====++++−−−− xxx EckGgÁTaMgBIEckGgÁTaMgBIEckGgÁTaMgBIEckGgÁTaMgBIrénsmIkarnwg rénsmIkarnwg rénsmIkarnwg rénsmIkarnwg 04 ≠≠≠≠x eK)an eK)an eK)an eK)an ³³³³
08
23
3023
27
0846
3049
27
2
====++++
−−−−
====++++
−−−−
xx
xx
tag tag tag tag 023 >>>>
====x
t enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr ³³³³
9216225';083027 2 ====−−−−====∆∆∆∆====++++−−−− tt eKTaj eKTaj eKTaj eKTaj
32
27315
;94
27315
21 ====++++========−−−−==== tt
eroberogeday lwm pl:ún nig Esn Bisidæ
- 32 -
----cMeBaH cMeBaH cMeBaH cMeBaH 94
1 ====t eK)an eK)an eK)an eK)an 94
23 ====
x
naM[ naM[ naM[ naM[ 2−−−−====x
----cMeBaH cMeBaH cMeBaH cMeBaH 32
2 ====t eK)an eK)an eK)an eK)an 32
23 ====
x
naM[ naM[ naM[ naM[ 1−−−−====x
dUcenH dUcenH dUcenH dUcenH 1;2 21 −−−−====−−−−==== xx . . . . X> X> X> X> (((( )))) (((( )))) 102323 ====−−−−++++++++ xx tag tag tag tag (((( )))) 023 >>>>++++==== x
t enaH enaH enaH enaH (((( ))))t
x 123 ====−−−−
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr 101 ====++++t
t b¤ b¤ b¤ b¤ 01102 ====++++−−−− tt
(((( )))) 06224125'2 >>>>========−−−−====∆∆∆∆
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s (((( ))))(((( )))) (((( ))))
++++====−−−−====−−−−====
++++====++++====−−−−22
2
21
2323625
23625
t
t
----cMeBaH cMeBaH cMeBaH cMeBaH (((( ))))223 ++++====t
eK)an eK)an eK)an eK)an (((( )))) (((( ))))22323 ++++====++++ x naM[ naM[ naM[ naM[ 2====x
----cMeBaH cMeBaH cMeBaH cMeBaH (((( )))) 223
−−−−++++====t eK)an eK)an eK)an eK)an (((( )))) (((( )))) 2
2323−−−−++++====++++ x naM[ naM[ naM[ naM[ 2−−−−====x
dUcenH dUcenH dUcenH dUcenH 2;2 21 −−−−======== xx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 33 -
g> g> g> g> (((( )))) (((( )))) 143232 ====++++++++−−−−xx
tag tag tag tag (((( )))) 032 >>>>++++====x
t enaH enaH enaH enaH (((( ))))t
x 132 ====−−−−
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr 141 ====++++t
t b¤ b¤ b¤ b¤ 01142 ====++++−−−− tt 2)34(48149' ========−−−−====∆∆∆∆
eKTajb¤seKTajb¤seKTajb¤seKTajb¤s (((( ))))21 32347 ++++====++++====t
ehIy ehIy ehIy ehIy (((( )))) 222 32)32(347
−−−−++++====−−−−====−−−−====t ----cMeBaH cMeBaH cMeBaH cMeBaH (((( ))))2
1 32 ++++====t eK)an eK)an eK)an eK)an (((( )))) (((( ))))2
3232 ++++====++++x naM[ naM[ naM[ naM[ 4====x . . . .
----cMeBaH cMeBaH cMeBaH cMeBaH (((( )))) 21 32
−−−−++++====t eK)an eK)an eK)an eK)an (((( )))) (((( )))) 2
3232−−−−++++====++++
x naM[ naM[ naM[ naM[ 4−−−−====x . . . . dUcenH dUcenH dUcenH dUcenH 4;4 21 −−−−======== xx . . . .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 34 -
lMhat´TI6lMhat´TI6lMhat´TI6lMhat´TI6 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
x
xx
x
xx
x
xx
x
xx
6
6
5
5
4
4
3
32222 −−−−−−−−−−−−−−−−
====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
x
xx
x
xx
x
xx
x
xx
6
6
5
5
4
4
3
32222 −−−−−−−−−−−−−−−−
====++++++++ xxxxxxxx 2222 2222
6543 −−−−−−−−−−−−−−−− ====++++++++ tag tag tag tag 11)1(2 22 −−−−≥≥≥≥−−−−−−−−====−−−−==== xxxt smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ (((( ))))itttt 6543 ====++++++++ ----ebI ebI ebI ebI 3====t enaHsmIkar enaHsmIkar enaHsmIkar enaHsmIkar (((( ))))i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³
2161256427
6543 3333
====++++++++====++++++++
216216==== epÞógpÞat; epÞógpÞat; epÞógpÞat; epÞógpÞat; dUcenH dUcenH dUcenH dUcenH 3====t Cab¤srbs;smIkar Cab¤srbs;smIkar Cab¤srbs;smIkar Cab¤srbs;smIkar )(i . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 35 -
----ebI ebI ebI ebI 31 <<<<≤≤≤≤−−−− t eKman eKman eKman eKman ³³³³
333
65
65
;64
64
;63
63
>>>>
>>>>
>>>>
ttt
eK)an eK)an eK)an eK)an 333
65
64
63
65
64
63
++++
++++
>>>>
++++
++++
ttt
16
543 >>>>++++++++t
ttt
naM[ naM[ naM[ naM[ tttt 6543 >>>>++++++++
dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar (((( ))))i Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH 31 <<<<≤≤≤≤−−−− t . . . .
----ebI ebI ebI ebI 3>>>>t eKman eKman eKman eKman ³³³³
333
65
65
;64
64
;63
63
<<<<
<<<<
<<<<
ttt
eK)an eK)an eK)an eK)an 333
65
64
63
65
64
63
++++
++++
<<<<
++++
++++
ttt
16
543 <<<<++++++++t
ttt
naM[ naM[ naM[ naM[ tttt 6543 <<<<++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 36 -
dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar (((( ))))i Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH 3>>>>t . . . .
srubmksmIkar srubmksmIkar srubmksmIkar srubmksmIkar (((( ))))i manb¤seTal manb¤seTal manb¤seTal manb¤seTal 3====t
cMeBaH cMeBaH cMeBaH cMeBaH 3====t eK)an eK)an eK)an eK)an 322 ====−−−− xx
b¤ b¤ b¤ b¤ 0322 ====−−−−−−−− xx naM[ naM[ naM[ naM[ 3;1 21 ====−−−−==== xx
dUcenH dUcenH dUcenH dUcenH 3;1 21 ====−−−−==== xx . . . .
����������������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 37 -
lMhat´TI7lMhat´TI7lMhat´TI7lMhat´TI7 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
8822642)1(2)1(2)1( ++++++++====++++ −−−−−−−−−−−− xxx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
8822642)1(2)1(2)1( ++++++++====++++ −−−−−−−−−−−− xxx
tag tag tag tag (((( )))) 0221 >>>>==== −−−−xX enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr ³³³³
(((( ))))iXXXXX
XXX
)42)(2(6
82622
32
++++−−−−++++====++++−−−−
++++++++====++++
tag tag tag tag 2++++==== Xu nig nig nig nig 222 ++++−−−−==== xxv smIkar smIkar smIkar smIkar (((( ))))ii Gacsresr Gacsresr Gacsresr Gacsresr ³³³³
uvvu 2====++++ b¤ b¤ b¤ b¤ (((( )))) 02 ====−−−− vu
naM[ naM[ naM[ naM[ vu ==== b¤ b¤ b¤ b¤ 222 2 ++++−−−−====++++ XXX b¤ b¤ b¤ b¤ 0232 ====++++−−−− XX eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 2;1 21 ======== XX . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 11 ====X enaH enaH enaH enaH (((( )))) 12
21 ====−−−−x naM[ naM[ naM[ naM[ 1====x . . . . ----cMeBaHcMeBaHcMeBaHcMeBaH 22 ====X enaH enaH enaH enaH (((( )))) 22
21 ====−−−−x smmUl smmUl smmUl smmUl 1)1( 2 ====−−−−x eKTaj eKTaj eKTaj eKTaj 01 ====x b¤ b¤ b¤ b¤ 22 ====x . dUcenH . dUcenH . dUcenH . dUcenH }2;1;0{∈∈∈∈x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 38 -
lMhat´TI8lMhat´TI8lMhat´TI8lMhat´TI8 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
(((( )))) 222210114 6333 xxxxx −−−−====++++++++ −−−−−−−−−−−−
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIedaHRsaysmIedaHRsaysmIedaHRsaysmIkar kar kar kar ³³³³
(((( )))) 222210114 6333 xxxxx −−−−====++++++++ −−−−−−−−−−−− eKman eKman eKman eKman 222 )3(9)69(96 xxxxx −−−−−−−−====++++−−−−−−−−====−−−− eday eday eday eday 0)3( 2 ≥≥≥≥−−−− x enaHeK)an enaHeK)an enaHeK)an enaHeK)an (((( ))))ixx 96 2 ≤≤≤≤−−−− müa:geTottamvismPaB müa:geTottamvismPaB müa:geTottamvismPaB müa:geTottamvismPaB GMAM −−−− eKman eKman eKman eKman ³³³³
(((( )))) (((( ))))
(((( )))) (((( ))))iixxx
xxxxxx
xxxxxx
9333
33333
3.3.33333
22210114
3 2221011422210114
3 2)2(2101142)2(210114
≥≥≥≥++++++++
≥≥≥≥++++++++
≥≥≥≥++++++++
−−−−−−−−−−−−
−−−−++++−−−−++++−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−−−−−−−
tam tam tam tam (((( ))))i nig nig nig nig (((( ))))ii smIkarsmmUl smIkarsmmUl smIkarsmmUl smIkarsmmUl ³³³³ (((( ))))
(((( )))) (((( ))))
====++++++++
====−−−−−−−−−−−−−−−− 29333
19622210114
2
xxx
xx
bnÞab;BIedaHRsayeK)anb¤s bnÞab;BIedaHRsayeK)anb¤s bnÞab;BIedaHRsayeK)anb¤s bnÞab;BIedaHRsayeK)anb¤s 3====x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 39 -
lMhat´TI9lMhat´TI9lMhat´TI9lMhat´TI9 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
04882)14(4 ====++++−−−−−−−−++++ xx xx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
04882)14(4 ====++++−−−−−−−−++++ xx xx tag tag tag tag 02 >>>>==== tx smIkarGacsrsr smIkarGacsrsr smIkarGacsrsr smIkarGacsrsr ³³³³
(((( ))))2
2
2
2
2
2
44
1923219628
)488)(1(4)14(
0488)14(
++++====
++++++++====
−−−−++++++++−−−−====
++++−−−−−−−−−−−−====∆∆∆∆
====++++−−−−−−−−++++
x
xx
xxx
xx
xtxt
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 82
)2()14( 2
1 ====++++++++−−−−−−−−
====xx
t
b¤ b¤ b¤ b¤ 62
)2()14( 2
2 ++++−−−−====++++−−−−−−−−−−−−
==== xxx
t ----cMeBaH cMeBaH cMeBaH cMeBaH 8====t eK)an eK)an eK)an eK)an 3282 ========x eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 3====x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 40 -
----cMeBaH cMeBaH cMeBaH cMeBaH 6++++−−−−==== xt eK)an eK)an eK)an eK)an 62 ++++−−−−==== xx CasmIkarGCasmIkarGCasmIkarGCasmIkarGab;sIuscMnucRbsBVrvagExSekag ab;sIuscMnucRbsBVrvagExSekag ab;sIuscMnucRbsBVrvagExSekag ab;sIuscMnucRbsBVrvagExSekag (((( )))) xyc 2: ==== CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; (((( )))) 6: ++++−−−−==== xyd . . . . tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag (((( ))))c nigbnÞat; nigbnÞat; nigbnÞat; nigbnÞat; )(d kat;Kña kat;Kña kat;Kña kat;Kña Rtg;cMnucmanGab;sIus Rtg;cMnucmanGab;sIus Rtg;cMnucmanGab;sIus Rtg;cMnucmanGab;sIus 2====x EdlCab¤srbs;s EdlCab¤srbs;s EdlCab¤srbs;s EdlCab¤srbs;smIkar .mIkar .mIkar .mIkar . dUcenHsmIkarmanb¤sBIrKW dUcenHsmIkarmanb¤sBIrKW dUcenHsmIkarmanb¤sBIrKW dUcenHsmIkarmanb¤sBIrKW 2;3 21 ======== xx . . . .
2 3 4 5 6 7 8-1-2-3-4-5
2
3
4
5
6
7
-1
0 1
1
x
y
(((( )))) xyc 2: ====
(((( )))) 6: ++++−−−−==== xyd
eroberogeday lwm pl:ún nig Esn Bisidæ
- 41 -
lMhat´TI10lMhat´TI10lMhat´TI10lMhat´TI10 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
04
21215
7443
2.34.9 ====−−−−++++
−−−−−−−− −−−−−−−− xxxx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
04
21215
7443
2.34.9 11 ====−−−−++++
−−−−−−−− ++++−−−−++++−−−− xxxx
tag tag tag tag 02.3 1 >>>>==== ++++−−−− xt smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ 0
421
215
74432 ====−−−−++++
−−−−−−−− xtxt
−−−−−−−−
−−−−====∆∆∆∆4
21215
47443 2
xx
22
2
437
716
13692
25949
2130492
30116
1849
−−−−====++++−−−−====
++++−−−−++++−−−−====
xx
x
xxx
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s
++++−−−−====
====
10743
2
1
xt
t
eroberogeday lwm pl:ún nig Esn Bisidæ
- 42 -
----cMeBaH cMeBaH cMeBaH cMeBaH 43====t eK)an eK)an eK)an eK)an
43
2.3 1 ====++++−−−− x b¤ b¤ b¤ b¤ 21 22 −−−−++++−−−− ====x naM[ naM[ naM[ naM[ 3====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 107 ++++−−−−==== xt eK)an eK)an eK)an eK)an 1072.3 1 ++++−−−−====++++−−−− xx b¤ b¤ b¤ b¤
35
67
2 ++++−−−−====−−−− xx CasmIkarGab;sIuscMnucRbsBVrvag CasmIkarGab;sIuscMnucRbsBVrvag CasmIkarGab;sIuscMnucRbsBVrvag CasmIkarGab;sIuscMnucRbsBVrvag
ExSekag ExSekag ExSekag ExSekag (((( )))) xyc −−−−==== 2: nwgbnÞat; nwgbnÞat; nwgbnÞat; nwgbnÞat; (((( ))))35
67
: ++++−−−−==== xyd . . . .
FFFF tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag (((( ))))c nigbnÞat; nigbnÞat; nigbnÞat; nigbnÞat; )(d kat;Kña kat;Kña kat;Kña kat;Kña Rtg;cMnucmanGRtg;cMnucmanGRtg;cMnucmanGRtg;cMnucmanGab;sIus ab;sIus ab;sIus ab;sIus 1;2 ====−−−−==== xx EdlCab¤srbs;smIkar EdlCab¤srbs;smIkar EdlCab¤srbs;smIkar EdlCab¤srbs;smIkar dUcenHsmIkarmanb¤sbI KW dUcenHsmIkarmanb¤sbI KW dUcenHsmIkarmanb¤sbI KW dUcenHsmIkarmanb¤sbI KW 3;1;2 321 ========−−−−==== xxx . . . .
2 3 4-1-2-3
2
3
4
5
0 1
1
x
y
eroberogeday lwm pl:ún nig Esn Bisidæ
- 43 -
lMhat´TI11lMhat´TI11lMhat´TI11lMhat´TI11 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
32423233
31
3 −−−−++++−−−−−−−− ++++====++++ xxxxx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
32423233
31
3 −−−−++++−−−−−−−− ++++====++++ xxxxx smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³
(((( )))) (((( ))))
(((( )))) 0271
393
093271
933
031
333
42
42
324232
====
−−−−−−−−
====−−−−−−−−−−−−
====++++−−−−−−−−
−−−−
−−−−
−−−−++++−−−−−−−−
xxx
xxxx
xxxxx
eK)an eK)an eK)an eK)an 093 ====−−−−x naM[ naM[ naM[ naM[ 2====x 0
271
3 22====−−−−−−−− xx naM[ naM[ naM[ naM[ 0342 ====++++−−−− xx
manb¤s manb¤s manb¤s manb¤s 3;1 21 ======== xx . . . . dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s ]3;2;1{∈∈∈∈x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 44 -
lMhat´TI12lMhat´TI12lMhat´TI12lMhat´TI12 edaHRsayedaHRsayedaHRsayedaHRsayRbBnæ½RbBnæ½RbBnæ½RbBnæ½smIkar smIkar smIkar smIkar ³³³³
====
====
4321
43
721
32
yx
yx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar ³³³³
(((( ))))
(((( ))))
====
====
ii
i
yx
yx
4321
43
721
32
smIkar smIkar smIkar smIkar (((( ))))i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³
(((( ))))(((( ))))13log233log
3.2log3log
721
log)3.2(log
22
2322
2
−−−−−−−−====++++====++++
====
−−−−−−−−
yx
yx
yx
smIkar smIkar smIkar smIkar (((( ))))ii Gacsresr Gacsresr Gacsresr Gacsresr ³³³³ (((( )))) (((( ))))
(((( ))))4333
33
2.3log4log
432log43log−−−−−−−−====++++
====
yx
yx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 45 -
(((( ))))22log432log2 33 −−−−−−−−====++++ yx dksmIkar dksmIkar dksmIkar dksmIkar (((( ))))1 nwg nwg nwg nwg (((( ))))2 GgÁ nwg GgÁeK)an GgÁ nwg GgÁeK)an GgÁ nwg GgÁeK)an GgÁ nwg GgÁeK)an ³³³³
(((( )))) (((( ))))3log2log222log23log
3log22log42log23log
2332
2332
−−−−====−−−−−−−−====−−−−
y
yy
EckGgÁTaMgBIrnwg EckGgÁTaMgBIrnwg EckGgÁTaMgBIrnwg EckGgÁTaMgBIrnwg 2log23log 32 −−−− eKTajeKTajeKTajeKTaj)an )an )an )an 2−−−−====y . . . . yktémø yktémø yktémø yktémø 2−−−−====y CMnYskñúg CMnYskñúg CMnYskñúg CMnYskñúg (((( ))))2 eK)an eK)an eK)an eK)an ³³³³
2log432log4 33 −−−−−−−−====−−−−x naM[ naM[ naM[ naM[ 2−−−−====x dUcenH dUcenH dUcenH dUcenH 3;2 −−−−====−−−−==== yx . . . .
������������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 46 -
lMhat´TI13lMhat´TI13lMhat´TI13lMhat´TI13 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
323.2 11
2 ====++++−−−−
−−−− xx
x
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
323.2 11
2 ====++++−−−−
−−−− xx
x smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa 01 ≠≠≠≠++++x b¤ b¤ b¤ b¤ 1−−−−≠≠≠≠x smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³
(((( ))))
0)3log1
11)(3(
03log21
11
)3(
3log21
13log11
2
32log3.2log
2
2
22
211
22
====++++
++++−−−−
====
−−−−++++−−−−++++−−−−
++++====++++−−−−++++−−−−
====
++++−−−−
−−−−
xx
xx
x
xx
x
xx
x
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ;31 ====x 3log1 22 −−−−−−−−====x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 47 -
lMhat´TI14lMhat´TI14lMhat´TI14lMhat´TI14 eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
10 ====a nig TMnak;TMngkMeNIn nig TMnak;TMngkMeNIn nig TMnak;TMngkMeNIn nig TMnak;TMngkMeNIn (((( ))))nanana ++++
++++ ++++==== 121 24log
Edl Edl Edl Edl ...;2;1;0====n . . . . k>eKtag k>eKtag k>eKtag k>eKtag na
nb 21++++==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa 21 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n ....
x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa n
nb 23==== . . . . K>TajrktY K>TajrktY K>TajrktY K>TajrktY na énsIVút énsIVút énsIVút énsIVút (((( ))))na CaGnuKmn_é CaGnuKmn_é CaGnuKmn_é CaGnuKmn_én n n n n . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> bgðajfa k> bgðajfa k> bgðajfa k> bgðajfa 2
1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n eKman eKman eKman eKman na
nb 21++++==== eK)an eK)an eK)an eK)an 1
1 21 ++++++++ ++++==== na
nb eday eday eday eday (((( ))))nanana ++++
++++ ++++==== 121 24log
++++++++
++++ ++++====nana
nb1242log
1 21
(((( )))) (((( ))))(((( )))) 22
2
1
12
1222
241
nna
nana
nana
b====++++====
++++++++====
++++++++==== ++++
dUcenH dUcenH dUcenH dUcenH 21 nn bb ====++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 48 -
x>edayeRbIGnumanrYmKNitviTüax>edayeRbIGnumanrYmKNitviTüax>edayeRbIGnumanrYmKNitviTüax>edayeRbIGnumanrYmKNitviTüaRsayfa Rsayfa Rsayfa Rsayfa n
nb 23==== cMeBaH cMeBaH cMeBaH cMeBaH 0====n eK)an eK)an eK)an eK)an 32121 0
0 ====++++====++++==== ab Bit Bit Bit Bit ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH kn ==== KW KW KW KW k
kb 23==== Bit Bit Bit Bit eyIgnwgRsaeyIgnwgRsaeyIgnwgRsaeyIgnwgRsayfavaBitcMeBaH yfavaBitcMeBaH yfavaBitcMeBaH yfavaBitcMeBaH 1++++==== kn KW KW KW KW 12
1 3++++
++++ ====k
kb BitBitBitBit eyIgman eyIgman eyIgman eyIgman 2
1 kk bb ====++++ eday eday eday eday k
kb 23==== enaH enaH enaH enaH (((( )))) 12
22
1 33++++
++++ ========kk
kb Bit Bit Bit Bit dUcenH dUcenH dUcenH dUcenH n
nb 23==== . . . . K>TajrktY K>TajrktY K>TajrktY K>TajrktY na énsIVút énsIVút énsIVút énsIVút (((( ))))na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ³ ³ ³ ³ eyIgman eyIgman eyIgman eyIgman na
nb 21++++==== eKTaj eKTaj eKTaj eKTaj (((( ))))1log2 −−−−==== nn ba eday eday eday eday n
nb 23==== dUcenH dUcenH dUcenH dUcenH (((( ))))13log 2
2 −−−−====n
na . . . .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 49 -
lMhat´TI15lMhat´TI15lMhat´TI15lMhat´TI15 eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday 10 ====a nig nig nig nigTMnak;TMngkMeNIn TMnak;TMngkMeNIn TMnak;TMngkMeNIn TMnak;TMngkMeNIn (((( ))))nbnana
na ++++++++++++ ++++++++==== 121
31 3327log Edl Edl Edl Edl ...;2;1;0====n . . . . k>eKtag k>eKtag k>eKtag k>eKtag na
nb 31++++==== . . . . bgðajfa bgðajfa bgðajfa bgðajfa 3
1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n .... x>x>x>x>KNna KNna KNna KNna nb nig nig nig nig na CaGCaGCaGCaGnuKmn_én nuKmn_én nuKmn_én nuKmn_én n . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> k> k> k> bgðajfa bgðajfa bgðajfa bgðajfa 3
1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n eyIgman eyIgman eyIgman eyIgman na
nb 31++++==== naM[ naM[ naM[ naM[ 1
1 31 ++++++++ ++++==== na
nb eday eday eday eday (((( ))))nbnana
na ++++++++++++ ++++++++==== 121
31 3327log eK)an eK)an eK)an eK)an nbnana
nb ++++++++++++ ++++++++++++==== 121
1 33271
(((( )))) (((( )))) (((( ))))(((( )))) 33
1
23
1
13
133333
nna
n
nananan
bb
b
====++++====
++++++++++++====
++++
++++
dUcenH dUcenH dUcenH dUcenH 31 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n ....
eroberogeday lwm pl:ún nig Esn Bisidæ
- 50 -
x>KNna x>KNna x>KNna x>KNna nb nig nig nig nig na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ³ ³ ³ ³ eKman eKman eKman eKman 3
1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n eK)an eK)an eK)an eK)an (((( )))) (((( ))))nn bb ln3ln 1 ====++++ tag tag tag tag (((( ))))nn bc ln==== enaH enaH enaH enaH nn cc 31 ====++++ naM[ naM[ naM[ naM[ (((( ))))nc CasIVútFrNImaRtmanersug CasIVútFrNImaRtmanersug CasIVútFrNImaRtmanersug CasIVútFrNImaRtmanersug 3====q nigtY nigtY nigtY nigtY (((( )))) (((( )))) 4ln31lnln 0
00 ====++++======== abc .... tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ 4ln30
nnn qcc ====××××====
eKTaj eKTaj eKTaj eKTaj (((( )))) 4ln3ln nnb ==== naM[ naM[ naM[ naM[ n
nb 34==== ehIyeday ehIyeday ehIyeday ehIyeday na
nb 31++++==== naM[ naM[ naM[ naM[ (((( )))) (((( ))))14log1log 3
33 −−−−====−−−−====n
nn ba dUcenH dUcenH dUcenH dUcenH n
nb 34==== nig nig nig nig (((( ))))14log 33 −−−−====
n
na . . . .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 51 -
lMhat´TI16lMhat´TI16lMhat´TI16lMhat´TI16 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_
++++−−−−====
xx
xf11
lg)(
k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit 11 <<<<<<<<−−−− a nig nig nig nig 11 <<<<<<<<−−−− b cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa (((( )))) (((( ))))
++++++++====++++abba
fbfaf1
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>rkEdnkMnt;énGnuKmn_ k>rkEdnkMnt;énGnuKmn_ k>rkEdnkMnt;énGnuKmn_ k>rkEdnkMnt;énGnuKmn_ f
++++−−−−====
xx
xf11
lg)(
GnuKmn_mann½ykalNa GnuKmn_mann½ykalNa GnuKmn_mann½ykalNa GnuKmn_mann½ykalNa
−−−−≠≠≠≠
>>>>++++−−−−
1
011
xxx
dUcenH dUcenH dUcenH dUcenH [1;1] −−−−====fD . . . . x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa ³³³³ (((( )))) (((( ))))
++++++++====++++abba
fbfaf1
eroberogeday lwm pl:ún nig Esn Bisidæ
- 52 -
eyIgman eyIgman eyIgman eyIgman (((( ))))
++++−−−−====
aa
af11
lg nig nig nig nig (((( ))))
++++−−−−====
bb
bf11
lg
eyIg)an eyIg)an eyIg)an eyIg)an (((( )))) (((( ))))
++++−−−−++++
++++−−−−====++++
bb
aa
bfaf11
lg11
lg
++++++++++++++++++++−−−−====
++++++++−−−−−−−−====
abbaabba
baba
)(1)(1
lg
)1)(1()1)(1(
lg
(((( )))) (((( )))) (((( ))))(((( )))) (((( ))))i
abbaabba
bfaf
++++++++++++++++++++−−−−====++++
11
lg
müa:geTot müa:geTot müa:geTot müa:geTot
++++++++++++
++++++++−−−−
====
++++++++
abba
abba
abba
f
11
11
lg1
(((( ))))iiabbaabba
abba
f
++++++++++++++++++++−−−−====
++++++++
)(1)(1
lg1
tam tam tam tam (((( ))))i nig nig nig nig (((( ))))ii eKTaj eKTaj eKTaj eKTaj (((( )))) (((( ))))
++++++++====++++abba
fbfaf1
. . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 53 -
lMhat´TI17lMhat´TI17lMhat´TI17lMhat´TI17 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit (((( )))) 0≥≥≥≥nnu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
au ====0 nig nig nig nig (((( )))) nunn uu 2log
1 ====++++ Edl Edl Edl Edl 2>>>>a . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n . . . . x>eKtag x>eKtag x>eKtag x>eKtag nn uV 2log==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa 2
1 nn VV ====++++ . . . . K>eKtag K>eKtag K>eKtag K>eKtag nn VW 2log==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa (((( ))))nW CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt X>KNna X>KNna X>KNna X>KNna nn VW ; nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n cMeBaH cMeBaH cMeBaH cMeBaH 0====n eK)an eK)an eK)an eK)an 20 >>>>==== au Bit Bit Bit Bit ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH kn ==== KW KW KW KW 2>>>>ku Bit Bit Bit Bit eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH 1++++==== kn KW KW KW KW 21 >>>>++++ku Bit Bit Bit Bit eyIgman eyIgman eyIgman eyIgman (((( )))) ku
kk uu 2log1 ====++++
eday eday eday eday 2>>>>ku enaH enaH enaH enaH 1log2 >>>>ku naM[ naM[ naM[ naM[ (((( )))) 22log
1 >>>>>>>>====++++ kku
kk uuu Bit Bit Bit Bit dUcenH dUcenH dUcenH dUcenH 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 54 -
x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 21 nn VV ====++++
eyIgman eyIgman eyIgman eyIgman nn uV 2log==== naM[ naM[ naM[ naM[ 121 log ++++++++ ==== nn uV eday eday eday eday (((( )))) nu
nn uu 2log1 ====++++ eK)an eK)an eK)an eK)an ³³³³
(((( ))))[[[[ ]]]]2
22
2log21
.log.log
log
nnnnn
nunn
VVVuu
uV
============
====++++
dUcenH dUcenH dUcenH dUcenH 21 nn VV ====++++ . . . .
K> bgðajK> bgðajK> bgðajK> bgðajfa fa fa fa (((( ))))nW CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt ³ ³ ³ ³ eKman eKman eKman eKman nn VW 2log==== naM[ naM[ naM[ naM[ 121 log ++++++++ ==== nn VW eday eday eday eday 2
1 nn VV ====++++ eK)an eK)an eK)an eK)an ³³³³ (((( )))) nnn VVW 2
221 log2log ========++++
dUcenH dUcenH dUcenH dUcenH (((( ))))nW CasIVútFrNImaRtmanpleFobrYm CasIVútFrNImaRtmanpleFobrYm CasIVútFrNImaRtmanpleFobrYm CasIVútFrNImaRtmanpleFobrYm 2====q . . . . X>X>X>X>KNna KNna KNna KNna nn VW ; nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ³ ³ ³ ³ tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ n
n qWW ××××==== 0 eday eday eday eday (((( )))) (((( ))))auVW 22022020 logloglogloglog ============ dUcenH dUcenH dUcenH dUcenH (((( ))))aW n
n 22 loglog2==== . . . . ehIy ehIy ehIy ehIy (((( )))) n
nWn aV 2
2log2 ======== nig nig nig nig (((( ))))n
anVnu
22log22 ======== ....
eroberogeday lwm pl:ún nig Esn Bisidæ
- 55 -
lMhatlMhatlMhatlMhat´TI18´TI18´TI18´TI18 eK[ eK[ eK[ eK[ x nig nig nig nig y CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; 12.544 −−−−++++====++++ yxyx . . . .
cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa
++++++++====++++3
22log21 2
yx
yx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay
Rsayfa Rsayfa Rsayfa Rsayfa
++++++++====++++3
22log1 2
yx
yx
eyIgman eyIgman eyIgman eyIgman 12.544 −−−−++++====++++ yxyx
(((( )))) (((( ))))(((( ))))(((( ))))(((( )))) (((( ))))(((( ))))
1
2
12
212
112
12
23
22
2.922
25222
22.522
2.522222
−−−−++++
−−−−++++
−−−−++++
++++++++−−−−++++
−−−−++++
====
++++
====++++
++++====++++
++++====++++
====−−−−++++
yxyx
yxyx
yxyx
yxyxyx
yxyxyx
eKTaj eKTaj eKTaj eKTaj
++++====
++++====−−−−++++3
22log2
322
log1 2
2
2
yxyx
yx
dUcenH dUcenH dUcenH dUcenH
++++++++====++++3
22log21 2
yx
yx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 56 -
lMhat´TI19lMhat´TI19lMhat´TI19lMhat´TI19 edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) 0
4loglog 2
22 ====
++++ xx
x> x> x> x> (((( )))) 04
loglog3
5,02
2 ====
++++ x
x
K> K> K> K> 4log1log2 xx ++++==== X> X> X> X> (((( ))))(((( )))) (((( )))) 0212log312log 3
2
3 ====++++++++−−−−++++ xx g> g> g> g> 03437.87 2log12
2log ====++++−−−− ++++ xx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) 0
4loglog 2
22 ====
++++ xx lkçxNÐ lkçxNÐ lkçxNÐ lkçxNÐ 0>>>>x
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³ ³ ³ ³ (((( )))) 04logloglog 222
2 ====−−−−++++ xx (((( )))) 02loglog 2
22 ====−−−−++++ xx
tag tag tag tag xt 2log==== eK)an eK)an eK)an eK)an 022 ====−−−−++++ tt eday eday eday eday 0====++++++++ cba eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 2;1 21 −−−−======== tt
eroberogeday lwm pl:ún nig Esn Bisidæ
- 57 -
----cMeBaH cMeBaH cMeBaH cMeBaH 1====t eK)an eK)an eK)an eK)an 1log2 ====x naM[ naM[ naM[ naM[ 2====x ----cMeBaH cMeBaH cMeBaH cMeBaH 2−−−−====t eK)an eK)an eK)an eK)an 2log2 −−−−====x naM[ naM[ naM[ naM[
41====x
dUcenH dUcenH dUcenH dUcenH 41
;2 21 ======== xx . . . .
x> x> x> x> (((( )))) 04
loglog3
5,02
2 ====
++++ x
x lkçxNÐ lkçxNÐ lkçxNÐ lkçxNÐ 0>>>>x
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ (((( ))))
(((( )))) 02log3log
04logloglog
22
2
21
3
21
22
====++++−−−−
====−−−−++++
xx
xx
tag tag tag tag xt 2log==== smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ 0232 ====++++−−−− tt naM[ naM[ naM[ naM[ 11 ====t b¤ b¤ b¤ b¤ 22 ====t
----cMeBaH cMeBaH cMeBaH cMeBaH 1====t eK)an eK)an eK)an eK)an 1log2 ====x naM[ naM[ naM[ naM[ 2====x ----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 2log2 ====x naM[ naM[ naM[ naM[ 4====x dUcenH dUcenH dUcenH dUcenH 4;2 21 ======== xx . . . . K> K> K> K> 4log1log2 xx ++++==== lkçxNÐ lkçxNÐ lkçxNÐ lkçxNÐ 0>>>>x ni ni ni nig g g g 1≠≠≠≠x smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr 2log21log2 xx ++++==== KuNGgÁTaMgBIrénsmIkarnwg KuNGgÁTaMgBIrénsmIkarnwg KuNGgÁTaMgBIrénsmIkarnwg KuNGgÁTaMgBIrénsmIkarnwg 0log2 ≠≠≠≠x eK)an eK)an eK)an eK)an ³³³³
eroberogeday lwm pl:ún nig Esn Bisidæ
- 58 -
(((( ))))(((( )))) 02loglog
2loglog
22
2
22
2
====−−−−−−−−
++++====
xx
xx
tag tag tag tag xt 2log==== eK)an eK)an eK)an eK)an 022 ====−−−−−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 2;1 21 ====−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 1−−−−====t eK)an eK)an eK)an eK)an 1log2 −−−−====x naM[ naM[ naM[ naM[
21====x
----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 2log2 ====x naM[ naM[ naM[ naM[ 4====x dUcenH dUcenH dUcenH dUcenH 4;
21
21 ======== xx . . . . X> X> X> X> (((( ))))(((( )))) (((( )))) 0212log312log 3
2
3 ====++++++++−−−−++++ xx tag tag tag tag (((( ))))12log3 ++++==== xt smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³
0232 ====++++−−−− tt naM[ naM[ naM[ naM[ 11 ====t b¤ b¤ b¤ b¤ 22 ====t ----cMeBaH cMeBaH cMeBaH cMeBaH 11 ====t eK)an eK)an eK)an eK)an (((( )))) 112log3 ====++++x naM[ naM[ naM[ naM[ 312 ====++++x b¤ b¤ b¤ b¤ 22 ====x naM[ naM[ naM[ naM[ 1====x . . . . ----cMeBacMeBacMeBacMeBaH H H H 22 ====t eK)an eK)an eK)an eK)an (((( )))) 212log3 ====++++x naM[ naM[ naM[ naM[ 912 ====++++x b¤ b¤ b¤ b¤ 82 ====x naM[ naM[ naM[ naM[ 3====x . . . . dUcenH dUcenH dUcenH dUcenH 3;1 ======== xx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 59 -
g> g> g> g> 03437.87 2log122log ====++++−−−− ++++ xx
smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa 0>>>>x . . . . smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³
03437.567 2log2log2 ====++++−−−− xx tag tag tag tag xt 2log==== smIkarkøayCa smIkarkøayCa smIkarkøayCa smIkarkøayCa ³³³³
441343784';0343562 ====−−−−====∆∆∆∆====++++−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 492128;72128 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 7====t eK)an eK)an eK)an eK)an 77 2log ====x b¤ b¤ b¤ b¤ 1log2 ====x naM[ naM[ naM[ naM[ 1====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 49====t eK)an eK)an eK)an eK)an 497 2log ====x b¤ b¤ b¤ b¤ 2log2 ====x naM[ naM[ naM[ naM[ 4====x . . . . dUcenH dUcenH dUcenH dUcenH 4;1 21 ======== xx . . . .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 60 -
lMhat´TI20lMhat´TI20lMhat´TI20lMhat´TI20 edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) (((( )))) 06log5log 2
22 ====−−−−++++−−−−++++ xxxx
x> x> x> x> (((( )))) 0216log)5(2log 32
3 ====++++−−−−−−−−++++ xxxx
K> K> K> K> (((( )))) 044log3log21
2
21 ====−−−−++++++++++++
xxxx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) (((( )))) 06log5log 2
22 ====−−−−++++−−−−++++ xxxx
tag tag tag tag xt 2log==== Edl Edl Edl Edl 0>>>>x . smIkarkøayCa . smIkarkøayCa . smIkarkøayCa . smIkarkøayCa ³³³³ (((( )))) 0652 ====−−−−++++−−−−++++ xtxt
KNna KNna KNna KNna (((( )))) )6(45 2 −−−−−−−−−−−−====∆∆∆∆ xx
(((( ))))22
2
74914
2442510
−−−−====++++−−−−====
++++−−−−++++−−−−====
xxx
xxx
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s
++++−−−−====++++−−−−++++−−−−====
−−−−====−−−−++++++++−−−−====
62
75
12
75
2
1
xxx
t
xxt
eroberogeday lwm pl:ún nig Esn Bisidæ
- 61 -
----cMeBaH cMeBaH cMeBaH cMeBaH 11 −−−−====t eK)an eK)an eK)an eK)an 1log2 −−−−====x naM[ naM[ naM[ naM[ 21====x
----cMeBaH cMeBaH cMeBaH cMeBaH 62 ++++−−−−==== xt eK)an eK)an eK)an eK)an 6log2 ++++−−−−==== xx CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag (((( )))) xyc 2log: ==== CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; (((( )))) 6: ++++−−−−==== xyd . . . .
2 3 4 5 6 7 8-1-2
2
3
4
5
6
7
-1
-2
-3
0 1
1
x
y
A
(d)
(c)
tamRkahViktamRkahViktamRkahViktamRkahVikeyIgeXIjfa eyIgeXIjfa eyIgeXIjfa eyIgeXIjfa (((( ))))c kat; kat; kat; kat; (((( ))))d Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc A manGab;sIus manGab;sIus manGab;sIus manGab;sIus 4====x EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . dUcenH dUcenH dUcenH dUcenH 4;
21
21 ======== xx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 62 -
x> x> x> x> (((( )))) 0216log)5(2log 32
3 ====++++−−−−−−−−++++ xxxx tag tag tag tag xt 3log==== Edl Edl Edl Edl 0>>>>x . smIkarGacsresr . smIkarGacsresr . smIkarGacsresr . smIkarGacsresr ³³³³
(((( )))) 0216522 ====++++−−−−−−−−++++ xtxt (((( )))) (((( )))) (((( ))))22 22165' −−−−====++++−−−−−−−−−−−−====∆∆∆∆ xxx
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 72;3 21 ++++−−−−======== xtt ----cMeBaH cMeBaH cMeBaH cMeBaH 31 ====t eK)an eK)an eK)an eK)an 3log3 ====x naM[ naM[ naM[ naM[ 27====x ----cMeBaH cMeBaH cMeBaH cMeBaH 722 ++++−−−−==== xt eK)an eK)an eK)an eK)an 72log3 ++++−−−−==== xx CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag (((( )))) xyc 3log: ==== CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; (((( )))) 72: ++++−−−−==== xyd . . . .
2 3 4 5 6 7 8-1-2
2
3
4
5
6
7
-1
-2
-3
0 1
1
x
y
A
(d)
(c)
eroberogeday lwm pl:ún nig Esn Bisidæ
- 63 -
tamRkahVikeyIgeXIjfa tamRkahVikeyIgeXIjfa tamRkahVikeyIgeXIjfa tamRkahVikeyIgeXIjfa (((( ))))c kat; kat; kat; kat; (((( ))))d Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc A manGab;sIus manGab;sIus manGab;sIus manGab;sIus 3====x EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar .
