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z.IE l eRecED.Cefz e3Pt3iO henz3e3Pt3i0.3Pt3i03PRele e cosczo feel e costacosta
ISo for 30E f Iz t 21h cos 8 o
O E n t t E CBOCso OIT E6 It
Iii niShaded region has cosho O
7T and the unshaded regionI6 has cosczo LO
fG I grows as 171 70 with cosc30 OIft l decays as 121 30 with CBC303 0
y I 1 h CBCIfcz oscillates if cos 03 0
The asymptotic behavior willnot change if we consider Pcz7esince there is an R 0 and C o sit for all El RI.tt ns1PCz3l E C 121 where n degCPFor any fixed 0if cos3070
limeIPA est 7 figg f EP eeoso lime
e'Pasospz eeti0 p pp o P
if cos30 so
fing IPA e l E Liff c een.eeosoc.ligyee3Poso nIo
as Iipm Paso np D since ouzo sotoo
if cos30 0 then le I L and lim le't Past limpest p12120 12136So the grow or decaybehavior is roughly unchanged except in the
interface where oscillation occurs
2 2 D or as E w o is an essential
singularity since as 171 70 Hal is neither boundedor simply go to hence 2 P is neither a removable
singularity nor a pole hence 2 0 is an essential singularity
2 fez On E faz has
Zeros 2 1 2 3poles Z Z 2 D
each of them has order 1
Near 2 L we have Taylor expansion
let u Z l then
flatus taffeta u 1 atu 2 u l UU 2 a Hutu'tU 212112W J y city inu 2 1 ut
2ut U2
Near 2 P use w Yz and expand around w othose fits i E IFw
To I w l 3W l12W t DIwc I 2W1th 2 t
FI Find the winding number around 0 for the followingcurve
a f f C C unit circle fc z3 EZus r fez HII7 45
Solin the winding number is
a ner or if two dw dzwEfcc
numberof zeros of food inside Cof poles of f inside c
3 O 3since the roots are 7 0 2 IFL
2 her o zero of polding of f in C2 2 0
Ht For any point 2 ED away from the poles we havea small disk DECZ that is freefrompoles ZiZz 3
2a
polesZeo
El Zz Zz f23 3
Dada i
Then for any Z E DIZ such that12 Znl E A Enlant
Then Ifczyl E En TEY E T En tant kAgAnd If't't'll E EE Tain e IT A CD
Hence f is holomorphic in a neighborhood Dscz of Lfor any E ID I Zi Zz iZn
gleio joiI y Let Flo Z be a functionon 0,212 X D then by Thur5.2 in Ch2 we know
fat f dei z f et.fi ei0idoOoJo HO Z do
is a holomorphic function on D
Similarly view FcQZ as a function on0,21T x IC DT is hot'c in Z continuous in
both variables hence fit is also holk on 14213
Let C Eff 1gal then for any121 1 we haveg
f glfczsltfswyp.IT nteuIIuI e o as so
2 and CD The story goes as follows
we first consider 8th Z for ZEC ItuNEIL
then forn 30 fez
HIEIO 121 I
mo fcHeo
zn 12171
In general for ZOE G we have the difference ofthe two boundary values from inside and outside being ga7 fcr.to Liff fcr.to gtfo
This is clear if gas is a finite linear combinationof Z got Az En't Ar Zn't AN 2nThis is also true more generally for any gotsmooth function on C or even just continuous function