K> K> K> K> (((( )))) 044log3log21
2
21 ====−−−−++++++++++++
xxxx
tag tag tag tag xt21log==== enaH enaH enaH enaH 044)3(2 ====−−−−++++++++++++ xtxt
manb¤s manb¤s manb¤s manb¤s 1;4 21 ++++−−−−====−−−−==== xtt . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 41 −−−−====t enaH enaH enaH enaH 4log
21 −−−−====x naM[ naM[ naM[ naM[ 16====x
----cMeBaH cMeBaH cMeBaH cMeBaH 12 ++++−−−−==== xt enaH enaH enaH enaH 1log21 ++++−−−−==== xx
2 3-1
2
-1
0 1
1
x
y
A
B
tamRkahVikeKTajb¤s tamRkahVikeKTajb¤s tamRkahVikeKTajb¤s tamRkahVikeKTajb¤s 2;1 ======== xx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 64 -
lMhat´TI21lMhat´TI21lMhat´TI21lMhat´TI21 k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa AaBa BA loglog ==== Edl Edl Edl Edl BAa ;; CabIcMnY CabIcMnY CabIcMnY CabIcMnYnBitviC¢man nig xusBI nBitviC¢man nig xusBI nBitviC¢man nig xusBI nBitviC¢man nig xusBI 1 . . . . x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar 6867 72log2log ====++++ xx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>RsaybBa¢ak;fa k>RsaybBa¢ak;fa k>RsaybBa¢ak;fa k>RsaybBa¢ak;fa AaBa BA loglog ==== tag tag tag tag (((( ))))iAt Balog==== eK)an eK)an eK)an eK)an Bt aA loglog ==== eday eday eday eday
At
tB
BA log
loglog ====
At
ABt
BAt
aB
BaB
aB
B
loglog
log.loglog
logloglog
========
====
eKTaj eKTaj eKTaj eKTaj (((( ))))iiBt Aalog==== tam tam tam tam (((( ))))i nig nig nig nig (((( ))))ii eK)an eK)an eK)an eK)an AaBa BA loglog ==== . . . . x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar 6867 72log2log ====++++ xx tamrUbmnþxagelIeK)an tamrUbmnþxagelIeK)an tamrUbmnþxagelIeK)an tamrUbmnþxagelIeK)an xx 2log72log 7====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 65 -
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³
823log
77
3437
6867.2
68677
32
32log
2log
2log
2log2log
========⇒⇒⇒⇒====
====
====
====
====++++
xx
x
x
x
xx
dUcenH dUcenH dUcenH dUcenH 8====x Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 66 -
lMhat´TI22lMhat´TI22lMhat´TI22lMhat´TI22 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar (((( )))) (((( )))) 2loglogloglog 2442 ====++++ xx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³
smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa
>>>>>>>>>>>>
0
0log
0log
2
4
x
x
x
b¤ b¤ b¤ b¤ 1>>>>x
smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ (((( )))) 2loglog
21
log21
log 2222 ====++++
xx
(((( )))) (((( ))))
(((( ))))(((( ))))(((( ))))
16
4log
2loglog
6loglog3
2loglog23
1
2loglog21
loglog21
log
2
22
22
22
22222
================
====++++−−−−
====++++++++
x
x
x
x
x
xx
dUcenH dUcenH dUcenH dUcenH 16====x Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 67 -
lMhat´TI23lMhat´TI23lMhat´TI23lMhat´TI23 eK[sIVút eK[sIVút eK[sIVút eK[sIVút )21(log 2
2
n
nu ++++==== Edl Edl Edl Edl INn ∈∈∈∈ KNna KNna KNna KNna (((( )))) n
n
kkn uuuuuS ++++++++++++++++======== ∑∑∑∑
====....210
0 . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNna KNna KNna KNna (((( )))) n
n
kkn uuuuuS ++++++++++++++++======== ∑∑∑∑
====....210
0
eKman eKman eKman eKman )21(log 22
n
nu ++++====
−−−−
−−−−====++++
12
12log
2
12
2 n
n
eK)an eK)an eK)an eK)an ∑∑∑∑====
++++
−−−−
−−−−====n
kk
k
nS0 2
12
212
12log
(((( ))))12log
12
12....
15127
315
13
log
12
12log
122
2
12
2
0 2
12
2
−−−−====
−−−−
−−−−××××××××××××××××====
−−−−
−−−−====
++++
++++
====
++++
∏∏∏∏
n
n
n
n
kk
k
dUcenH dUcenH dUcenH dUcenH )12(log12
2 −−−−====++++n
nS . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 68 -
lMhat´TI24lMhat´TI24lMhat´TI24lMhat´TI24 eK[sIVút eK[sIVút eK[sIVút eK[sIVút
−−−−==== 12
cos2ln nn
xu Edl Edl Edl Edl INn ∈∈∈∈
KNna KNna KNna KNna (((( )))) n
n
kkn uuuuuS ++++++++++++++++======== ∑∑∑∑
====....210
0 . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNnaKNnaKNnaKNna nS ³ ³ ³ ³ eKman eKman eKman eKman 1cos22cos 2 −−−−==== aa eK)an eK)an eK)an eK)an
(((( ))))(((( ))))1cos21cos21cos412cos2 2 ++++−−−−====−−−−====++++ aaaa eKTaj eKTaj eKTaj eKTaj
1cos21cos2
1cos2++++++++====−−−−
aa
a edayyk edayyk edayyk edayyk n
xa
2====
eK)an eK)an eK)an eK)an
++++
++++====
−−−−====1
2cos2
122
cos2ln1
2cos2ln
n
n
nn x
xx
u
eyIg)an eyIg)an eyIg)an eyIg)an ∑∑∑∑====
++++
++++====
++++
++++====
n
knk
k
n xx
x
x
S0 1
2cos2
12cos2ln
12
cos2
122
cos2ln
dUcenH dUcenH dUcenH dUcenH (((( ))))
++++−−−−++++==== 12
cos2ln12cos2ln nn
xxS ....
eroberogeday lwm pl:ún nig Esn Bisidæ
- 69 -
lMhat´TI25lMhat´TI25lMhat´TI25lMhat´TI25 edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar
====
====
9001
6.5
4001
5.4
yx
yx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar ³³³³
====
====
9001
6.5
4001
5.4
yx
yx
eyIgman eyIgman eyIgman eyIgman )4001
ln()5.4(ln ====yx b¤ b¤ b¤ b¤ )1(5ln24ln25ln4ln −−−−−−−−====++++ yx ehIy ehIy ehIy ehIy )
9001
ln()6.5ln( ====yx b¤ b¤ b¤ b¤ )2(5ln26ln26ln5ln −−−−−−−−====++++ yx tam tam tam tam )2(&)1( eK)anRbBnæ½ eK)anRbBnæ½ eK)anRbBnæ½ eK)anRbBnæ½ ³³³³
)5ln(
)6ln(
5ln26ln26ln5ln
5ln24ln25ln4ln −−−−
−−−−−−−−====++++−−−−−−−−====++++
yx
yx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 70 -
−−−−−−−−====++++
++++====−−−−−−−−
)4(5ln25ln.6ln26ln.5ln5ln
)3(6ln.5ln26ln.4ln26ln.5ln6ln.4ln22 yx
yx
bUksmIkar bUksmIkar bUksmIkar bUksmIkar )4(&)3( eK)an eK)an eK)an eK)an ³³³³ )5ln6ln.4(ln2)6ln.4ln5(ln 22 −−−−====−−−− x naM[ naM[ naM[ naM[ 2−−−−====x
tamsmIkar tamsmIkar tamsmIkar tamsmIkar 4001
5.4 ====yx eKTaj eKTaj eKTaj eKTaj 4001
5.4 2 ====−−−− y naM[eKTaj naM[eKTaj naM[eKTaj naM[eKTaj 2−−−−====y . . . . dUcenHRbB½næsmIkarmanKUcemøIy dUcenHRbB½næsmIkarmanKUcemøIy dUcenHRbB½næsmIkarmanKUcemøIy dUcenHRbB½næsmIkarmanKUcemøIy 2,2 −−−−====−−−−==== yx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 71 -
lMhat´TI26lMhat´TI26lMhat´TI26lMhat´TI26 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³ ³ ³ ³ 0logloglog30 3
42
5,03 ====−−−−++++ xxx xxx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³ ³ ³ ³
0logloglog30 34
25,0
3 ====−−−−++++ xxx xxx
lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð
≠≠≠≠≠≠≠≠
≠≠≠≠>>>>
14
15.0
1
0
x
x
x
x
naMeGay naMeGay naMeGay naMeGay
≠≠≠≠
≠≠≠≠≠≠≠≠>>>>
41
2
1
0
x
x
x
x
smIkarxagelIGacsresr smIkarxagelIGacsresr smIkarxagelIGacsresr smIkarxagelIGacsresr ³³³³
04lnln
ln32lnln
ln210
0)4ln(
ln)5,0ln(
lnln
ln30
323
====++++
−−−−−−−−
++++
====−−−−++++
xx
xx
xx
xx
xx
tag tag tag tag xt ln==== eyIg)an eyIg)an eyIg)an eyIg)an 0
4ln3
2ln2
10 ====++++
−−−−−−−−
++++t
tt
t
eroberogeday lwm pl:ún nig Esn Bisidæ
- 72 -
2ln.3612ln722ln289
02ln22ln179
02ln332ln422ln22ln1010
0)2ln(3)2ln2(2)2ln2)(2ln(10
222
22
2222
====++++====∆∆∆∆
====−−−−++++
====++++−−−−++++++++−−−−++++
====−−−−−−−−++++++++++++−−−−
tt
tttttt
tttttt
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ³³³³
2ln4
92ln192ln17
2ln92
92ln192ln17
2
1
−−−−====−−−−−−−−====
====++++−−−−====
t
t
eday eday eday eday xt ln==== eKTaj eKTaj eKTaj eKTaj
−−−−====
====
2ln4ln
2ln92
ln
x
x
naM[ naM[ naM[ naM[ 161
;49 ======== xx
dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s 161
,4 29
1 ======== xx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 73 -
lMhat´TI27lMhat´TI27lMhat´TI27lMhat´TI27 edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar ³³³³
====
====++++
64
26)loglog(5
yx
yx xy
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar
====
====++++
)2(64
)1(26)loglog(5
yx
yx xy
lkçx½NÐ lkçx½NÐ lkçx½NÐ lkçx½NÐ
≠≠≠≠≠≠≠≠>>>>>>>>1,1
0,0
yx
yx
smIkar smIkar smIkar smIkar )1( Gacsresr Gacsresr Gacsresr Gacsresr 526
log1
log ====++++x
xy
y
b¤ b¤ b¤ b¤ 01log.526
)(log 2 ====++++−−−− xx yy tag tag tag tag xt ylog====
eK)an eK)an eK)an eK)an 25
1441
25169
',015262 ====−−−−====∆∆∆∆====++++−−−− tt
naM[ naM[ naM[ naM[ 5512
513
,51
512
513
21 ====++++========−−−−==== tt
---- cMeBaH cMeBaH cMeBaH cMeBaH 51====t eK)an eK)an eK)an eK)an
51
log ====xy
naM[ naM[ naM[ naM[ )3(51
yx ====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 74 -
yk ¬#¦ CYskñúg ¬@¦ eK)an yk ¬#¦ CYskñúg ¬@¦ eK)an yk ¬#¦ CYskñúg ¬@¦ eK)an yk ¬#¦ CYskñúg ¬@¦ eK)an 64. 51
====yy
naM[ naM[ naM[ naM[ 32====y ehIytam ¬#¦ eK)an ehIytam ¬#¦ eK)an ehIytam ¬#¦ eK)an ehIytam ¬#¦ eK)an 2)32( 51
========x .... ---- cMeBaH cMeBaH cMeBaH cMeBaH 5====t eK)an eK)an eK)an eK)an 5log ====xy naM[ naM[ naM[ naM[ )4(5yx ==== yk ¬$¦ CYskñúg ¬@¦ eK)an yk ¬$¦ CYskñúg ¬@¦ eK)an yk ¬$¦ CYskñúg ¬@¦ eK)an yk ¬$¦ CYskñúg ¬@¦ eK)an 64. 5 ====yy naM[ naM[ naM[ naM[ 2====y ehIytam ¬$¦ eK)an ehIytam ¬$¦ eK)an ehIytam ¬$¦ eK)an ehIytam ¬$¦ eK)an 3225 ========x . . . . dUcenHRbB½næmanKUcemøIy dUcenHRbB½næmanKUcemøIy dUcenHRbB½næmanKUcemøIy dUcenHRbB½næmanKUcemøIy ³³³³ )32,2( ======== yx b¤ b¤ b¤ b¤ )2,32( ======== yx . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 75 -
lMhat´TI28lMhat´TI28lMhat´TI28lMhat´TI28 eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½
06)54(log5)54(log 36
326 ====++++−−−−−−−−−−−− −−−−−−−− xx xx ¿¿¿¿
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½
)1(06)54(log5)54(log 36
326 ====++++−−−−−−−−−−−− −−−−−−−− xx xx ¿¿¿¿
l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ
≠≠≠≠−−−−>>>>−−−−
>>>>−−−−
16
06
054 3
x
x
x
smmUl smmUl smmUl smmUl
≠≠≠≠<<<<<<<<
5
6
23 3
x
x
x
¦ ¦ ¦ ¦
≠≠≠≠<<<<
5
23 3
x
x
tag tag tag tag )54(log 36 xy x −−−−==== −−−− smIkar smIkar smIkar smIkar )1( Gacsresr ½Gacsresr ½Gacsresr ½Gacsresr ½
12425,0652 ====−−−−====∆∆∆∆====++++−−−− yy eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 3,2 21 ======== yy ----cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ 2====y eKán eKán eKán eKán 2)54(log 3
6 ====−−−−−−−− xx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 76 -
0)62)(3(
01812
123654
)6(54
2
23
23
23
====−−−−−−−−++++
====−−−−−−−−++++
++++−−−−====−−−−
−−−−====−−−−
xxx
xxx
xxx
xx
eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 71,71,3 321 ++++====−−−−====−−−−==== xxx ----cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ 3====y eKán eKán eKán eKán 3)54(log 3
6 ====−−−−−−−− xx
0)3(18
016210818
)6()54(
2
2
33
====−−−−
====++++−−−−
−−−−====−−−−
x
xx
xx
eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 3====x . . . . dUcenHsMNMub¤srbs;smIkarKW dUcenHsMNMub¤srbs;smIkarKW dUcenHsMNMub¤srbs;smIkarKW dUcenHsMNMub¤srbs;smIkarKW ³³³³
}71;71;3;3{ ++++−−−−−−−−∈∈∈∈x . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 77 -
lMhat´TI29lMhat´TI29lMhat´TI29lMhat´TI29 eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½
)9(log31
)9(log1 323
322)3( 3 xxxx xx
−−−−====−−−−++++ −−−−−−−−¿¿¿¿
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½
)1()9(log31
)9(log1 323
322)3( 3 xxxx xx
−−−−====−−−−++++ −−−−−−−−¿¿¿¿
l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ
≠≠≠≠−−−−>>>>−−−−
>>>>−−−−
13
03
09 32
x
x
xx
smmUl smmUl smmUl smmUl
≠≠≠≠<<<<
<<<<≠≠≠≠
2
3
9,0
x
x
xx
¦ ¦ ¦ ¦
≠≠≠≠≠≠≠≠<<<<
2
0
3
x
x
x
smIkar smIkar smIkar smIkar )1( Gacsresr ½Gacsresr ½Gacsresr ½Gacsresr ½
[[[[ ]]]] 03)9(log
09)9(log6)9(log
)9(log32
)9(log91
1
2323
323
3223
323
3223
====−−−−−−−−
====++++−−−−−−−−−−−−
−−−−====−−−−++++
−−−−
−−−−−−−−
−−−−−−−−
xx
xxxx
xxxx
x
xx
xx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 78 -
12727
02727
927279
)3(9
3)9(log
3232
332
323
========
====−−−−−−−−++++−−−−====−−−−
−−−−====−−−−
====−−−−−−−−
x
x
xxxxx
xxx
xxx
dUcen¼smIkarman¦s dUcen¼smIkarman¦s dUcen¼smIkarman¦s dUcen¼smIkarman¦s 1====x . . . .
lMhat´TI30lMhat´TI30lMhat´TI30lMhat´TI30
eK[GnuKmeK[GnuKmeK[GnuKmeK[GnuKmn_ n_ n_ n_
++++++++
++++++++====
1344
log.1
13log)(
2
222 xx
x
xxf
k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar
++++++++====
++++++++
1344
log1
13log
2
222 xx
x
x
x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam
++++++++
1
13log
22 x
x nig nig nig nig
++++++++1344
log2
2 xx viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña .
K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ )(xf elIcenøa¼ elIcenøa¼ elIcenøa¼ elIcenøa¼ ]3;0[ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 79 -
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k¿ eda¼RsaysmIkar ½k¿ eda¼RsaysmIkar ½k¿ eda¼RsaysmIkar ½k¿ eda¼RsaysmIkar ½
++++++++====
++++++++
1344
log1
13log
2
222 xx
x
x
lkç&x&NÐ lkç&x&NÐ lkç&x&NÐ lkç&x&NÐ 31−−−−≥≥≥≥x
eyIgán eyIgán eyIgán eyIgán 1344
1
13 2
2 ++++++++====
++++++++
xx
x
x
222 )1(4)13( ++++====++++ xx smmUl smmUl smmUl smmUl
−−−−−−−−====++++
++++====++++
2213
22132
2
xx
xx ¦ ¦ ¦ ¦
====++++++++
====++++−−−−
0332
01322
2
xx
xx
eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 21
,1 21 ======== xx . . . . x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x
edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam
++++++++
1
13log
22 x
x nig nig nig nig
++++++++1344
log2
2 xx
viC¢manRBmKñalu¼RtaEt viC¢manRBmKñalu¼RtaEt viC¢manRBmKñalu¼RtaEt viC¢manRBmKñalu¼RtaEt ³³³³
≥≥≥≥++++++++
≥≥≥≥++++++++
11344
11
13
2
2
xx
x
x
smmUl smmUl smmUl smmUl
≥≥≥≥++++
++++−−−−
≥≥≥≥++++++++−−−−
013
334
01
3
2
2
2
xxx
x
xx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 80 -
naM[ naM[ naM[ naM[
−−−−>>>>
≤≤≤≤≤≤≤≤
31
30
x
x ¦ ¦ ¦ ¦ 30 ≤≤≤≤≤≤≤≤ x . . . .
dUcen¼ dUcen¼ dUcen¼ dUcen¼ ]3;0[∈∈∈∈x . . . . K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ )(xf elIcenøa¼ elIcenøa¼ elIcenøa¼ elIcenøa¼ ]3;0[
cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ ]3;0[∈∈∈∈x eKman eKman eKman eKman
≥≥≥≥
++++++++
≥≥≥≥
++++++++
01344
log
01
13log
2
2
22
xx
x
x
tamvismPaBkUsIueKman tamvismPaBkUsIueKman tamvismPaBkUsIueKman tamvismPaBkUsIueKman 0;;2
.2
≥≥≥≥∀∀∀∀
++++≤≤≤≤ baba
ba
eyIgman eyIgman eyIgman eyIgman
++++++++
++++++++====
1344
log.1
13log)(
2
222 xx
x
xxf
eyIgán eyIgán eyIgán eyIgán ³³³³ 22
222 )1344
(log21
)1
13(log
21
)(
++++++++++++
++++++++≤≤≤≤
xx
x
xxf
(((( )))) [[[[ ]]]]3;0;44log
41
)(
)1344
()1
13(log
41
)(
22
22
22
∈∈∈∈∀∀∀∀====≤≤≤≤
++++++++
++++++++≤≤≤≤
xxf
xx
x
xxf
dUcen¼témøFMbMputénGnuKmn_ dUcen¼témøFMbMputénGnuKmn_ dUcen¼témøFMbMputénGnuKmn_ dUcen¼témøFMbMputénGnuKmn_ f esµInwg esµInwg esµInwg esµInwg 4====M . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 81 -
lMhat´TI31lMhat´TI31lMhat´TI31lMhat´TI31 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ IRxxxxf ∈∈∈∈++++++++==== ,)1ln()( 2 cUrRsaybBa¢ak´fa cUrRsaybBa¢ak´fa cUrRsaybBa¢ak´fa cUrRsaybBa¢ak´fa )
11()
1(
bb
aa
fba
baf
++++++++
++++<<<<
++++++++++++
cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb 0,0 >>>>>>>> ba .... dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay RsaybBa¢ak´fa RsaybBa¢ak´fa RsaybBa¢ak´fa RsaybBa¢ak´fa )
11()
1(
bb
aa
fba
baf
++++++++
++++<<<<
++++++++++++
eyIgman eyIgman eyIgman eyIgman IRxxxxf ∈∈∈∈++++++++==== ,)1ln()( 2
eyIgán eyIgán eyIgán eyIgán 2
2
2
2
1
12
21
1
)'1()('
xx
x
x
xx
xxxf
++++++++++++
++++====
++++++++
++++++++====
222
2
1
1
)1)(1(
1
xxxx
xx
++++====
++++++++++++
++++++++====
edayRKb edayRKb edayRKb edayRKb 01: 2 >>>>++++∈∈∈∈ xIRx naM[ naM[ naM[ naM[ IRx
xxf ∈∈∈∈∀∀∀∀>>>>
++++==== ,0
1
1)('
2
dUcen¼GnuKmn_ dUcen¼GnuKmn_ dUcen¼GnuKmn_ dUcen¼GnuKmn_ )(xf CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI IR.... müa¨geTotcMeBa¼RKb müa¨geTotcMeBa¼RKb müa¨geTotcMeBa¼RKb müa¨geTotcMeBa¼RKb 0,0 >>>>>>>> ba eKman ½eKman ½eKman ½eKman ½
eroberogeday lwm pl:ún nig Esn Bisidæ
- 82 -
aa
baa
++++<<<<
++++++++ 11 nig nig nig nig
bb
bab
++++<<<<
++++++++ 11
eKTaj eKTaj eKTaj eKTaj b
ba
aba
bba
a++++
++++++++
<<<<++++++++
++++++++++++ 1111
¦ ¦ ¦ ¦
bb
aa
baba
++++++++
++++<<<<
++++++++++++
111
edaysarEtGnuKmn_ edaysarEtGnuKmn_ edaysarEtGnuKmn_ edaysarEtGnuKmn_ )(xf CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI IR ehtuen¼tamlkçN¼GnuKmn_ekIneKán ehtuen¼tamlkçN¼GnuKmn_ekIneKán ehtuen¼tamlkçN¼GnuKmn_ekIneKán ehtuen¼tamlkçN¼GnuKmn_ekIneKán ³³³³
bb
aa
baba
++++++++
++++<<<<
++++++++++++
111
naM[ naM[ naM[ naM[ )11
()1
(b
ba
af
baba
f++++
++++++++
<<<<++++++++
++++ dUcen¼ dUcen¼ dUcen¼ dUcen¼ )
11()
1(
bb
aa
fba
baf
++++++++
++++<<<<
++++++++++++
cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb 0,0 >>>>>>>> ba . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 83 -
lMhat´TI32lMhat´TI32lMhat´TI32lMhat´TI32 eK[ eK[ eK[ eK[ 1≥≥≥≥a nig nig nig nig 1≥≥≥≥b . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )
2(log2loglog 222
baba
++++≤≤≤≤++++
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgHajfa bgHajfa bgHajfa bgHajfa )
2(log2loglog 222
baba
++++≤≤≤≤++++ cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ 1≥≥≥≥a nig nig nig nig 1≥≥≥≥b eyIgman eyIgman eyIgman eyIgman baba .2≥≥≥≥++++ ( vismPaBkUsIu )( vismPaBkUsIu )( vismPaBkUsIu )( vismPaBkUsIu ) ¦ ¦ ¦ ¦ ba
ba.
2≥≥≥≥
++++
naM[ naM[ naM[ naM[ )loglog(21
)2
(log 222 baba ++++≥≥≥≥
++++
¦ ¦ ¦ ¦ )1()2
(log2loglog 222ba
ba++++≤≤≤≤++++
müa¨geToteKman müa¨geToteKman müa¨geToteKman müa¨geToteKman baba 2222 log.log2loglog ≥≥≥≥++++
)2()loglog(
21
loglog
)loglog()log(log2
loglog.log2log)log(log2
22222
22222
222222
baba
baba
bbaaba
++++≥≥≥≥++++
++++≥≥≥≥++++
++++++++≥≥≥≥++++
tamTMnak´TMng tamTMnak´TMng tamTMnak´TMng tamTMnak´TMng )1( nig nig nig nig )2( eyIgTaj½ eyIgTaj½ eyIgTaj½ eyIgTaj½
eroberogeday lwm pl:ún nig Esn Bisidæ
- 84 -
)2
(log2loglog
)2
(log4)loglog(
)2
(log2)loglog(21
222
22
22
22
22
baba
baba
baba
++++≤≤≤≤++++
++++≤≤≤≤++++
++++≤≤≤≤++++
dUcen¼ dUcen¼ dUcen¼ dUcen¼ )2
(log2loglog 222ba
ba++++≤≤≤≤++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 85 -
lMhat´TI33lMhat´TI33lMhat´TI33lMhat´TI33 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
(((( ))))nnn
un ++++
++++==== 222 log.
11log
Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ++++++++++++++++==== ....321 CaGnuKm CaGnuKm CaGnuKm CaGnuKmn_én n_én n_én n_én n K>kMnt; K>kMnt; K>kMnt; K>kMnt; n edIm,I[ edIm,I[ edIm,I[ edIm,I[ 100====nS . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman (((( ))))nn
nun ++++
++++==== 222 log.
11log
eday eday eday eday (((( )))) (((( ))))nnn
nn 2222 log1log
1log
11log −−−−++++====
++++====
++++
nig nig nig nig (((( )))) [[[[ ]]]] (((( ))))1log)(log)1(loglog 2222
2 ++++++++====++++====++++ nnnnnn eK)an eK)an eK)an eK)an (((( ))))[[[[ ]]]] (((( ))))[[[[ ]]]] 2
2
2
2 log1log nnun −−−−++++==== cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman (((( ))))nn 22 log)1(log >>>>++++ dUcenH dUcenH dUcenH dUcenH 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n . . . . x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ++++++++++++++++==== ....321
eroberogeday lwm pl:ún nig Esn Bisidæ
- 86 -
eKman eKman eKman eKman (((( ))))[[[[ ]]]] (((( ))))[[[[ ]]]] 2
2
2
2 log1log nnun −−−−++++==== ----ebI ebI ebI ebI [[[[ ]]]] [[[[ ]]]]2
2
2
21 1log2log:1 −−−−======== un ----ebI ebI ebI ebI [[[[ ]]]] [[[[ ]]]]2
2
2
22 2log3log:2 −−−−======== un ----ebI ebI ebI ebI [[[[ ]]]] [[[[ ]]]]2
2
2
23 3log4log:3 −−−−======== un -------------------------------------------------------------------------------------------------------------------------------------------------------- ----ebI ebI ebI ebI [[[[ ]]]] (((( ))))[[[[ ]]]]2
2
2
2 log)1(log: nnunn n −−−−++++======== eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an ³³³³ dUcenH dUcenH dUcenH dUcenH (((( ))))[[[[ ]]]] 2
2 1log ++++==== nSn . . . . K>kMnt; K>kMnt; K>kMnt; K>kMnt; n ed ed ed edIm,I[ Im,I[ Im,I[ Im,I[ 100====nS eyIg)an eyIg)an eyIg)an eyIg)an (((( ))))[[[[ ]]]] 1001log 2
2 ====++++n (((( )))) 101log2 ====++++n 102421 10 ========++++n dUcenH dUcenH dUcenH dUcenH 1023====n . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 87 -
lMhat´TI34lMhat´TI34lMhat´TI34lMhat´TI34 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xxxf 2log2)( ++++==== Edl Edl Edl Edl 0>>>>x eKtag eKtag eKtag eKtag (((( )))) (((( ))))(((( )))))(;)(;)( 321 xfffuxffuxfu ============ nig nig nig nig (((( ))))(((( ))))(((( ))))....)(.... xffffu nn ==== . . . . k>eKyk k>eKyk k>eKyk k>eKyk nn uV 2log1++++==== . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa 2
1 nn VV ====++++ xxxx>eRbIGnumanrYmKNitviTüacUrRsayfa >eRbIGnumanrYmKNitviTüacUrRsayfa >eRbIGnumanrYmKNitviTüacUrRsayfa >eRbIGnumanrYmKNitviTüacUrRsayfa n
VVn2
1==== . . . . KKKK>KNna >KNna >KNna >KNna nV nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig x . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----bgðajfa bgðajfa bgðajfa bgðajfa 2
1 nn VV ====++++ eKman eKman eKman eKman nn uV 2log1++++==== enaH enaH enaH enaH 121 log1 ++++++++ ++++==== nn uvV eday eday eday eday (((( ))))(((( ))))(((( )))) (((( ))))1....)(.... −−−−======== nnn ufxffffu eK)an eK)an eK)an eK)an (((( )))) nu
nnn uufu 2log21 )( ++++
++++ ======== ehtuenH ehtuenH ehtuenH ehtuenH (((( ))))[[[[ ]]]]nu
nn uV 2log221 log1 ++++
++++ ++++====
(((( ))))22
21
221
)log1(
loglog21
nnn
nnn
VuV
uuV
====++++====
++++++++====
++++
++++
dUcenH dUcenH dUcenH dUcenH 21 nn VV ====++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 88 -
x>eRbIGnumanrYmKNitviTüaRsayfa x>eRbIGnumanrYmKNitviTüaRsayfa x>eRbIGnumanrYmKNitviTüaRsayfa x>eRbIGnumanrYmKNitviTüaRsayfa n
VVn2
1==== eKman eKman eKman eKman 2
1 nn VV ====++++ ¬ tamsRmayxagelI ¦ ¬ tamsRmayxagelI ¦ ¬ tamsRmayxagelI ¦ ¬ tamsRmayxagelI ¦ eK)an eK)an eK)an eK)an 12
12
12 VVV ======== Bit Bit Bit Bit 22
14
12
23 VVVV ============ Bit Bit Bit Bit ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH kn ==== KW KW KW KW k
VVk2
1==== eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH 1++++==== kn KW KW KW KW 12
11
++++====++++
k
VVk eKman eKman eKman eKman 2
1 kk VV ====++++ eday eday eday eday k
VVk2
1==== eK)an eK)an eK)an eK)an 12
1
22
11
++++====
====++++
kk
VVVk Bit Bit Bit Bit dUdUdUdUcenH cenH cenH cenH n
VVn2
1==== . . . . K>KNna K>KNna K>KNna K>KNna nV nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig x ³ ³ ³ ³ eKman eKman eKman eKman n
VVn2
1==== eday eday eday eday (((( )))) (((( ))))2
2log2
2121 log1log1log1 2 xxuV x ++++====++++====++++==== ++++ dUcenH dUcenH dUcenH dUcenH (((( )))) 12
2log1++++
++++====n
xVn ehIy ehIy ehIy ehIy nn uV 2log1++++==== naM[ naM[ naM[ naM[ 12 −−−−==== nV
nu
dUcenH dUcenH dUcenH dUcenH (((( ))))(((( )))) 1log112
22 −−−−++++++++
====n
xnu . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 89 -
lMhat´TI35lMhat´TI35lMhat´TI35lMhat´TI35 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
(((( ))))2log )1( ++++==== ++++ nu nn Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa (((( ))))nu CasIV CasIV CasIV CasIVútcuHCanic©cMeBaHRKb; útcuHCanic©cMeBaHRKb; útcuHCanic©cMeBaHRKb; útcuHCanic©cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ln....lnlnln 321 ++++++++++++++++==== CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa (((( ))))nu CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman (((( ))))2log )1( ++++==== ++++ nu nn nig nig nig nig (((( ))))3log )2(1 ++++==== ++++++++ nu nn cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman 1)2()2( 22 −−−−++++>>>>++++ nn b¤ b¤ b¤ b¤ )3)(1()12)(12()2( 2 ++++++++====++++++++−−−−++++>>>>++++ nnnnn eK)an eK)an eK)an eK)an (((( )))) [[[[ ]]]])3)(1(log2log )2(
2
)2( ++++++++>>>>++++ ++++++++ nnn nn b¤ b¤ b¤ b¤ (((( )))) (((( )))) (((( ))))13log1log2 )2()2( ++++++++++++>>>> ++++++++ nn nn tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eKman eKman eKman eKman ³³³³
(((( )))) (((( )))) (((( )))) (((( ))))2log1log22log1log )1()2()1()2( ++++++++≥≥≥≥++++++++++++ ++++++++++++++++ nnnn nnnn
b¤ b¤ b¤ b¤ (((( )))) (((( )))) (((( ))))222log1log )1()2( ≥≥≥≥++++++++++++ ++++++++ nn nn
eroberogeday lwm pl:ún nig Esn Bisidæ
- 90 -
tam tam tam tam (((( ))))1 nig nig nig nig (((( ))))2 eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³ (((( )))) (((( )))) (((( )))) (((( ))))
(((( )))) (((( ))))1
)2()1(
)2()2()1()2(
3log2log
3log1log2log1log
++++
++++++++
++++++++++++++++
>>>>
++++>>>>++++
++++++++++++>>>>++++++++++++
nn
nn
nnnn
uu
nn
nnnn
dUcenH dUcenH dUcenH dUcenH (((( ))))nu CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; 1≥≥≥≥n . . . . x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ln....lnlnln 321 ++++++++++++++++==== eKman eKman eKman eKman (((( ))))
)1ln()2ln(
2log )1( ++++++++====++++==== ++++ n
nnu nn
eK)an eK)an eK)an eK)an (((( )))) (((( ))))
======== ∏∏∏∏∑∑∑∑
========
n
kk
n
kkn uuS
11lnln
eday eday eday eday (((( )))) (((( ))))(((( ))))∏∏∏∏∏∏∏∏
========
++++++++====
n
k
n
kk n
nu
11 1ln2ln
)2(log
2ln)2ln(
)1ln()2ln(
.....4ln5ln
.3ln4ln
.2ln3ln
2 ++++====++++====
++++++++====
nn
nn
dUcenH dUcenH dUcenH dUcenH (((( ))))[[[[ ]]]]2logln 2 ++++==== nSn . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 91 -
lMhat´TI36lMhat´TI36lMhat´TI36lMhat´TI36 eK[ eK[ eK[ eK[ xxxxf 2424 sin4coscos4sin)( ++++−−−−++++==== k> sRmYl k> sRmYl k> sRmYl k> sRmYl )(xf x>kMnt; x>kMnt; x>kMnt; x>kMnt; x edIm,I[ edIm,I[ edIm,I[ edIm,I[
21
)( ====xf dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> sRmYl k> sRmYl k> sRmYl k> sRmYl )(xf eKman eKman eKman eKman xxxxf 2424 sin4coscos4sin)( ++++−−−−++++==== eday eday eday eday xx 22 sin1cos −−−−==== nig nig nig nig xx 22 cos1sin −−−−====
)cos1(4cos)sin1(4sin)( 2424 xxxxxf −−−−++++−−−−−−−−++++====
xxxx
xx
xxxx
2222
2222
4242
sincos)cos2()sin2(
)cos2()sin2(
coscos44sinsin44
−−−−====−−−−−−−−−−−−====
−−−−−−−−−−−−====
++++−−−−−−−−++++−−−−====
dUcenH dUcenH dUcenH dUcenH xxf 2cos)( ==== . . . . x>kMnt; x>kMnt; x>kMnt; x>kMnt; x edIm,I[ edIm,I[ edIm,I[ edIm,I[
21
)( ====xf
eK)an eK)an eK)an eK)an 21
2cos ====x naM[ naM[ naM[ naM[ ππππππππkx 2
3++++±±±±==== Edl Edl Edl Edl Zk ∈∈∈∈ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 92 -
lMhat´TI37lMhat´TI37lMhat´TI37lMhat´TI37 cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; INn∈∈∈∈ eK[ eK[ eK[ eK[
12sin
12cos
ππππππππ nnnS ++++====
k> KNnatémø k> KNnatémø k> KNnatémø k> KNnatémø 12
cosππππ nig nig nig nig
12sin
ππππ x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 0624 12 ====++++−−−− ++++++++ nnn SSS dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> KNnatémø k> KNnatémø k> KNnatémø k> KNnatémø
12cos
ππππ nig nig nig nig 12
sinππππ
426
22
.23
22
.21
4sin
3sin
4cos
3cos
)43
cos(12
cos
++++====++++====
++++====
−−−−====
ππππππππππππππππ
ππππππππππππ
426
21
.22
22
.23
3cos
4sin
4cos
3sin
)43
sin(12
sin
−−−−====−−−−====
−−−−====
−−−−====
ππππππππππππππππ
ππππππππππππ
dUcenH dUcenH dUcenH dUcenH 12
2612
sin;12
2612
cos−−−−====++++==== ππππππππ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 93 -
x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 0624 12 ====++++−−−− ++++++++ nnn SSS eKman eKman eKman eKman
12sin
12cos
ππππππππ nnnS ++++====
tag tag tag tag 12
sin;12
cos 21ππππππππ ======== xx enaH enaH enaH enaH nn
n xxS 21 ++++====
eKman eKman eKman eKman 26
426
426
21 ====−−−−++++++++====++++ xx
ehIy ehIy ehIy ehIy 41
1626
426
.4
26. 21 ====−−−−====−−−−++++====xx
eK)an eK)an eK)an eK)an 1x nig nig nig nig 2x Cab¤ssmIkar Cab¤ssmIkar Cab¤ssmIkar Cab¤ssmIkar 041
262 ====++++−−−− xx
b¤ b¤ b¤ b¤ 01624 2 ====++++−−−− xx
eKTaj eKTaj eKTaj eKTaj
====++++−−−−
====++++−−−−
01624
01624
22
2
12
1
xx
xx
b¤ b¤ b¤ b¤
====++++−−−−
====++++−−−−++++++++
++++++++
)(0624
)(0624
21
22
2
11
12
1
iixxx
ixxxnnn
nnn
bUksmIkar bUksmIkar bUksmIkar bUksmIkar )(i nig nig nig nig )(ii GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an ³³³³ 0)()(62)(4 21
12
11
22
21 ====++++++++++++−−−−++++ ++++++++++++++++ nnnnnn xxxxxx
dUcenH dUcenH dUcenH dUcenH 0624 12 ====++++−−−− ++++++++ nnn SSS . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 94 -
lMhat´TI38lMhat´TI38lMhat´TI38lMhat´TI38 cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa
21
73
cos7
2cos
7cos ====++++−−−−
ππππππππππππ (IMO 1963)
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa
21
73
cos7
2cos
7cos ====++++−−−− ππππππππππππ
tag tag tag tag 7
3cos
72
cos7
cosππππππππππππ ++++−−−−====S
KuNGgÁTaMgBIrénvismPaBnwg KuNGgÁTaMgBIrénvismPaBnwg KuNGgÁTaMgBIrénvismPaBnwg KuNGgÁTaMgBIrénvismPaBnwg 7
sin2ππππ eK)an eK)an eK)an eK)an ³³³³
7sin
73
cos27
sin7
2cos2
72
sin7
sin2ππππππππππππππππππππππππ ++++−−−−====S
7sin
)7
2sin
73
(sin)7
sin7
3(sin
72
sin
)7
2sin
74
(sin)7
sin7
3(sin
72
sin
ππππ
ππππππππππππππππππππ
ππππππππππππππππππππ
====
−−−−++++−−−−−−−−====
−−−−++++−−−−−−−−====
eKTaj eKTaj eKTaj eKTaj 21====S
dUcenH dUcenH dUcenH dUcenH 21
73
cos7
2cos
7cos ====++++−−−− ππππππππππππ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 95 -
lMhat´TI39lMhat´TI39lMhat´TI39lMhat´TI39 KNna KNna KNna KNna )
53
sin41()5
sin41( 22 ππππππππ −−−−−−−−====A dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNna KNna KNna KNna )
53
sin41()5
sin41( 22 ππππππππ −−−−−−−−====A eKman eKman eKman eKman xxx cos3cos43cos 3 −−−−==== eKTaj eKTaj eKTaj eKTaj
xx
xcos
3cos3cos4 2 ====−−−−
xx
x
xx
x
cos3cos
sin41
cos3cos
3)sin1(4
2
2
====−−−−
====−−−−−−−−
eK)an eK)an eK)an eK)an 5
cos
59
cos
53
cos
59
cos
5cos
53
cos
ππππ
ππππ
ππππ
ππππ
ππππ
ππππ
====××××====A
eday eday eday eday 5
cos)5
2cos(5
9cos
ππππππππππππππππ ====−−−−==== eKTaj)an eKTaj)an eKTaj)an eKTaj)an 1====A . . . . dUcenH dUcenH dUcenH dUcenH 1)
53
sin41()5
sin41( 22 ====−−−−−−−− ππππππππ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 96 -
lMhat´TI40lMhat´TI40lMhat´TI40lMhat´TI40
eK[ eK[ eK[ eK[ 144
)( 2
3
−−−−++++====
xx
xxf . KNnatémø . KNnatémø . KNnatémø . KNnatémø )
7(cos
ππππf . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNnatémø KNnatémø KNnatémø KNnatémø )
7(cos
ππππf
eK)an eK)an eK)an eK)an )(1
7cos4
7cos4
7cos
)7
(cos2
3
if−−−−++++
==== ππππππππ
ππππππππ
eKman eKman eKman eKman )7
3sin(
74
sinππππππππππππ −−−−====
)()17
cos47
cos4(81
7cos
)7
cos1(437
cos47
cos8
7sin43)1
7cos2(
7cos4
7sin4
7sin3)1
7cos2(
7cos
7sin4
73
sin7
2cos
72
sin2
23
23
22
32
ii−−−−++++====
−−−−−−−−====−−−−
−−−−====−−−−
−−−−====−−−−
====
ππππππππππππ
ππππππππππππ
ππππππππππππ
ππππππππππππππππππππ
ππππππππππππ
tam tam tam tam )(i nig nig nig nig )(ii eKTaj eKTaj eKTaj eKTaj 81
)7
(cos ====ππππf . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 97 -
lMhat´TI41lMhat´TI41lMhat´TI41lMhat´TI41 eK[ eK[ eK[ eK[ )cos(sin
1)( xx
kxf kk
k ++++==== Edl Edl Edl Edl ...;3;2;1====k cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa
121
)()( 64 ====−−−− xfxf . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bbbbgðajfa gðajfa gðajfa gðajfa
121
)()( 64 ====−−−− xfxf
eKman eKman eKman eKman )cos(sin41
)( 444 xxxf ++++====
(((( ))))xx
xxxx
22
22222
cossin2141
]cossin2)cos(sin[41
−−−−====
−−−−++++====
ehIy ehIy ehIy ehIy )cos(sin61
)( 666 xxxf ++++====
[[[[ ]]]])cos(sincossin3)cos(sin61
)( 22223226 xxxxxxxf ++++−−−−++++====
(((( ))))xx 22 cossin3161 −−−−====
eK)an eK)an eK)an eK)an 121
61
41
)()( 64 ====−−−−====−−−− xfxf
dUcenH dUcenH dUcenH dUcenH 121
)()( 64 ====−−−− xfxf . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 98 -
lMhat´TI42lMhat´TI42lMhat´TI42lMhat´TI42 cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati n nig cMeBaHRKb;cMnYnBit nig cMeBaHRKb;cMnYnBit nig cMeBaHRKb;cMnYnBit nig cMeBaHRKb;cMnYnBit
tk
x2
ππππ≠≠≠≠ Edl Edl Edl Edl nt ....,,2,1,0==== nig nig nig nig k CacMnYnKt;eKman CacMnYnKt;eKman CacMnYnKt;eKman CacMnYnKt;eKman
xxxxx
nn 2cotcot
2sin
1.....
4sin1
2sin1 −−−−====++++++++++++
(IMO 1966)
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfabgðajfabgðajfabgðajfa
xxxxx
nn 2cotcot
2sin
1.....
4sin1
2sin1 −−−−====++++++++++++
tag tag tag tag ∑∑∑∑====
====++++++++++++====n
kknn
xxxxS
1 2sin
1
2sin
1.....
4sin1
2sin1
eKman eKman eKman eKman a
aaa
aaa 2sin
2coscos22sincossin
2sin1 222 −−−−====++++====
aaaa
aa
a2cotcot
2sin2cos
2sincos2
2sin1 2
−−−−====−−−−====
eK)an eK)an eK)an eK)an (((( ))))∑∑∑∑====
−−−− −−−−====−−−−====n
k
nkkn xxxxS
1
1 2cotcot2cot2cot
dUcenH dUcenH dUcenH dUcenH xxxxx
nn 2cotcot
2sin
1.....
4sin1
2sin1 −−−−====++++++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 99 -
lMhat´TI43lMhat´TI43lMhat´TI43lMhat´TI43 eK[ eK[ eK[ eK[ dcba ,,, CacMnYnenAkñúgcenøaH CacMnYnenAkñúgcenøaH CacMnYnenAkñúgcenøaH CacMnYnenAkñúgcenøaH ];0[ ππππ edaydwgfa edaydwgfa edaydwgfa edaydwgfa
++++====++++++++====++++
)cos2(cos4cos7cos
)sin2(sin4sin7sin
dcba
dcba
cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )cos(7)cos(2 cbda −−−−====−−−− . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa )cos(7)cos(2 cbda −−−−====−−−− eKman eKman eKman eKman
++++====++++++++====++++
)cos2(cos4cos7cos
)sin2(sin4sin7sin
dcba
dcba
b¤ b¤ b¤ b¤
−−−−====−−−−−−−−====−−−−
bcda
bcda
cos7cos4cos8cos
sin7sin4sin8sin
b¤ b¤ b¤ b¤
−−−−====−−−−
−−−−====−−−−
)()cos7cos4()cos8cos(
)()sin7sin4()sin8sin(22
22
iibcda
ibcda
bUksmIkar bUksmIkar bUksmIkar bUksmIkar )(i nig nig nig nig )(ii GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an ³³³³
)cos(56)cos(16
)cos(5665)cos(1665
cbda
cbda
−−−−−−−−====−−−−−−−−−−−−−−−−====−−−−−−−−
dUcenH dUcenH dUcenH dUcenH )cos(7)cos(2 cbda −−−−====−−−− . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 100 -
lMhat´TI44lMhat´TI44lMhat´TI44lMhat´TI44 cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam ³³³³
)sin()sin()sin( xzzyyx −−−−++++−−−−++++−−−− dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay sresrCaplKuNk+tþa sresrCaplKuNk+tþa sresrCaplKuNk+tþa sresrCaplKuNk+tþa tag tag tag tag )sin()sin()sin( xzzyyxT −−−−++++−−−−++++−−−−==== man man man man
22
cos2
sin2)sin()sin(zyxzx
zyyx++++−−−−−−−−====−−−−++++−−−−
ehIy ehIy ehIy ehIy 2
cos2
sin2)sin(xzxz
xz−−−−−−−−====−−−−
−−−−−−−−++++−−−−−−−−====2
cos22
cos2
sin2xzzyxzx
T
2
sin2
sin2
sin4
2sin
2sin
2sin4
xzzyyx
yxyzzx
−−−−−−−−−−−−−−−−====
−−−−−−−−−−−−−−−−====
dUcenH dUcenH dUcenH dUcenH
2sin
2sin
2sin4)sin()sin()sin(
xzzyyxxzzyyx
−−−−−−−−−−−−−−−−====−−−−++++−−−−++++−−−−
eroberogeday lwm pl:ún nig Esn Bisidæ
- 101 -
lMhat´TI45lMhat´TI45lMhat´TI45lMhat´TI45 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag
2cba
p++++++++==== CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN .
kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa bc
cpbpA ))((2
sin−−−−−−−−==== rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng
BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH . xxxx----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa
81
2sin
2sin
2sin ≤≤≤≤CBA
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----bgðajfa bgðajfa bgðajfa bgðajfa
bccpbpA ))((
2sin
−−−−−−−−==== tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs Abccba cos2222 −−−−++++==== eKTaj eKTaj eKTaj eKTaj
bcacb
A2
cos222 −−−−++++====
eday eday eday eday 2cos1
2sin2 AA −−−−====
bc
cbacbabc
cbabc
acbbc
4))((
4)(
4)(2 22222
−−−−++++++++−−−−====
−−−−−−−−====−−−−++++−−−−====
eday eday eday eday 2
cbap
++++++++==== enaH enaH enaH enaH
−−−−====−−−−++++−−−−====++++−−−−
)(2
)(2
cpcba
bpcba
eroberogeday lwm pl:ún nig Esn Bisidæ
- 102 -
eK)an eK)an eK)an eK)an bc
cpbpbc
cpbpA ))((4
))((42
sin2 −−−−−−−−====−−−−−−−−====
dUcenH dUcenH dUcenH dUcenH bc
cpbpA ))((2
sin−−−−−−−−==== . . . .
TMnak;TMngBIreTOtEdlRsedogKñaenHKW TMnak;TMngBIreTOtEdlRsedogKñaenHKW TMnak;TMngBIreTOtEdlRsedogKñaenHKW TMnak;TMngBIreTOtEdlRsedogKñaenHKW ³³³³
accpapB ))((
2sin
−−−−−−−−==== nig nig nig nig ab
bpapC ))((2
sin−−−−−−−−====
xxxx----Rsayfa Rsayfa Rsayfa Rsayfa 81
2sin
2sin
2sin ≤≤≤≤CBA
eKman eKman eKman eKman bc
cpbpA ))((2
sin−−−−−−−−====
accpapB ))((
2sin
−−−−−−−−==== nig nig nig nig ab
bpapC ))((2
sin−−−−−−−−====
eK)an eK)an eK)an eK)an )())()((
2sin
2sin
2sin i
abccpbpapCBA −−−−−−−−−−−−====
tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eKman eKman eKman eKman
))((2
))((2
))((22
))((2)()(
bpapc
bpapbacba
bpapbap
bpapbpap
−−−−−−−−≥≥≥≥
−−−−−−−−≥≥≥≥−−−−−−−−++++++++
−−−−−−−−≥≥≥≥−−−−−−−−
−−−−−−−−≥≥≥≥−−−−++++−−−−
eKTaj)an eKTaj)an eKTaj)an eKTaj)an )1(4
))((2c
bpap ≤≤≤≤−−−−−−−−
dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr )2(4
))((2a
cpbp ≤≤≤≤−−−−−−−−
eroberogeday lwm pl:ún nig Esn Bisidæ
- 103 -
nig nig nig nig )3(4
))((2b
apcp ≤≤≤≤−−−−−−−− eFIvplKuNTMnak;TMng eFIvplKuNTMnak;TMng eFIvplKuNTMnak;TMng eFIvplKuNTMnak;TMng )2(;)1( nig nig nig nig )3( eK)an eK)an eK)an eK)an ³³³³
64)()()(
222222 cba
cpbpap ≤≤≤≤−−−−−−−−−−−−
eKTaj eKTaj eKTaj eKTaj )(81))()((
iiabc
cpbpap ≤≤≤≤−−−−−−−−−−−−
tamTMnak;TMtamTMnak;TMtamTMnak;TMtamTMnak;TMng ng ng ng )(i nig nig nig nig )(ii eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³
81
2sin
2sin
2sin ≤≤≤≤CBA . . . . lMhat´TI46lMhat´TI46lMhat´TI46lMhat´TI46 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC ehIyeKtag ehIyeKtag ehIyeKtag ehIyeKtag r nig nig nig nig R erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg; carwkkñúg nig kaMrgVg;carwkeRkAénRcarwkkñúg nig kaMrgVg;carwkeRkAénRcarwkkñúg nig kaMrgVg;carwkeRkAénRcarwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .tIekaN .tIekaN .tIekaN . cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa
Rr
CBA ++++====++++++++ 1coscoscos dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay Rsayfa Rsayfa Rsayfa Rsayfa
Rr
CBA ++++====++++++++ 1coscoscos
eKman eKman eKman eKman 2
cos2
cos2coscosBABA
BA−−−−++++====++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 104 -
2
cos2
sin2
2cos)
22cos(2
BAC
BAC
−−−−====
−−−−−−−−==== ππππ
ehIy ehIy ehIy ehIy 2
sin21cos 2 CC −−−−====
)2
cos2
(sin2
sin21coscoscosBACC
CBA−−−−−−−−−−−−====++++++++
eday eday eday eday 2
cos2
cos2
cos2
sinBABABAC −−−−−−−−++++====−−−−−−−−
2
sin2
sin2BA−−−−====
eK)an eK)an eK)an eK)an )(
2sin
2sin
2sin41coscoscos i
CBACBA ++++====++++++++
eKdwgfa eKdwgfa eKdwgfa eKdwgfa ³³³³
bccpbpA ))((
2sin
−−−−−−−−==== ; ac
cpapB ))((2
sin−−−−−−−−====
nig nig nig nig ab
bpapC ))((2
sin−−−−−−−−====
abccpbpap
CBA))()((
41coscoscos−−−−−−−−−−−−++++====++++++++
tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug ³³³³
Rabc
prcpbpappS4
))()(( ========−−−−−−−−−−−−====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 105 -
eKTaj eKTaj eKTaj eKTaj Srprp
Scpbpap ============−−−−−−−−−−−− 2
2
))()((
ehIy ehIy ehIy ehIy prRabc 4==== eK)an eK)an eK)an eK)an
Rr
prRSr
CBA ++++====++++====++++++++ 14
.41coscoscos
dUcenH dUcenH dUcenH dUcenH Rr
CBA ++++====++++++++ 1coscoscos . . . . lMhat´TI47lMhat´TI47lMhat´TI47lMhat´TI47 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag
2cba
p++++++++==== Caknø Caknø Caknø CaknøHbrimaRténRtIekaNehIy HbrimaRténRtIekaNehIy HbrimaRténRtIekaNehIy HbrimaRténRtIekaNehIy r nig nig nig nig R
erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrprcabcab 422 ++++++++====++++++++ xxxx----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrrpcba 822 22222 −−−−−−−−====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----bgðajfa bgðajfa bgðajfa bgðajfa Rrprcabcab 422 ++++++++====++++++++ ttttamrUbmnþehrug amrUbmnþehrug amrUbmnþehrug amrUbmnþehrug ³³³³
prcpbpappS ====−−−−−−−−−−−−==== ))()(( eKTaj eKTaj eKTaj eKTaj
pcpbpap
r))()(( −−−−−−−−−−−−====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 106 -
pabcpcabcabpp
r
pabcpcabcabpcbap
r
pcpbpap
r
−−−−++++++++++++−−−−====
−−−−++++++++++++++++++++−−−−====
−−−−−−−−−−−−====
)(2
)()(
))()((
332
232
2
pabc
prcabcab
pabcpcabcabp
r
++++++++====++++++++
−−−−++++++++++++−−−−====
22
32 )(
eday eday eday eday R
abcprS
4======== eKTaj eKTaj eKTaj eKTaj rR
pabc
4====
dUcenH dUcenH dUcenH dUcenH Rrprcabcab 422 ++++++++====++++++++ . . . . xxxx----bgðajfa bgðajfa bgðajfa bgðajfa Rrrpcba 822 22222 −−−−−−−−====++++++++ tamsmPaB tamsmPaB tamsmPaB tamsmPaB )(2)( 2222 cabcabcbacba ++++++++++++++++++++====++++++++ eKTaj eKTaj eKTaj eKTaj )(2)( 2222 cabcabcbacba ++++++++−−−−++++++++====++++++++ eday eday eday eday pcba 2====++++++++ nig nig nig nig Rrprcabcab 422 ++++++++====++++++++ eK)an eK)an eK)an eK)an )4(24 222222 Rrprpcba ++++++++−−−−====++++++++ dUcenH dUcenH dUcenH dUcenH Rrrpcba 822 22222 −−−−−−−−====++++++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 107 -
lMhat´TI48lMhat´TI48lMhat´TI48lMhat´TI48 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag
2cba
p++++++++==== CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy r
CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa ³³³³ k> k> k> k> r
Ccp
Bbp
Aap ====−−−−====−−−−====−−−−
2tan)(
2tan)(
2tan)(
x> x> x> x> p
rRCBA ++++====++++++++4
2tan
2tan
2tan
K> K> K> K> rpCBA ====
2tan
2tan
2tan
X> X> X> X> rpCBA ====++++++++
2cot
2cot
2cot
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> k> k> k> r
Ccp
Bbp
Aap ====−−−−====−−−−====−−−−
2tan)(
2tan)(
2tan)(
A
'A B C
'B 'C
I
eroberogeday lwm pl:ún nig Esn Bisidæ
- 108 -
eKman eKman eKman eKman pCABCAB 2====++++++++ eday eday eday eday '''' BAACBCACAB ++++====++++==== ¬ eRBaH ¬ eRBaH ¬ eRBaH ¬ eRBaH '' BABC ==== ¦ ¦ ¦ ¦ ehIy ehIy ehIy ehIy '''' ACCAABCBCA ++++====++++==== ¬ eRBaH ¬ eRBaH ¬ eRBaH ¬ eRBaH CACB '' ==== nig nig nig nig '' ACAB ==== ¦ ¦ ¦ ¦ eK)an eK)an eK)an eK)an pACCABCBAAC 2'''' ====++++++++++++++++
paAC
pBCAC
pBCBCAC
PBCCABAAC
22'2
22'2
2'2
2)''('2
====++++====++++
====++++++++====++++++++++++
eKTaj eKTaj eKTaj eKTaj apABAC −−−−======== '' kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg AIC ' eKman eKman eKman eKman
apr
ACICA
−−−−========
''
2tan
eKTaj eKTaj eKTaj eKTaj rA
ap ====−−−−2
tan)( . . . . dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr
bprB−−−−
====2
tan b¤ b¤ b¤ b¤ rB
bp ====−−−−2
tan)(
nig nig nig nig cp
rC−−−−
====2
tan b¤ b¤ b¤ b¤ rC
cp ====−−−−2
tan)(
dUcenH dUcenH dUcenH dUcenH rC
cpB
bpA
ap ====−−−−====−−−−====−−−−2
tan)(2
tan)(2
tan)(
eroberogeday lwm pl:ún nig Esn Bisidæ
- 109 -
x> x> x> x> p
rRCBA ++++====++++++++ 42
tan2
tan2
tan
tag tag tag tag 2
tan2
tan2
tanCBA
T ++++++++====
eday eday eday eday cp
rCbp
rBap
rA−−−−
====−−−−
====−−−−
====2
tan;2
tan;2
tan
eK)an eK)an eK)an eK)an ³³³³
cpr
bpr
apr
T−−−−
++++−−−−
++++−−−−
====
[[[[ ]]]]
(((( ))))
(((( ))))
(((( ))))
prcabcabp
Scabcabpp
S
cabcabpcbapS
cpbpapabacbcpbapcapcbppr
cpbpapbpapcpapcpbpr
++++++++++++−−−−====
++++++++++++−−−−====
++++++++++++++++++++−−−−====
−−−−−−−−−−−−++++++++++++++++−−−−++++−−−−++++−−−−====
−−−−−−−−−−−−−−−−−−−−++++−−−−−−−−++++−−−−−−−−====
2
22
2
2
2
43
)(23.
))()(()()()(3
))()(())(())(())((
tamrUbmnþtamrUbmnþtamrUbmnþtamrUbmnþehrug ehrug ehrug ehrug ³³³³
prcpbpappS ====−−−−−−−−−−−−==== ))()(( eKTaj eKTaj eKTaj eKTaj
pcpbpap
r))()(( −−−−−−−−−−−−====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 110 -
pabcpcabcabpp
r
pabcpcabcabpcbap
r
pcpbpap
r
−−−−++++++++++++−−−−====
−−−−++++++++++++++++++++−−−−====
−−−−−−−−−−−−====
)(2
)()(
))()((
332
232
2
pabc
prcabcab
pabc
cabcabpr
pabcpcabcabp
r
++++++++====++++++++
−−−−++++++++++++−−−−====
−−−−++++++++++++−−−−====
22
22
32 )(
eday eday eday eday R
abcprS
4======== eKTaj eKTaj eKTaj eKTaj rR
pabc
4====
eKTaj eKTaj eKTaj eKTaj Rrprcabcab 422 ++++++++====++++++++ ehtuenH ehtuenH ehtuenH ehtuenH
pRr
prrRprp
T44222 ++++====++++++++++++−−−−====
dUcenH dUcenH dUcenH dUcenH p
rRCBA ++++====++++++++4
2tan
2tan
2tan . . . .
K> K> K> K> prCBA ====
2tan
2tan
2tan
eKman eKman eKman eKman cp
rCbp
rBap
rA−−−−
====−−−−
====−−−−
====2
tan;2
tan;2
tan
eroberogeday lwm pl:ún nig Esn Bisidæ
- 111 -
eK)an eK)an eK)an eK)an ))()((2
tan2
tan2
tan3
cpbpaprCBA
−−−−−−−−−−−−====
pr
prr
Sr
S
rS
cpbpapppr
================
−−−−−−−−−−−−====
22
2
2
3
))()((
dUcenH dUcenH dUcenH dUcenH prCBA ====
2tan
2tan
2tan . . . .
X> X> X> X> rpCBA ====++++++++
2cot
2cot
2cot
eK)an eK)an eK)an eK)an r
cpr
bpr
apCBA −−−−++++−−−−++++−−−−====++++++++2
cot2
cot2
cot
rp
rppr
cbap
====−−−−====
++++++++−−−−====
23
)(3
dUcenH dUcenH dUcenH dUcenH rpCBA ====++++++++
2cot
2cot
2cot . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 112 -
lMhat´TI49lMhat´TI49lMhat´TI49lMhat´TI49 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag S CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 22
coscoscosR
abcCcBbAa ====++++++++
x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa SCcBbAa 4cotcotcot 222 ====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa 22
coscoscosR
abcCcBbAa ====++++++++
tag tag tag tag CcBbAaT coscoscos ++++++++==== tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs R
Cc
Bb
Aa
2sinsinsin
============
eKTaj eKTaj eKTaj eKTaj
============
CRc
BRb
ARa
sin2
sin2
sin2
eK)an eK)an eK)an eK)an )2sin2sin2(sin CBART ++++++++====
[[[[ ]]]][[[[ ]]]]
[[[[ ]]]]CBAR
BABACR
BACBACR
CCBABAR
sinsinsin4
)cos()cos(sin2
)cos(sin2)cos(sin2
cossin2)cos()sin(2
====++++−−−−−−−−====
++++−−−−−−−−====++++−−−−++++====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 113 -
eday eday eday eday Rc
CRb
BRa
A2
sin;2
sin;2
sin ============
eK)an eK)an eK)an eK)an 23 28.4
R
abc
R
abcRT ========
dUcenH dUcenH dUcenH dUcenH 22coscoscos
R
abcCcBbAa ====++++++++ . . . .
x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa SCcBbAa 4cotcotcot 222 ====++++++++
tag tag tag tag CcBbAa cotcotcot 222 ++++++++====∑∑∑∑
SR
abcR
abc
R
abcR
CcBbAaR
CRcBRbARa
CcC
cBb
Bb
AaA
a
44
.42
.2
)coscoscos(2
cos2cos2cos2
cossin
cossin
cossin
2 ================
++++++++====++++++++====
++++++++====
dUcenH dUcenH dUcenH dUcenH SCcBbAa 4cotcotcot 222 ====++++++++ ....
eroberogeday lwm pl:ún nig Esn Bisidæ
- 114 -
lMhat´TI50lMhat´TI50lMhat´TI50lMhat´TI50 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag r CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )(2cotcotcot RrCcBbAa ++++====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa )(2cotcotcot RrCcBbAa ++++====++++++++ tag tag tag tag CcBbAaT cotcotcot ++++++++====
)()coscos(cos2
cos2cos2cos2
cossin
cossin
cossin
iCBAR
CRBRAR
CC
cB
Bb
AA
a
++++++++====++++++++====
++++++++====
eKman eKman eKman eKman 2
cos2
cos2coscosBABA
BA−−−−++++====++++
2
cos2
sin2
2cos)
22cos(2
BAC
BAC
−−−−====
−−−−−−−−==== ππππ
ehIy ehIy ehIy ehIy 2
sin21cos 2 CC −−−−====
)2
cos2
(sin2
sin21coscoscosBACC
CBA−−−−−−−−−−−−====++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 115 -
eday eday eday eday 2
cos2
cos2
cos2
sinBABABAC −−−−−−−−
++++====−−−−−−−−
2
sin2
sin2BA−−−−====
eK)an eK)an eK)an eK)an )(2
sin2
sin2
sin41coscoscos iCBA
CBA ++++====++++++++ eKdwgfa eKdwgfa eKdwgfa eKdwgfa ³³³³
bccpbpA ))((
2sin
−−−−−−−−==== ; ac
cpapB ))((2
sin−−−−−−−−====
nig nig nig nig ab
bpapC ))((2
sin−−−−−−−−====
abccpbpap
CBA))()((
41coscoscos−−−−−−−−−−−−++++====++++++++
tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug ³³³³
Rabc
prcpbpappS4
))()(( ========−−−−−−−−−−−−==== eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³
Srprp
Scpbpap ============−−−−−−−−−−−− 2
2
))()(( nig nig nig nig prRabc 4====
eK)an eK)an eK)an eK)an )(14
.41coscoscos iiRr
prRSr
CBA ++++====++++====++++++++
tam tam tam tam )(i nig nig nig nig )(ii eK)an eK)an eK)an eK)an )(2)1(2 RrRr
RT ++++====++++==== dUcenH dUcenH dUcenH dUcenH )(2cotcotcot RrCcBbAa ++++====++++++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 116 -
lMhat´TI51lMhat´TI51lMhat´TI51lMhat´TI51 bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman
2tan
2tan
CBAbaba −−−−====
++++−−−−
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa
2tan
2tan
CBAbaba −−−−====
++++−−−−
eKman eKman eKman eKman ARa sin2==== nig nig nig nig BRb sin2==== eK)an eK)an eK)an eK)an )sin(sin2 BARba −−−−====−−−−
)(
2sin
2sin4
2cos
2sin4
iCBA
Rba
BABARba
−−−−====−−−−
++++−−−−====−−−−
ehIy ehIy ehIy ehIy )sin(sin2 BARba ++++====++++
)(
2cos
2cos4
2cos
2sin4
iiBAC
Rba
BABARba
−−−−====++++
−−−−++++====++++
eFIviFIEck eFIviFIEck eFIviFIEck eFIviFIEck )(i nig nig nig nig )(ii GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an ³³³³
2tan
2tan
CBAbaba −−−−====
++++−−−− . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 117 -
lMhat´TI52lMhat´TI52lMhat´TI52lMhat´TI52 bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman
)2
sin2
sin2
(sin4 222222 CBA
abc
cab
bca ++++++++≥≥≥≥++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay
bgðajfa bgðajfa bgðajfa bgðajfa )2
sin2
sin2
(sin4 222222 CBA
abc
cab
bca ++++++++≥≥≥≥++++++++
tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs Abccba cos2222 −−−−++++==== eKTaj eKTaj eKTaj eKTaj A
bc
cb
bca
cos22
−−−−++++==== eday eday eday eday 2≥≥≥≥++++bc
cb
eK)an eK)an eK)an eK)an 2
sin4cos22 22 A
Abca ====−−−−≥≥≥≥
dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr 2
sin4 22 B
cab ≥≥≥≥ nig nig nig nig
2sin4 2
2 Cabc ≥≥≥≥
eK)an eK)an eK)an eK)an 2
sin42
sin42
sin4 222222 CBA
abc
cab
bca ++++++++≥≥≥≥++++++++
dUcenH dUcenH dUcenH dUcenH )2
sin2
sin2
(sin4 222222 CBA
abc
cab
bca ++++++++≥≥≥≥++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 118 -
lMhat´TI53lMhat´TI53lMhat´TI53lMhat´TI53 bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman
3333 16
81coscoscos
pc
C
b
B
a
A ≥≥≥≥++++++++
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa 3333 16
81coscoscos
pc
C
b
B
a
A ≥≥≥≥++++++++
tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs Abccba cos2222 −−−−++++==== eKTaj eKTaj eKTaj eKTaj )1(1
cos22
2
2
2
2 −−−−++++====a
c
a
b
a
Abc
dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr )2(1cos2
2
2
2
2
2 −−−−++++====b
a
b
c
b
Bca
nig nig nig nig )3(1cos2
2
2
2
2
2 −−−−++++====c
b
c
a
c
Aab bUksmIkar bUksmIkar bUksmIkar bUksmIkar )2(;)1( nig nig nig nig )3( eK)an eK)an eK)an eK)an ³³³³
33222
3
cos2cos2cos2
2
2
2
2
2
2
2
2
2
2
2
2
222
====−−−−++++++++≥≥≥≥
−−−−++++++++++++++++++++====
++++++++====
b
c
c
b
c
a
a
c
b
a
a
b
c
Cab
b
Bca
a
AbcT
eKTaj eKTaj eKTaj eKTaj 23coscoscos
222 ≥≥≥≥++++++++c
Cab
b
Bca
a
Abc
eroberogeday lwm pl:ún nig Esn Bisidæ
- 119 -
b¤ b¤ b¤ b¤ )(2
3coscoscos333 i
abcc
C
b
B
a
A ≥≥≥≥++++++++ eKman eKman eKman eKman 332 abccbap ≥≥≥≥++++++++==== eKTaj eKTaj eKTaj eKTaj
278 3p
abc ≤≤≤≤ naM[ naM[ naM[ naM[ )(16
812
33 ii
pabc≥≥≥≥
tam tam tam tam )(i nig nig nig nig )(ii eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³
3333 16
81coscoscos
pc
C
b
B
a
A ≥≥≥≥++++++++ . . . .
lMhat´TI54lMhat´TI54lMhat´TI54lMhat´TI54 bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyEdl mYyEdl mYyEdl mYyEdl CBA ≠≠≠≠≠≠≠≠ eKmaneKmaneKmaneKman k> k> k> k> 0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa x> x> x> x> abc
BAACCBBAcACbCBa
3)sin()sin()sin(
)(sin)(sin)(sin 333333
====−−−−−−−−−−−−
−−−−++++−−−−++++−−−−
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa 0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa
A
H B C
eroberogeday lwm pl:ún nig Esn Bisidæ
- 120 -
kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg ABH eKman eKman eKman eKman c
BHABBH
B ========cos eKTaj eKTaj eKTaj eKTaj BcBH cos==== kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg ACH eKman eKman eKman eKman
bHC
ACHC
C ========cos eKTaj eKTaj eKTaj eKTaj CbHC cos==== eday eday eday eday HCBHBCa ++++======== naM[ naM[ naM[ naM[ BcCba coscos ++++==== dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr AcCab coscos ++++==== nig nig nig nig AbBac coscos ++++==== tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs
Bb
Aa
sinsin====
eK)an eK)an eK)an eK)an B
AcCaA
BcCbsin
coscossin
coscos ++++====++++
b¤ b¤ b¤ b¤
CBAcCAaBb
BABAccAaBb
BAc
CAaBb
AAcACaBBcBCb
cos)sin(cos)sinsin(
)cos()sin(cos)sinsin(
)2sin2(sin2
cos)sinsin(
sincossincoscossinsincos
−−−−−−−−====−−−−++++−−−−====−−−−
−−−−====−−−−
++++====++++
eKTaj eKTaj eKTaj eKTaj )(sinsin)sin( iBbAaBAc −−−−====−−−− dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr )(sinsin)sin( iiCcBbCBa −−−−====−−−− )(sinsin)sin( iiiAaCcACb −−−−====−−−− bUkTMnak;TMng bUkTMnak;TMng bUkTMnak;TMng bUkTMnak;TMng )(&)(;)( iiiiii eK)an eK)an eK)an eK)an ³³³³
0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 121 -
x> x> x> x> abcBAACCB
BAcACbCBa3
)sin()sin()sin()(sin)(sin)(sin 333333
====−−−−−−−−−−−−
−−−−++++−−−−++++−−−−
tag tag tag tag )sin(;)sin(;)sin( BAczACbyCBax −−−−====−−−−====−−−−==== eK)an eK)an eK)an eK)an 0====++++++++ zyx b¤ b¤ b¤ b¤ zyx −−−−====++++ eK)an eK)an eK)an eK)an 33)( zyx −−−−====++++
xyzzyx
zyxyzx
zyyxxyx
zyxyyxx
3
3
)(3
33
333
333
333
33223
====++++++++
−−−−====++++−−−−
−−−−====++++++++++++
−−−−====++++++++++++
CMnYs CMnYs CMnYs CMnYs )sin(;)sin(;)sin( BAczACbyCBax −−−−====−−−−====−−−−==== rYcEckGgÁTaMgBIrnwg rYcEckGgÁTaMgBIrnwg rYcEckGgÁTaMgBIrnwg rYcEckGgÁTaMgBIrnwg )sin()sin()sin( BAACCB −−−−−−−−−−−−
abcBAACCB
BAcACbCBa3
)sin()sin()sin()(sin)(sin)(sin 333333
====−−−−−−−−−−−−
−−−−++++−−−−++++−−−−
������������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 122 -
lMhlMhlMhlMhat´TI55at´TI55at´TI55at´TI55 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³
4sin.2
ππππnU
nn ==== Edl Edl Edl Edl *INn ∈∈∈∈
k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 4
sin4
cos4
)1(cos.2
ππππππππππππ nnn−−−−====
++++ x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa
4)1(
cos)2(4
cos)2( 1 ππππππππ ++++−−−−==== ++++ nnU nn
n K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk nn UUUUS ++++++++++++++++==== .........321 CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>k>k>k>bgðajfa bgðajfa bgðajfa bgðajfa
4sin
4cos
4)1(
cos.2ππππππππππππ nnn
−−−−====++++
tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ bababa sinsincoscos)cos( −−−−====++++ eyIg)an eyIg)an eyIg)an eyIg)an ³³³³
)4
sin4
sin4
cos4
(cos2
)44
cos(24
)1(cos2
ππππππππππππππππ
ππππππππππππ
ni
n
nn
−−−−====
++++====++++
4
sin4
cos
)4
sin22
4cos
22
(2
ππππππππ
ππππππππ
nn
nn
−−−−====
−−−−====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 123 -
dUcenH dUcenH dUcenH dUcenH 4
sin4
cos4
)1(cos.2
ππππππππππππ nnn−−−−====
++++ . . . .
x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa 4
)1(cos)2(
4cos)2( 1 ππππππππ ++++−−−−==== ++++ nn
U nnn
eeeeyIgman yIgman yIgman yIgman 4
sin4
cos4
)1(cos.2
ππππππππππππ nnn−−−−====
++++
naM[ naM[ naM[ naM[ 4
)1(cos2
4cos
4sin
ππππππππππππ ++++−−−−====nnn
KuNGgÁTaMgBIrnwg KuNGgÁTaMgBIrnwg KuNGgÁTaMgBIrnwg KuNGgÁTaMgBIrnwg n)2( eK)an eK)an eK)an eK)an
4)1(
cos)2(4
cos)2(4
sin)2( 1 ππππππππππππ ++++−−−−==== ++++ nnn nnn
dUcenH dUcenH dUcenH dUcenH 4
)1(cos)2(
4cos)2( 1 ππππππππ ++++−−−−==== ++++ nn
U nnn . . . .
K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk nn UUUUS ++++++++++++++++==== .........321 eyIg)an eyIg)an eyIg)an eyIg)an ∑∑∑∑
========++++++++++++++++====
n
kknn UUUUUS
1321 )(...........
4
)1(cos)2(
4cos2
4)1(
cos)2(4
cos)2(
1
1
1
ππππππππ
ππππππππ
++++−−−−====
++++−−−−====
++++
====
++++∑∑∑∑
n
kk
n
n
k
kk
dUcenH dUcenH dUcenH dUcenH 4
)1(cos)2(1 1 ππππ++++
−−−−==== ++++ nS n
n . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 124 -
lMhat´TI56lMhat´TI56lMhat´TI56lMhat´TI56 eK[ eK[ eK[ eK[ x CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl 0217160 2 <<<<++++−−−− xx . . . . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 0
13sin <<<<
−−−−xππππ . . . .
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa 0
13sin <<<<
−−−−xππππ
tag tag tag tag 217160)( 2 ++++−−−−==== xxxf eeeebI bI bI bI 02171600)( 2 ====++++−−−−⇔⇔⇔⇔==== xxxf
150405041)21)(60(4)71( 2 ====−−−−====−−−−−−−−====∆∆∆∆ eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s
53
1203971
,127
120171
21 ====++++========
−−−−==== xx eyIg)an eyIg)an eyIg)an eyIg)an 0217160)( 2 <<<<++++−−−−==== xxxf naM[ naM[ naM[ naM[
53
127 <<<<<<<< x b¤ b¤ b¤ b¤
59
347 <<<<<<<< x
54
1343 <<<<−−−−<<<< x naM[ naM[ naM[ naM[
34
131
54 <<<<
−−−−<<<<
x
eKTaj eKTaj eKTaj eKTaj 3
4135
4 ππππππππππππ <<<<−−−−
<<<<x
naM[ naM[ naM[ naM[ 013
sin <<<<
−−−−xππππ . . . .
dUcenH ebI dUcenH ebI dUcenH ebI dUcenH ebI x CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl 0217160 2 <<<<++++−−−− xx eK)an eK)an eK)an eK)an 0
13sin <<<<
−−−−xππππ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 125 -
lMhat´TI57lMhat´TI57lMhat´TI57lMhat´TI57 eK[ eK[ eK[ eK[
20
ππππ<<<<<<<< x . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa ³³³³
21cos
11
sin1
1 ++++≥≥≥≥
++++
++++xx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay RsaybBa¢ak;RsaybBa¢ak;RsaybBa¢ak;RsaybBa¢ak;
21cos
11
sin1
1 ++++≥≥≥≥
++++
++++xx
eyIgman eyIgman eyIgman eyIgman ³³³³
xxxxxx cossin1
cos1
sin1
1cos
11
sin1
1 ++++++++++++====
++++
++++
eKman eKman eKman eKman xxxx cossin
12
cos1
sin1 ≥≥≥≥++++
eKTaj eKTaj eKTaj eKTaj xxxxxx cossin
1cossin1
21cos
11
sin1
1 ++++++++≥≥≥≥
++++
++++
eday eday eday eday 21
2sin21
cossin ≤≤≤≤==== xxx enaH enaH enaH enaH 2cossin1 ≥≥≥≥
xx
eK)an eK)an eK)an eK)an 2)21(2221cos
11
sin1
1 ++++====++++++++≥≥≥≥
++++
++++xx
dUcenH dUcenH dUcenH dUcenH 21cos
11
sin1
1 ++++≥≥≥≥
++++
++++xx
. . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 126 -
lMhat´TI58lMhat´TI58lMhat´TI58lMhat´TI58 eK[eK[eK[eK[RtIekaN RtIekaN RtIekaN RtIekaN ABC . . . . 1111----cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; [,0], ππππ∈∈∈∈yx cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³
)2
(sin2sinsinyx
yx++++≤≤≤≤++++ . . . .
2222----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa 2
33sinsinsin ≤≤≤≤++++++++ CBA . . . .
3333----Tajbgðajfa Tajbgðajfa Tajbgðajfa Tajbgðajfa ³³³³ k/ k/ k/ k/
833
sin.sin.sin ≤≤≤≤CBA
x/ x/ x/ x/ 8
332
cos2
cos2
cos ≤≤≤≤CBA
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay 1111----Rsayfa Rsayfa Rsayfa Rsayfa )
2(sin2sinsin
yxyx
++++≤≤≤≤++++
eyIgman eyIgman eyIgman eyIgman
−−−−
++++====++++2
cos2
sin2sinsinyxyx
yx
eday eday eday eday [,0], ππππ∈∈∈∈yx eKTaj eKTaj eKTaj eKTaj
222,
20
ππππππππππππ <<<<−−−−<<<<−−−−<<<<
++++<<<<yxyx
ehIy ehIy ehIy ehIy 02
sin >>>>
++++ yx nig nig nig nig 12
cos ≤≤≤≤
−−−− yx
eroberogeday lwm pl:ún nig Esn Bisidæ
- 127 -
eKTaj eKTaj eKTaj eKTaj
++++≤≤≤≤
−−−−
++++====++++2
sin22
cos2
sin2sinsinyxyxyx
yx
dUcenH dUcenH dUcenH dUcenH )2
(sin2sinsinyx
yx++++≤≤≤≤++++ RKb; RKb; RKb; RKb; [,0], ππππ∈∈∈∈yx . . . .
2222----RRRRsayfa sayfa sayfa sayfa 2
33sinsinsin ≤≤≤≤++++++++ CBA
tamsRmayxagelIeKman tamsRmayxagelIeKman tamsRmayxagelIeKman tamsRmayxagelIeKman )2
(sin2sinsinyx
yx++++≤≤≤≤++++
eday eday eday eday CBA ,, CamMukñúgénRtIekaN CamMukñúgénRtIekaN CamMukñúgénRtIekaN CamMukñúgénRtIekaN ABC enaHeKman enaHeKman enaHeKman enaHeKman [,0],, ππππ∈∈∈∈CBA .... eyIg)an eyIg)an eyIg)an eyIg)an )(
2sin2sinsin a
BABA
++++≤≤≤≤++++
)(6
4sin2
)2
3sin(23
sinsin
bCBA
CBACCBA
C
++++++++<<<<
++++++++++++≤≤≤≤
++++++++++++
bUkvismIkar bUkvismIkar bUkvismIkar bUkvismIkar )(a nig nig nig nig )(b GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an ³³³³ )()
64
sin()2
sin(23
sinsinsinsin cCBABACBA
CBA
++++++++++++++++≤≤≤≤
++++++++++++++++++++
edayeKman edayeKman edayeKman edayeKman )2
64
2sin(2)6
4sin()
2sin(
CBABACBABA
++++++++++++++++
≤≤≤≤++++++++++++
++++ )()
3sin(2)
64
sin()2
sin( dCBACBABA ++++++++≤≤≤≤
++++++++++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 128 -
tamTMnak;TMng tamTMnak;TMng tamTMnak;TMng tamTMnak;TMng )(c nig nig nig nig )(d eKTaj)an eKTaj)an eKTaj)an eKTaj)an
233
3sin3sinsinsin
3sin4
3sinsinsinsin
====
++++++++≤≤≤≤++++++++
++++++++≤≤≤≤
++++++++++++++++++++
CBACBA
CBACBACBA
dUcenH dUcenH dUcenH dUcenH 2
33sinsinsin ≤≤≤≤++++++++ CBA . . . .
3333----TajbgðajTajbgðajTajbgðajTajbgðaj k/ k/ k/ k/
833
sin.sin.sin ≤≤≤≤CBA
tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman 2
33sinsinsin ≤≤≤≤++++++++ CBA
tamvismPaBkUsIu tamvismPaBkUsIu tamvismPaBkUsIu tamvismPaBkUsIu 3 sinsinsin3sinsinsin CBACBA ≥≥≥≥++++++++ eKTaj eKTaj eKTaj eKTaj
233
sinsinsin33 ≤≤≤≤CBA
naM[ naM[ naM[ naM[ 8
33sinsinsin ≤≤≤≤CBA
dUcenH dUcenH dUcenH dUcenH 8
33sin.sin.sin ≤≤≤≤CBA . . . .
x/ x/ x/ x/ 8
332
cos2
cos2
cos ≤≤≤≤CBA
eyIgman eyIgman eyIgman eyIgman 2
cos2
sin22
cos2
sin2sinsinsinCCBABA
CBA ++++−−−−++++====++++++++
eday eday eday eday ππππ====++++++++ CBA b¤ b¤ b¤ b¤ 222CBA −−−−====
++++ ππππ
eroberogeday lwm pl:ún nig Esn Bisidæ
- 129 -
eKman eKman eKman eKman 2
sin)22
cos(2
cos,2
cos)22
sin(2
sinCCBACCBA ====−−−−====
++++====−−−−====++++ ππππππππ
eKeKeKeK)an)an)an)an2
cos2
cos22
cos2
cos2sinsinsinCBABAC
CBA++++++++
−−−−====++++++++ )
2cos
2(cos
2cos2sinsinsin
BABACCBA
++++++++−−−−====++++++++
2
cos2
cos2
cos4sinsinsinCBA
CBA ====++++++++
tamsRtamsRtamsRtamsRmayxagelIeyIgman mayxagelIeyIgman mayxagelIeyIgman mayxagelIeyIgman 2
33sinsinsin ≤≤≤≤++++++++ CBA
eyIgTaj eyIgTaj eyIgTaj eyIgTaj 2
332
cos2
cos2
cos4 ≤≤≤≤CBA
dUcendUcendUcendUcenH H H H 8
332
cos2
cos2
cos ≤≤≤≤CBA . . . .
��������������������
eroberogeday lwm pl:ún nig Esn Bisidæ
- 130 -
lMhat´TI59lMhat´TI59lMhat´TI59lMhat´TI59 eK[sIVúténcMnYnkMupøic eK[sIVúténcMnYnkMupøic eK[sIVúténcMnYnkMupøic eK[sIVúténcMnYnkMupøic )( nZ kMnteday kMnteday kMnteday kMnteday ³³³³
(((( ))))
∈∈∈∈++++====
++++====
++++ INnZZZ
iZ
nnn ;||212
31
1
0
( ( ( ( || nZ Cam Cam Cam Cam¨UDulén ¨UDulén ¨UDulén ¨UDulén nZ ) . ) . ) . ) . snµtfa snµtfa snµtfa snµtfa INniZ nnnn ∈∈∈∈∀∀∀∀++++==== ,)sin.(cos θθθθθθθθρρρρ Edl Edl Edl Edl IRnnn ∈∈∈∈>>>> θθθθρρρρρρρρ ;,0 . . . . kkkk----rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag nθθθθ nignignignig 1++++nθθθθ ehIy ehIy ehIy ehIy nρρρρ nignignignig 1++++nρρρρ . . . . xxxx----rkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVút )( nθθθθ rYcKNnarYcKNnarYcKNnarYcKNna nθθθθ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . KKKK----cUrbgHajfacUrbgHajfacUrbgHajfacUrbgHajfa
2cos....
2cos
2coscos 121
00−−−−==== n
nθθθθθθθθθθθθθθθθρρρρρρρρ
rYbBa¢ak rYbBa¢ak rYbBa¢ak rYbBa¢ak nρρρρ GnuKmn_én GnuKmn_én GnuKmn_én GnuKmn_én n .... dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----TMnak´TMngrvag TMnak´TMngrvag TMnak´TMngrvag TMnak´TMngrvag nθθθθ nig nig nig nig 1++++nθθθθ ehIy ehIy ehIy ehIy nρρρρ nig nig nig nig 1++++nρρρρ eyIgman eyIgman eyIgman eyIgman )sin.(cos nnnn iZ θθθθθθθθρρρρ ++++==== naM[ naM[ naM[ naM[ )sin(cos 1111 ++++++++++++++++ ++++==== nnnn iZ θθθθθθθθρρρρ
eroberogeday lwm pl:ún nig Esn Bisidæ
- 131 -
eday eday eday eday )||(21
1 nnn ZZZ ++++====++++ ehIy ehIy ehIy ehIy nnZ ρρρρ====|| eKán eKán eKán eKán ³³³³
[[[[ ]]]]nnnnnnn ii ρρρρθθθθθθθθρρρρθθθθθθθθρρρρ ++++++++====++++ ++++++++++++ )sin.(cos21
)sin(cos 111
)2
sin.2
(cos2
cos)sin.(cos
)sin.cos1(21
)sin.(cos
111
111
nnnnnnn
nnnnnn
ii
ii
θθθθθθθθθθθθρρρρθθθθθθθθρρρρ
θθθθθθθθρρρρθθθθθθθθρρρρ
++++====++++
++++++++====++++
++++++++++++
++++++++++++
eKTaján eKTaján eKTaján eKTaján 2
cos1n
nnθθθθρρρρρρρρ ====++++ nig nig nig nig
21n
nθθθθθθθθ ====++++
dUcen¼ dUcen¼ dUcen¼ dUcen¼ 21n
nθθθθθθθθ ====++++ nignignignig
2cos1
nnn
θθθθρρρρρρρρ ====++++ . . . . xxxx----RbePTénsIVút RbePTénsIVút RbePTénsIVút RbePTénsIVút )( nθθθθ nigKNna nigKNna nigKNna nigKNna nθθθθ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ½ ½ ½ ½ tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman nn θθθθθθθθ
21
1 ====++++ naM[ naM[ naM[ naM[ (((( ))))nθθθθ CasIVútFrNImaRtmanersugesµI CasIVútFrNImaRtmanersugesµI CasIVútFrNImaRtmanersugesµI CasIVútFrNImaRtmanersugesµI
21====q . . . .
tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ nn q××××==== 0θθθθθθθθ
eday eday eday eday 3
sin.3
cos2
31)sin(cos 0000
ππππππππθθθθθθθθρρρρ ii
iZ ++++====++++====++++====
eKTaján eKTaján eKTaján eKTaján 3
;1 00ππππθθθθρρρρ ========
dUcen¼ dUcen¼ dUcen¼ dUcen¼ nn
2
1.
3ππππθθθθ ==== . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 132 -
KKKK----bgHajfa bgHajfa bgHajfa bgHajfa 2
cos....2
cos2
coscos 2100
nn
θθθθθθθθθθθθθθθθρρρρρρρρ ==== tamtamtamtamsRmayxagelIeKman sRmayxagelIeKman sRmayxagelIeKman sRmayxagelIeKman ³³³³
2cos1
nnn
θθθθρρρρρρρρ ====++++ ¦ ¦ ¦ ¦ 2
cos1 n
n
n θθθθρρρρ
ρρρρ====++++
eKán eKán eKán eKán ∏∏∏∏ ∏∏∏∏−−−−====
====
−−−−====
====
++++
====
1
0
1
0
1 )2
cos(nk
k
nk
k
k
k
k θθθθρρρρ
ρρρρ
2
cos.........2
cos.2
cos.cos 1210
0
−−−−==== nn θθθθθθθθθθθθθθθθρρρρρρρρ
dUcen¼ dUcen¼ dUcen¼ dUcen¼ 2
cos....2
cos2
coscos 12100
−−−−==== nn
θθθθθθθθθθθθθθθθρρρρρρρρ . . . .
müa¨geToteyIgman müa¨geToteyIgman müa¨geToteyIgman müa¨geToteyIgman ³³³³
2cossin2
2cos
2sin2sin 1
nn
nnn
θθθθθθθθθθθθθθθθθθθθ ++++======== ( eRBa¼( eRBa¼( eRBa¼( eRBa¼
21n
nθθθθθθθθ ====++++ ))))
eKTaj eKTaj eKTaj eKTaj 1sin
sin.
21
2cos
++++====
n
nn
θθθθθθθθθθθθ
ehtuen¼ ehtuen¼ ehtuen¼ ehtuen¼ n
nn
nnn θθθθ
θθθθθθθθ
θθθθθθθθθθθθ
θθθθθθθθρρρρ
sinsin
.2
1sin
sin.....
sinsin
.sinsin
.2
1 01
2
1
1
0 ======== −−−−
dUcen¼ dUcen¼ dUcen¼ dUcen¼ )
2
1.
3sin(
1.
2
3
)2
1.
3sin
3sin
2
11
n
n
n
nn ππππππππ
ππππ
ρρρρ ++++====((((
==== . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 133 -
lMhat´TI60lMhat´TI60lMhat´TI60lMhat´TI60 eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit )( nU kMntelI kMntelI kMntelI kMntelI n eday½eday½eday½eday½
10 ====U nig nig nig nig aaUUINn nn sincos: 1 ++++====∈∈∈∈∀∀∀∀ ++++ Edl Edl Edl Edl
20
ππππ<<<<<<<< a .... k¿ tag k¿ tag k¿ tag k¿ tag
2cot
aUV nn −−−−==== . . . .
cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )( nV CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy . x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt )....(lim 10 n
nVVV ++++++++++++
+∞+∞+∞+∞→→→→ nig nig nig nig n
nU
+∞+∞+∞+∞→→→→lim
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k¿ bgHajfa k¿ bgHajfa k¿ bgHajfa k¿ bgHajfa )( nV CasIVútFrNImaRtmYYy ½CasIVútFrNImaRtmYYy ½CasIVútFrNImaRtmYYy ½CasIVútFrNImaRtmYYy ½ man man man man
2cot
aUV nn −−−−==== naM[ naM[ naM[ naM[
2cot11
aUV nn −−−−==== ++++++++
Et Et Et Et aaUU nn sincos1 ++++====++++ eKán eKán eKán eKán
2cotsincos1
aaaUV nn −−−−++++====++++
)1
2tan
2cos
2sin2(
2cotcos
2cot
2cos
2sin2cos
−−−−++++====
−−−−++++====
aaaaaU
aaaaU
n
n
eroberogeday lwm pl:ún nig Esn Bisidæ
- 134 -
aVV
aa
Uaa
aUV
aaaUV
a
aaaa
aUV
nn
nnn
nn
nn
cos
cos)2
cot(cos2
cotcos
)12
sin2(2
cotcos
)1
2cos
2sin
2cos
2sin2(
2cotcos
1
1
21
1
====
−−−−====−−−−====
−−−−++++====
−−−−++++====
++++
++++
++++
++++
eday eday eday eday aVV nn cos1 ====++++ naM[ naM[ naM[ naM[ )( nV CasIVútFrNImaRtCasIVútFrNImaRtCasIVútFrNImaRtCasIVútFrNImaRtmanersug manersug manersug manersug acos nig tY nig tY nig tY nig tY
2cot1
2cot00
aaUV −−−−====−−−−==== . . . .
x¿x¿x¿x¿KNnalImItKNnalImItKNnalImItKNnalImIt )....(lim 10 nn
VVV +++++++++++++∞+∞+∞+∞→→→→
nig nig nig nig nn
U+∞+∞+∞+∞→→→→
lim eyIgmaneyIgmaneyIgmaneyIgman³³³³
aaa
VVVVnn
n cos1cos1
).2
cot1(1
1...
11
010 −−−−−−−−−−−−====
−−−−−−−−====++++++++++++
++++++++ eyIgán eyIgán eyIgán eyIgán
−−−−−−−−−−−−====++++++++++++
++++
+∞+∞+∞+∞→→→→+∞+∞+∞+∞→→→→ aaa
VVVn
nn
n cos1cos1
)2
cot1(lim)....(lim1
10
eday eday eday eday 0coslim 1 ====+++++∞+∞+∞+∞→→→→
an
n
dUcen¼ dUcen¼ dUcen¼ dUcen¼ a
a
VVV nn cos1
2cot1
)....(lim 10 −−−−
−−−−====++++++++++++
+∞+∞+∞+∞→→→→ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 135 -
müa¨geTot müa¨geTot müa¨geTot müa¨geTot 2
cota
UV nn −−−−==== naM[ naM[ naM[ naM[ 2
cota
VU nn ++++==== eday eday eday eday a
aqVV nn
n cos)2
cot1(0 −−−−====××××==== eKán eKán eKán eKán
2cotcos)
2cot1(
aa
aU n
n ++++−−−−==== nig nig nig nig
2cot
2cotcos)
2cot1(limlim
aaa
aU n
nn
n====
++++−−−−====+∞+∞+∞+∞→→→→+∞+∞+∞+∞→→→→
dUcen¼ dUcen¼ dUcen¼ dUcen¼ 2
cotlima
U nn
====+∞+∞+∞+∞→→→→
. . . . lMhat´TI61lMhat´TI61lMhat´TI61lMhat´TI61 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnteday ½kMnteday ½kMnteday ½kMnteday ½
1;0 10 ======== UU nig nig nig nig nnn UaUUINn −−−−====∈∈∈∈∀∀∀∀ ++++++++ cos2: 12 Edl Edl Edl Edl IRa ∈∈∈∈ .... k¿ tag k¿ tag k¿ tag k¿ tag INnUaiaUZ nnn ∈∈∈∈∀∀∀∀−−−−−−−−==== ++++ ,)sin(cos1 . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa nn ZaiaZ )sin(cos1 ++++====++++ rYcTajrk rYcTajrk rYcTajrk rYcTajrk
nZ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ n nig nig nig nig a . . . . x¿ Tajrk x¿ Tajrk x¿ Tajrk x¿ Tajrk nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n rYcTajrk rYcTajrk rYcTajrk rYcTajrk n
aU
0lim
→→→→ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 136 -
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k¿bgHajfa k¿bgHajfa k¿bgHajfa k¿bgHajfa nn ZaiaZ )sin(cos1 ++++====++++ eyIgman eyIgman eyIgman eyIgman nnn UaiaUZ )sin(cos1 −−−−−−−−==== ++++ eyIgán eyIgán eyIgán eyIgán 121 )sin(cos ++++++++++++ −−−−−−−−==== nnn UaiaUZ
[[[[ ]]]]
n
nn
nn
nn
nnn
Uaia
UaiaUaiaaia
UUaia
UUaia
UaiaUaU
)sin(cos
)sin(cos)sin(cos
)sincos
)(sin(cos
)sin(cos
)sin(coscos2
1
1
1
11
++++====−−−−−−−−++++====++++
−−−−++++====
−−−−++++====−−−−−−−−−−−−====
++++
++++
++++
++++++++
dUcen¼ dUcen¼ dUcen¼ dUcen¼ nn ZaiaZ )sin(cos1 ++++====++++ . . . . KNna KNna KNna KNna nZ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig a ½ ½ ½ ½ eday eday eday eday nn ZaiaZ )sin(cos1 ++++====++++ naM[ naM[ naM[ naM[ )( nZ CasIVútFrNImaRténcMnYnkMupøicEdlmanersugCasIVútFrNImaRténcMnYnkMupøicEdlmanersugCasIVútFrNImaRténcMnYnkMupøicEdlmanersugCasIVútFrNImaRténcMnYnkMupøicEdlmanersug aiaq sincos ++++==== nigtY nigtY nigtY nigtY 1)sin(cos 010 ====−−−−−−−−==== UaiaUZ tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ )sin()cos()sin(cos0 nainaaiaqZZ nn
n ++++====++++====××××==== dUcen¼ dUcen¼ dUcen¼ dUcen¼ )sin(.)cos( nainaZn ++++==== . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 137 -
x¿Tajrk x¿Tajrk x¿Tajrk x¿Tajrk nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ½ ½ ½ ½ eyIgman eyIgman eyIgman eyIgman )1()sin(cos1 nnn UaiaUZ −−−−−−−−==== ++++ nig nig nig nig )2()sin(cos1 nnn UaiaUZ ++++−−−−==== ++++ dksmIkar dksmIkar dksmIkar dksmIkar )1( nig nig nig nig )2( Gg:nwgGg:eKán ½Gg:nwgGg:eKán ½Gg:nwgGg:eKán ½Gg:nwgGg:eKán ½
nnn UaiZZ sin2====−−−− naM[ naM[ naM[ naM[ ai
ZZU nn
n sin2−−−−
==== Edl Edl Edl Edl 0sin ≠≠≠≠a . . . . eday eday eday eday )sin()cos( nainaZn ++++==== nig nig nig nig )sin()cos( nainaZn −−−−==== eKán eKán eKán eKán
ana
ainainanaina
Un sin)sin(
sin2)sin()cos()sin()cos( ====
++++−−−−++++====
dUcen¼ dUcen¼ dUcen¼ dUcen¼ a
naUn sin
)sin(==== . . . .
ehIy ehIy ehIy ehIy na
ana
nan
ana
Uaa
na
====××××========→→→→→→→→→→→→ sin)(
)sin(lim
sin)sin(
limlim000
dUcen¼ dUcen¼ dUcen¼ dUcen¼ nUn
a====
→→→→0lim . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 138 -
lMhat´TI62lMhat´TI62lMhat´TI62lMhat´TI62 eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½
223)12
cos()4
sin(4 ++++====++++++++ ππππππππxx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½
223)12
cos()4
sin(4 ++++====++++++++ ππππππππxx
tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ )sin()sin(cossin2 bababa −−−−++++++++==== smIkarxagelIGacsresrCabnþbnÞabxageRkam ½smIkarxagelIGacsresrCabnþbnÞabxageRkam ½smIkarxagelIGacsresrCabnþbnÞabxageRkam ½smIkarxagelIGacsresrCabnþbnÞabxageRkam ½
22
)3
2sin(
211)3
2sin(2
)21(6
sin)3
2sin(2
223)124
sin()124
sin(2
2
====++++
++++====++++++++
++++====
++++++++
++++====
−−−−−−−−++++++++++++++++++++
ππππ
ππππ
ππππππππ
ππππππππππππππππ
x
x
x
xxxx
eKTaj eKTaj eKTaj eKTaj ππππππππππππkx 2
432 ++++====++++ b¤b¤b¤b¤ ππππππππππππππππ
kx 243
2 ++++−−−−====++++
dUcen¼ dUcen¼ dUcen¼ dUcen¼ Zkkxkx ∈∈∈∈++++====++++−−−−==== ;245
;24
ππππππππππππππππ
eroberogeday lwm pl:ún nig Esn Bisidæ
- 139 -
lMhat´TI63lMhat´TI63lMhat´TI63lMhat´TI63 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;elI kMnt;elI kMnt;elI kMnt;elI IN eday eday eday eday ³³³³
22
0 ====U nig nig nig nig INnU
U nn ∈∈∈∈∀∀∀∀
−−−−−−−−====++++ ,
2
11 2
1 KNna KNna KNna KNna nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNna KNna KNna KNna nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n eyIgman eyIgman eyIgman eyIgman
4sin
22
0ππππ========U Bit Bit Bit Bit
]bmafavaBitdl;tYTI ]bmafavaBitdl;tYTI ]bmafavaBitdl;tYTI ]bmafavaBitdl;tYTI p KW KW KW KW 22
sin ++++====ppUππππ
eyIgnwgRsayfavaBitdl;tYTI eyIgnwgRsayfavaBitdl;tYTI eyIgnwgRsayfavaBitdl;tYTI eyIgnwgRsayfavaBitdl;tYTI )1( ++++p KW KW KW KW 31
2sin ++++++++ ====
ppUππππ Bit Bit Bit Bit
eyIgman eyIgman eyIgman eyIgman 2
11 2
1p
p
UU
−−−−−−−−====++++
tamkar]bma tamkar]bma tamkar]bma tamkar]bma 22
sin ++++====ppUππππ
eyIg)an eyIg)an eyIg)an eyIg)an 2
2sin11
22
1
++++++++
−−−−−−−−====
p
pU
ππππ
3
32
2
2sin
22
sin2
22
cos1
++++
++++++++========
−−−−====
p
pp ππππππππππππ
Bit Bit Bit Bit
dUcenH dUcenH dUcenH dUcenH 22
sin ++++====nnUππππ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 140 -
lMhat´TI64lMhat´TI64lMhat´TI64lMhat´TI64 eK[eK[eK[eK[
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
====++++++++
====++++
====
4
3
2
2cos2222
2cos222
2cos22
ππππ
ππππ
ππππ
BI]TaBI]TaBI]TaBI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg hrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg hrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg hrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay rkrUbmnþTUeTA ½rkrUbmnþTUeTA ½rkrUbmnþTUeTA ½rkrUbmnþTUeTA ½ eKman eKman eKman eKman
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
====++++++++
====++++
====
4
3
2
2cos2222
2cos222
2cos22
ππππ
ππππ
ππππ
tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½
1)(
2cos22.........222 ++++====++++++++++++++++
nn
ππππ44444 344444 21
. . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 141 -
RsaybBaØak´rUbmnþen¼ ½RsaybBaØak´rUbmnþen¼ ½RsaybBaØak´rUbmnþen¼ ½RsaybBaØak´rUbmnþen¼ ½ eyIgtag eyIgtag eyIgtag eyIgtag
4444 34444 21)(
2......222n
nA ++++++++++++++++====
cMeBa¼RcMeBa¼RcMeBa¼RcMeBa¼RKb Kb Kb Kb *INn ∈∈∈∈ . . . . eyIgman eyIgman eyIgman eyIgman
212
cos22ππππ========A Bit Bit Bit Bit
eyIg«bmafavaBitdltYTI eyIg«bmafavaBitdltYTI eyIg«bmafavaBitdltYTI eyIg«bmafavaBitdltYTI p KW KW KW KW ³³³³
1)(
2cos22......222 ++++====++++++++++++++++====
pp
pAππππ
4444 34444 21 Bit Bit Bit Bit
eyIgnwgRsayfavaBitdltYTI eyIgnwgRsayfavaBitdltYTI eyIgnwgRsayfavaBitdltYTI eyIgnwgRsayfavaBitdltYTI 1++++p KW KW KW KW 21
2cos2 ++++++++ ====
ppAππππ Bit Bit Bit Bit
eyIgman eyIgman eyIgman eyIgman pp AA ++++====++++ 21 edaytamkar«bma edaytamkar«bma edaytamkar«bma edaytamkar«bma
12cos2 ++++====
ppAππππ
eyIgáneyIgáneyIgáneyIgán22
211
2cos2
2cos4
2cos22 ++++++++++++++++ ========++++====
ppppAππππππππππππ Bit Bit Bit Bit
dUcen¼ dUcen¼ dUcen¼ dUcen¼ 1
)(2
cos22.........222 ++++====++++++++++++++++n
n
ππππ44444 344444 21
. . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 142 -
lMhat´TI65lMhat´TI65lMhat´TI65lMhat´TI65 ykykykyk a; b ;c CaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nig A,B,C CargVas;mMuTaMgbIénRtIekaN CargVas;mMuTaMgbIénRtIekaN CargVas;mMuTaMgbIénRtIekaN CargVas;mMuTaMgbIénRtIekaN ABCmYyEdl mYyEdl mYyEdl mYyEdl S CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH.
Rsayfa Rsayfa Rsayfa Rsayfa S
cbaCotCCotBCotA
4
222 ++++++++====++++++++ dMdMdMdMeNa¼eNa¼eNa¼eNa¼RsRsRsRsayayayay tamRTWsþIbTsIunus tamRTWsþIbTsIunus tamRTWsþIbTsIunus tamRTWsþIbTsIunus
(((( ))))(((( ))))(((( ))))3cot4
2cot4
1cot4
sincos
)sin21
(4
cos2
222
222
222
22
222
gCSabc
gBScab
gAScba
AA
Abccb
Abcba
−−−−++++====
−−−−++++====
−−−−++++====
−−−−++++====
−−−−++++====
bUk bUk bUk bUk (1),(2),(3) eyIg)an eyIg)an eyIg)an eyIg)an
Scba
gCgBgA
gCgBgAScbacba
4cotcotcot
)cotcot(cot4)(2222
222222
++++++++====++++++++
++++++++−−−−++++++++====++++++++
dUcenH dUcenH dUcenH dUcenH S
cbagCggA
4cotBcotcot
222 ++++++++====++++++++
eroberogeday lwm pl:ún nig Esn Bisidæ
- 143 -
lMhat´TI66lMhat´TI66lMhat´TI66lMhat´TI66 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cABbACaBC ============ ,, cUrRsaybBa¢k;facUrRsaybBa¢k;facUrRsaybBa¢k;facUrRsaybBa¢k;fa
abccba
cC
bB
aA
2coscoscos 222 ++++++++====++++++++
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman eKman eKman eKman ³³³³
Abccba cos2222 −−−−++++==== smmUl smmUl smmUl smmUl bc
acbA
2cos
222 −−−−++++====
Baccab cos2222 −−−−++++==== smmUl smmUl smmUl smmUl ac
bcaB
2cos
222 −−−−++++====
Cabbac cos2222 −−−−++++==== smmUl smmUl smmUl smmUl ab
cbaC
2cos
222 −−−−++++==== tag tag tag tag
cC
bB
aA
Mcoscoscos ++++++++====
abccba
abccbabcaacb
abccba
abcbca
abcacb
2
2
222
222
222222222
222222222
++++++++====
−−−−++++++++−−−−++++++++−−−−++++====
−−−−++++++++−−−−++++++++
−−−−++++====
dUcenH dUcenH dUcenH dUcenH abc
cbac
Cb
Ba
A2
coscoscos 222 ++++++++====++++++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 144 -
lMhat´TI67lMhat´TI67lMhat´TI67lMhat´TI67 eeeedaHRsaysmIkardaHRsaysmIkardaHRsaysmIkardaHRsaysmIkar k> k> k> k> (((( )))) 0)sin2(logsinlog2 3
22
2 ====++++ xx
x> x> x> x> 02coslog3coslog 2
2
22 ====++++++++
xx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkar k> k> k> k> (((( )))) 0)sin2(logsinlog2 3
22
2 ====++++ xx lkç½xNн lkç½xNн lkç½xNн lkç½xNн 0sin >>>>x smIkarGacsresmIkarGacsresmIkarGacsresmIkarGacsresr sr sr sr ³³³³
(((( ))))(((( )))) 01sinlog3sinlog2
02log)(sinlogsinlog2
22
2
23
22
2
====++++++++
====++++++++
xx
xx
tag tag tag tag xX sinlog2==== 0132 2 ====++++++++ XX manb¤s manb¤s manb¤s manb¤s
21
;1 21 −−−−====−−−−==== XX
----cMeBaH cMeBaH cMeBaH cMeBaH 1−−−−====X eK)an eK)an eK)an eK)an 1sinlog2 −−−−====x b¤ b¤ b¤ b¤ 21
sin ====x
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ππππππππkx 2
6++++==== b¤ b¤ b¤ b¤ )(2
65
Zkkx ∈∈∈∈++++==== ππππππππ
eroberogeday lwm pl:ún nig Esn Bisidæ
- 145 -
----cMeBaH cMeBaH cMeBaH cMeBaH 21−−−−====X eK)an eK)an eK)an eK)an
21
sinlog2 −−−−====x b¤ b¤ b¤ b¤ 22
sin ====x
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ππππππππkx 2
4++++==== b¤ b¤ b¤ b¤ )(2
43
Zkkx ∈∈∈∈++++==== ππππππππ
x> x> x> x> 02coslog3coslog 2
2
22 ====++++++++
xx
lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð 0cos >>>>x tag tag tag tag xt coslog
22==== enaH enaH enaH enaH tx −−−−====coslog 2
eK)an eK)an eK)an eK)an 0232 ====++++−−−− tt manb¤s manb¤s manb¤s manb¤s 2;1 21 ======== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 11 ====t eK)an eK)an eK)an eK)an 1coslog
22 ====x b¤ b¤ b¤ b¤
22
cos ====x
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s )(24
Zkkx ∈∈∈∈++++±±±±==== ππππππππ
----cMeBaH cMeBaH cMeBaH cMeBaH 22 ====t eK)an eK)an eK)an eK)an 2coslog22 ====x b¤ b¤ b¤ b¤
21
cos ====x
eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s )(23
Zkkx ∈∈∈∈++++±±±±==== ππππππππ
eroberogeday lwm pl:ún nig Esn Bisidæ
- 146 -
lMhat´TI68lMhat´TI68lMhat´TI68lMhat´TI68 edaHRsayedaHRsayedaHRsayedaHRsayRbB½næRbB½næRbB½næRbB½næsmIkarsmIkarsmIkarsmIkar
(((( )))) (((( ))))(((( ))))
====
====yx
yx
sinlnsinln
2ln2ln
22
sin2sin2
Edl Edl Edl Edl 2
0ππππ<<<<<<<< x nig nig nig nig
20
ππππ<<<<<<<< y . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar
(((( )))) (((( ))))(((( ))))
====
====
)(22
)(sin2sin2
sinlnsinln
2ln2ln
ii
iyx
yx
smIkasmIkasmIkasmIkar r r r )(i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³ (((( )))) (((( ))))
)(2ln43
sinln21
sinln
sinln21
2ln41
sinln2ln
)sinln2ln21
(2ln21
)sinln2(ln2ln
sin2lnsin2ln2ln2ln
ayx
yx
yx
yx
−−−−====−−−−
++++====++++
++++====++++
====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 147 -
smIkar smIkar smIkar smIkar )(ii Gacsresr Gacsresr Gacsresr Gacsresr ³³³³ (((( )))) (((( ))))
)(sinln2sinln
2lnsinln2lnsinln21
2lnsinln2lnsinln
2ln2ln sinlnsinln
byx
yx
yx
yx
====
====
====
====
yksmIkar yksmIkar yksmIkar yksmIkar )(b CYkñúg CYkñúg CYkñúg CYkñúg )(a eK)an eK)an eK)an eK)an ³³³³
====
−−−−====
−−−−====
−−−−====−−−−
22
lnsinln
2ln21
sinln
2ln43
sinln23
2ln43
sinln21
sinln2
y
y
y
yy
eKTaj eKTaj eKTaj eKTaj 22
sin ====y na na na naM[ M[ M[ M[ 4ππππ====y eRBaH eRBaH eRBaH eRBaH
20
ππππ<<<<<<<< y
tam tam tam tam )(b eKTaj eKTaj eKTaj eKTaj
====
22
ln2sinln x
eKTaj eKTaj eKTaj eKTaj 21
sin ====x naM[ naM[ naM[ naM[ 6ππππ====x eRBaH eRBaH eRBaH eRBaH
20
ππππ<<<<<<<< x
dUcenH dUcenH dUcenH dUcenH 6ππππ====x nig nig nig nig
4ππππ====y . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 148 -
lMhat´TI69lMhat´TI69lMhat´TI69lMhat´TI69 edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar
++++====++++
====++++
2222
23
sinsin
sinsin yx
yx
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar
++++====++++
====++++
)(2222
)(23
sinsin
sinsin ii
iyx
yx
tam tam tam tam )(i eKTaj eKTaj eKTaj eKTaj )(sin23
sin iiixy −−−−==== yk yk yk yk )(iii CYskñúg CYskñúg CYskñúg CYskñúg )(ii eK)an eK)an eK)an eK)an ³³³³
222.222
2222sinsin
sin23
sin
++++====++++
++++====++++−−−−
−−−−
xx
xx
tag tag tag tag 02sin >>>>==== xt eK)an eK)an eK)an eK)an 22
122 ++++====++++
tt
b¤ b¤ b¤ b¤ 022)22(2 ====++++++++−−−− tt manb¤s manb¤s manb¤s manb¤s 2;2 21 ======== tt
eroberogeday lwm pl:ún nig Esn Bisidæ
- 149 -
----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 22sin ====x naM[ naM[ naM[ naM[ 21
sin ====x eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ππππππππ
kx 26
++++==== b¤ b¤ b¤ b¤ )(26
5Zkkx ∈∈∈∈++++==== ππππππππ
ehIytam ehIytam ehIytam ehIytam )(iii eK)an eK)an eK)an eK)an 121
23
sin ====−−−−====y
eKTaj eKTaj eKTaj eKTaj Zkky ∈∈∈∈++++==== ;22
ππππππππ ----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 22sin ====x naM[ naM[ naM[ naM[ 1sin ====x eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s )(2
2Zkkx ∈∈∈∈++++==== ππππππππ
ehIytam ehIytam ehIytam ehIytam )(iii eK)an eK)an eK)an eK)an 21
23
sin ====−−−−====y
eKTaj eKTaj eKTaj eKTaj Zkkyky ∈∈∈∈++++====++++==== ;26
5;2
6ππππππππππππππππ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 150 -
lMhat´TI70lMhat´TI70lMhat´TI70lMhat´TI70 eK[ eK[ eK[ eK[
2a0
ππππ<<<<<<<< nig nig nig nig 2
b0ππππ<<<<<<<< . . . .
cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa 1bcosacos
bsinasin
2222
====
++++
lu¼RtaEt lu¼RtaEt lu¼RtaEt lu¼RtaEt ba ==== . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkkarbgHajarbgHajarbgHajarbgHaj
eKman eKman eKman eKman 1bcosacos
bsinasin
2222
====
++++
smmUl smmUl smmUl smmUl 1bcosacos
bsinasin
)bcosb(sin 2
4
2
422 ====
++++++++
0)acosbcosbsin
asinbsinbcos
1acosbcosbsin
asinbsinbcos
acosasin21
1acosbcosbsin
asinbsinbcos
acosasin
222
42
24
2
222
42
24
2
244
====
−−−−
====++++++++−−−−
====++++++++++++
eKTaj eKTaj eKTaj eKTaj acosbcosbsin
asinbsinbcos 22 ====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 151 -
smmUl smmUl smmUl smmUl bcosbsin
acosasin
2
2
2
2
==== smmUl smmUl smmUl smmUl btanatan 22 ==== eday eday eday eday
2a0
ππππ<<<<<<<< nig nig nig nig 2
b0ππππ<<<<<<<<
ena¼eKTena¼eKTena¼eKTena¼eKTaj aj aj aj ba ==== . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 152 -
lMhat´TI71lMhat´TI71lMhat´TI71lMhat´TI71 eK[ eK[ eK[ eK[ ABC CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ
AcosCsinBsin1CsinBsin 22 ++++====++++ . . . . bgHajfa bgHajfa bgHajfa bgHajfa ABC CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgHajfa bgHajfa bgHajfa bgHajfa ABC CaRtIekaNEkg CaRtIekaNEkg CaRtIekaNEkg CaRtIekaNEkg tamRTwsþIbTsIutamRTwsþIbTsIutamRTwsþIbTsIutamRTwsþIbTsIunUseKman nUseKman nUseKman nUseKman R2
Csinc
Bsinb
Asina ============
eKTaj eKTaj eKTaj eKTaj CsinR2c,BsinR2b,AsinR2a ============ eday eday eday eday Acosbc2cba 222 −−−−++++==== ( RTwsþIbTkUsIunUs ) ( RTwsþIbTkUsIunUs ) ( RTwsþIbTkUsIunUs ) ( RTwsþIbTkUsIunUs ) eKTaján ½eKTaján ½eKTaján ½eKTaján ½
)AcosCsinBsin2CsinB(sinR4AsinR4 22222 −−−−++++==== )1(AcosCsinBsin2CsinBsinAsin 222 −−−−++++==== Et Et Et Et )2(AcosCsinBsin1CsinBsin 22 ++++====++++ yksmIkar yksmIkar yksmIkar yksmIkar )2( CYsenAkñúg CYsenAkñúg CYsenAkñúg CYsenAkñúg )1( eKTaj eKTaj eKTaj eKTaj 1Asin2 ==== naM[ naM[ naM[ naM[ 090A ==== . dUcen¼ . dUcen¼ . dUcen¼ . dUcen¼ ABC CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 153 -
lMhat´TI72lMhat´TI72lMhat´TI72lMhat´TI72 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCúg mYymanRCúg mYymanRCúg mYymanRCúg c,b,a . . . . kMnt´RbePTénRtIekaN kMnt´RbePTénRtIekaN kMnt´RbePTénRtIekaN kMnt´RbePTénRtIekaN ABC ebIeK ebIeK ebIeK ebIeKdwgfa ½dwgfa ½dwgfa ½dwgfa ½ )
c1
b1
a1
(21
cCcos
bBcos
aAcos ++++++++====++++++++
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay RbePTénRtIekaN RbePTénRtIekaN RbePTénRtIekaN RbePTénRtIekaN ABC tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman ½ eKman ½ eKman ½ eKman ½
bc2acb
Acos222 −−−−++++==== naM[ naM[ naM[ naM[ )1(
abc2acb
aAcos 222 −−−−++++====
dUcKñaEdreKTaj dUcKñaEdreKTaj dUcKñaEdreKTaj dUcKñaEdreKTaj )2(abc2
bcab
Bcos 222 −−−−++++====
nig nig nig nig )3(abc2
cbac
Ccos 222 −−−−++++==== bUkTMnak´TMng bUkTMnak´TMng bUkTMnak´TMng bUkTMnak´TMng )3(,)2(,)1( Gg:nwgGg:eKán ½ Gg:nwgGg:eKán ½ Gg:nwgGg:eKán ½ Gg:nwgGg:eKán ½
abc2cba
cCcos
bBcos
aAcos 222 ++++++++====++++++++
eday eday eday eday )c1
b1
a1
(21
cCcos
bBcos
aAcos ++++++++====++++++++
eKTaján ½eKTaján ½eKTaján ½eKTaján ½
eroberogeday lwm pl:ún nig Esn Bisidæ
- 154 -
0)ac()cb()ba(
0)aac2c()cbc2b()bab2a(
ac2bc2ab2c2b2a2
acbcabcba
abc2abcabc
abc2cba
)c1
b1
a1
(21
abc2cba
222
222222
222
222
222
222
====−−−−++++−−−−++++−−−−
====++++−−−−++++++++−−−−++++++++−−−−
++++++++====++++++++
++++++++====++++++++
++++++++====++++++++
++++++++====++++++++
eKTajánsmPaB eKTajánsmPaB eKTajánsmPaB eKTajánsmPaB cba ======== . . . . dUcen¼ dUcen¼ dUcen¼ dUcen¼ ABC CaRtIekaNsmg&ß . CaRtIekaNsmg&ß . CaRtIekaNsmg&ß . CaRtIekaNsmg&ß .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 155 -
lMhat´TI73lMhat´TI73lMhat´TI73lMhat´TI73 kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN ABC mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½
rR
4
2C
sin
1
2B
sin
1
2A
sin
1 ≥≥≥≥++++++++
Edl Edl Edl Edl r nig nig nig nig R CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay Rsayfa Rsayfa Rsayfa Rsayfa (((( ))))1
rR
4
2C
sin
1
2B
sin
1
2A
sin
1 ≥≥≥≥++++++++
tag tag tag tag cAB,bAC,aBC ============ tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman ½ eKman ½ eKman ½ eKman ½
Acos.bc2cba 222 −−−−++++==== eday eday eday eday 2A
sin21Acos 2−−−−==== ena¼ ena¼ ena¼ ena¼ )
2A
sin21(bc2cba 2222 −−−−−−−−++++====
eKTaj eKTaj eKTaj eKTaj bc4
)cba)(cba(bc4
)cb(a2A
sin22
2 ++++−−−−−−−−++++====−−−−−−−−====
tag tag tag tag 2
cbap
++++++++==== ( knø¼brimaRténRtIekaN ) ( knø¼brimaRténRtIekaN ) ( knø¼brimaRténRtIekaN ) ( knø¼brimaRténRtIekaN ) eKán eKán eKán eKán )cp(2cba −−−−====−−−−++++ nig nig nig nig )bp(2cba −−−−====++++−−−− eKán eKán eKán eKán
bc)cp)(bp(
2A
sin2 −−−−−−−−====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 156 -
naM[ naM[ naM[ naM[ bc
)cp)(bp(2A
sin−−−−−−−−==== . dUcKñaEdreKTaj ½ . dUcKñaEdreKTaj ½ . dUcKñaEdreKTaj ½ . dUcKñaEdreKTaj ½
ab)bp)(ap(
2C
sin;ac
)cp)(ap(2B
sin−−−−−−−−====−−−−−−−−====
eKán eKán eKán eKán abc
)cp)(bp)(ap(2C
sin2B
sin2A
sin−−−−−−−−−−−−====
eKman eKman eKman eKman R4
abcpr)cp)(bp)(ap(pS ========−−−−−−−−−−−−====
eKTaj eKTaj eKTaj eKTaj S.R4abc ==== nig nig nig nig S.rpS
)cp)(bp)(ap(2
========−−−−−−−−−−−−
eKán eKán eKán eKán R4r
S.R4S.r
2C
sin2B
sin2A
sin ======== . . . . vismPaB vismPaB vismPaB vismPaB (((( ))))1 smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½
(((( ))))22
2C
sin
2B
sin2A
sin
2B
sin
2A
sin2C
sin
2A
sin
2C
sin2B
sin
2C
sin2B
sin2A
sin4
14
2C
sin
1
2B
sin
1
2A
sin
1
≥≥≥≥++++++++
≥≥≥≥++++++++
eday eday eday eday 2A
sina
apbca
)cp)(bp()ap(2C
sin2B
sin 2
2 −−−−====−−−−−−−−−−−−====
eroberogeday lwm pl:ún nig Esn Bisidæ
- 157 -
eKTaj eKTaj eKTaj eKTaj a
ap
2A
sin
2C
sin2B
sin −−−−==== . dUcKñaEdreKTaján ½ . dUcKñaEdreKTaján ½ . dUcKñaEdreKTaján ½ . dUcKñaEdreKTaján ½
bbp
2B
sin
2A
sin2C
sin −−−−==== nig nig nig nig c
cp
2C
sin
2B
sin2A
sin −−−−====
vismPaB vismPaB vismPaB vismPaB (((( ))))2 smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½ 2
ccp
bbp
aap ≥≥≥≥−−−−++++−−−−++++−−−−
tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eKán ½ eKán ½ eKán ½ eKán ½ a)ap(2a)ap(p −−−−≥≥≥≥++++−−−−==== naM[ naM[ naM[ naM[
p)ap(2
aap −−−−≥≥≥≥−−−−
dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr p
)bp(2b
bp −−−−≥≥≥≥−−−− nig nig nig nig
p)cp(2
ccp −−−−≥≥≥≥
−−−− eKán eKán eKán eKán
Bit2c
cpb
bpa
ap
p)cp()bp()ap(
2c
cpb
bpa
ap
≥≥≥≥−−−−++++−−−−++++−−−−
−−−−++++−−−−++++−−−−≥≥≥≥−−−−++++−−−−++++−−−−
dUcen¼dUcen¼dUcen¼dUcen¼ rR
4
2C
sin
1
2B
sin
1
2A
sin
1 ≥≥≥≥++++++++ . . . .
eroberogeday lwm pl:ún nig Esn Bisidæ
- 158 -
lMhat´TI74lMhat´TI74lMhat´TI74lMhat´TI74 k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa
aaa
a
2sin
1
sin2
1
2sin
2cos222 −−−−====
x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk ∑∑∑∑====
====n
kk
k
kn x
x
S0 2
2sin
2cos
.2
1
dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> Rsayfa k> Rsayfa k> Rsayfa k> Rsayfa
aaa
a
2sin
1
sin2
1
2sin
2cos222 −−−−====
eKman eKman eKman eKman 1cos22cos 2 −−−−==== aa nig nig nig nig aaa cossin22sin ==== eK)an eK)an eK)an eK)an
aa
a
a
a22
2
2 cossin4
1cos2
2sin
2cos −−−−====
aa
aaaa
a
2sin
1
sin2
1cossin4
1
cossin4
cos2
22
2222
2
−−−−====
−−−−====
dUcenH dUcenH dUcenH dUcenH aaa
a
2sin
1
sin2
1
2sin
2cos222 −−−−====
x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk ∑∑∑∑====
====n
kk
k
kn x
x
S0 2
2sin
2cos
.2
1
eroberogeday lwm pl:ún nig Esn Bisidæ
- 159 -
eKman eKman eKman eKman aaa
a
2sin
1
sin2
1
2sin
2cos222 −−−−====
yk yk yk yk 12 ++++==== kx
a eK)an eK)an eK)an eK)an ³³³³
kkk
k
xxx
x
2sin
1
2sin2
1
2sin
2cos
21
22−−−−====
++++
KuNGgÁTaMgBIr nwg KuNGgÁTaMgBIr nwg KuNGgÁTaMgBIr nwg KuNGgÁTaMgBIr nwg k2
1 eK)an eK)an eK)an eK)an ³³³³
kk
kk
k
k
k xxx
x
2sin2
1
2sin2
1
2sin
2cos
.2
12
1212
−−−−====
++++++++
xx
xxS
nn
n
kk
kk
kn
sin1
2sin2
12
sin2
1
2sin2
1
11
0 21
21
−−−−====
−−−−====
++++++++
====++++
++++∑∑∑∑
dUcenH dUcenH dUcenH dUcenH xx
S
nn
n sin1
2sin2
1
11
−−−−====
++++++++
. . . .