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8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
1/20
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
2/20
x = 1
1x+1
= 1−x+x2−x3+x4− ... 12 = 1−1+1−1+1−...
n 1−1 + 1−1 + 1− ...
12
1 + 2 + 3 + 4 + 5 + ... = −112
+∞i=1 ai
sn = n
i=1 ai
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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x = −1 11−x =1 + x + x2 + x3 + x4 + ...
1 − 1 + 1 − 1 + ... + (−1)n
−1
+ ... =
1
2 .
x = 2
1 + 2 + 4 + 8 + ... + 2n−1 + ... = −1. x = 1 1+1+1+1+1+... = 10 x = −2
1 − 2 + 4 − 8 + ... + (−2)n−1 + ... = 13
.
x = a |a| > 1
1 + a + a2 + a3 + ... + an−1 + ... = 1
1 − a.
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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x = 1 11+x = 1 − 2x + 3x2 − 4x3 + 5x4 − ...
1 − 2 + 3 − 4 + ... + (−n)n
−1
+ ... =
1
4 .
x = −1 1+2+3+4+5+... = 102 −1 +∞
ζ (s) = 1−s + 2−s + 3−s + 4−s + 5−s + ... (∗)
Re(s) > 1 2−s
(1
−21−s)
·(1−s + 2−s + 3−s + 4−s + 5−s + ...) = 1−s
−2−s + 3−s
−4−s + 5−s + ... (
∗∗)
s = −1 −3 · ( 1 + 2 + 3 +4 + 5 + ...) = 1 − 2 + 3 − 4 + ... −3 · (1 + 2 + 3 + 4 + 5 + ...) = 1
4
1 + 2 + 3 + 4 + ... + n + ... = −1
12 .
s = 0
1 + 1 + 1 + 1 + ... + n0 + ... = −1
2 .
2 + 3 + 4 + 5 + ... + (n + 1) + ... = −7
12 .
0 + 1 + 2 + 3 + ... + (n − 1) + ... = 512
.
1−s ln1 + 2−s ln2 + 3−s ln3 + 4−s ln 4 +5−s l n 5 + ... = −ζ (s) 1−s ln 1 − 2−s l n 2 + 3−s ln 3 − 4−s l n 4 + 5−s ln 5 − ... = (21−s −1)ζ (s) − 21−s ln 2 · ζ (s) s = 0
ln 1 + ln 2 + ln 3 + ln 4 + ... + ln(n) + ... = 1
2 ln(2π),
ln 1 − ln2 + ln 3 − ln4 + ... + (−1)n−1 ln(n) + ... = −12
ln(1
2π).
s = 1 + x + x2 + x3 + x4 + ... = 1 + x · s
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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s = 11−x 1 + x + x2 + x3 + x4 + ... = 11−x
x = 1 x = eiθ 0 < θ
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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s = 1
1−s + 2−s + 3−s + 4−s + ... + n−s + ... = ζ (s)(Re(s) < 1).
limn→+∞ s1+s2+...+snn s1 = 1, s2 = 1 + (−1) = 0, s3 = 1 + (−1) + 1 = 1, s4 =1 + (−1)+1+(−1) = 0,...
s11 =
11 = 1,
s1+s22 =
1+02 =
12 ,
s1+s2+s33 =
1+0+13 =
23 ,
s1+s2+s3+s44 =
1+0+1+04 =
12
, s1+s2+s3+s4+s55
= 1+0+1+0+15
= 35
,... 12
1 + (−1) + 0 + 1 + (−1) + 0 + ... = 23
,
1 + (−1) + 0 + 1 + (−1) + 0 + ... = 13
.
x = 1
1+x1+x+x2
+∞n=0(−1)n
= +∞n=0(−1)
n
· 1 =+∞n=0(−1)n·
+∞0 e
−ttndt
n! =
+∞n=0
+∞0 e
−t (−t)nn!
dt = +∞0 e
−t+∞n=0
(−t)nn!
dt = +∞0 e
−2tdt =12
1 − 1! + 2! − 3! + 4! − ... + (−1)n−1(n − 1)! + ... = 0, 596347...,
1 + 1! + 2! + 3! + 4! + ... + (n − 1)! + ... = 0, 697175...,
x
k=1 f (x) =
c + +∞0 f (t)dt + 12f (x) ++∞k=1 B2k(2k)!f (2k−1)(x)
c = − 12f (0) −+∞
k=1B2k(2k)!
f (2k−1)(0) f (t) 1, t , t + 1 t−1 c B2 = 16 x = −1 − ln(1+x) 1 + 12 +
13 +
14 +
15 + ... = − ln 0
1 + 1
2 +
1
3 +
1
4 +
1
5 + ... +
1
n + ... = γ = 0.57721566...
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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s =+∞
k=1 ak sn =
nk=1 ak
+∞
k=1
ak = limn→+∞
n
k=1
ak.
sn
s1+s2+...+snn
limn→+∞
sn = limn→+∞
s1 + s2 + ... + snn
.
sn sn ±∞
sn
sn =s(n) {1, 2, 3, ...n, } {ω(1), ω(2), ω(3), ...ω(n)}
ω (7)
1 + 1 + 1 + 1 + ...n0 + ... = −12
.
sn = s(n) = n
limDn→+∞ sω(n) = limDn→+∞ ω(n) = lim
Dn→+∞
ω(1)+ω(2)+ω(3)+...+ω(n)n
=s =
−12
ω(k) = − kn + 1
limDn→+∞1n
nk=1 ω(k) =
−12
limDn→+∞1n
nk=1 s(k) = lim
Dn→+∞
nk=1 s(− kn+1)· 1n =
0−1 s(k)dk ∈
R
D
limn→+∞
sn =
0−1
s(n)dn.
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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n −∞ limDn→−∞ sn =limDn→+∞ s−n =
0
−1 s(−n)dn D
limn→−∞
sn =
10
s(n)dn.
s(n) [1, +∞) ∞ [−1, 0) 0 s
[−1, 0) limn→−∞ sn s(n) [0, 1) I (x) = −1/x I : [1, +∞) → [−1, 0) I : [−∞, −1) → [0, 1)
(6)
1 + 2 + 3 + 4 + ... + n + ... = s.
sn = n(n+1)
2
s = Dlimn→+∞
sn = 0−1
s(n)dn = 0−1
n(n + 1)2
dn = −1/12.
f ∞ ∞ E R = {z ∈ C ||z | > R} ∞ BR = BR(∞) = E R
{∞} r > 0 a B(a; r) = {z |0
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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∞ f ◦ J 0 I (z ) = −J (z ) = − 1
z f
∞
f ◦ I E R f B(a; 1) P ∞(0, 0, 1) P 0(0, 0, −1) E R B(a; 1)
∞
∞
lα =
{r
·eiα
|r
∈ R
}, α
∈ [0, 2π),
∞
∞
∞ C f (z ) z 0 ω ∈ C
{∞} z n → z 0 f (z n) → ω n → +∞ ∞
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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+∞k=−∞ ck(z ) · 1zk
+∞k=−∞ ck(z ) · (z − a)k
a ∈ C
f (z ) ∞ f (z ) ∞ lα = {r · eiα|r ∈ R} limDz→∞(α) f (z ) f (z ) ∞ lα
D
limz→∞(α)+
f (z ) =D
limz→+eiα∞
f (z ) =D
limr→+∞
f (r · eiα),
D
limz→∞(α)−
f (z ) =D
limz→−eiα∞
f (z ) =D
limr→−∞
f (r · eiα),D
limz→∞(α)
f (z ) =D
limr→∞(0)
f (r · eiα),D
limz→∞(α)
f (z ) = 1
2(
D
limz→∞(α)+
f (z ) +D
limz→∞(α)−
f (z )).
f (z ) ∞
limDz→∞
(α) f (z ) α
∈ [0, 2π) limDz
→∞f (z )
α = 0 l0 = R
D
limx→∞(0)+
f (x) =D
limx→+∞
f (x),
D
limx→∞(0)−
f (x) =D
limx→−∞
f (x).
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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f (z ) a ∈ C f (z ) a lα,a = {a+r ·eiα|r ∈ R} limD
z→a(α)f (z )
f (z ) a lα,a
D
limz→a(α)+
f (z ) =D
limz→a+eiα0
f (z ) =D
limr→0+
f (a + r · eiα),
D
limz→a(α)−
f (z ) =D
limz→a−eiα0
f (z ) =D
limr→0−
f (a + r · eiα),D
limz→a(α)
f (z ) =D
limr→0(0)
f (a + r · eiα),
Dlimz→a(α)
f (z ) = 12
( Dlimz→a(α)+
f (z ) + Dlimz→a(α)−
f (z )).
f (z ) a
limDz→a(α) f (z ) α ∈ [0, 2π) limDz→a f (z ) α = 0 l0 = R
D
limx→a(0)+
f (x) =D
limx→a+
f (x),
D
limx→a(0)
−f (x) =
D
limx→a
−f (x).
f (z ) a ∈ C {∞}
D
limz→a
f (z ) =D
limz→a(α)±
f (z ) = limz→a
f (z ), α ∈ [0, 2π).
f (z ) = z n (n ∈ N ) z → ∞(0)
D
limz→+∞
z n = 0
−1z ndz,
D
limz→−∞
z n =
10
z ndz.
f (z ) = z −n (n ∈ N ) z → 0(0)
D
limz→0+
z −n = −1−∞
z −n−2dz,
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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D
limz→0−
z −n = +∞1
z −n−2dz.
f (z )
a ∈ C {∞}
a c0(z ) = f (z ) a
d(c0(z )) = c0(z ) f (z ) g(z ) a ∈ C {∞} c ∈ C
Dlim
z→a(α)±(f (z ) + g(z )) =
Dlim
z→a(α)±f (z ) +
Dlim
z→a(α)±g(z ),
D
limz→a(α)±
(c · f (z )) = c ·D
limz→a(α)±
f (z ).
ex sin(x) cos(x) ∞ limx→+∞ ex = +∞ limx→+∞ sin(x) limx→+∞ cos(x) +∞
e
x
+ 0
sin(x) + 0
cos(x) + 0
lim
D
x→+∞ ex
= lim
D
x→+∞ sin(x) =limDx→+∞ cos(x) = 0
∞k=0 anx
n 0
limDx→+∞ ax = 0(a = 0)
ln(x) ∞ limx→+∞ ln(x) =+∞ +∞ ln(x) + 0 limDx→+∞ ln(x) = 0 lim
Dx→+∞
n√
xm
=0, (m, n) = 1
limz→
+∞
H z = +∞
f (z ) = H z
∞ c0(z ) = H z ∞
d(H z)dz = H z
16
(π2 − 6H (2)z )dz = H z,
H n =n
k=11k
= ψ(0)(n + 1) + γ
H (s)n =
nk=1
1ks
= ζ (s) − ζ (s, n + 1) ζ (s, z) ≡+∞k=0 1(k+z)s
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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ψ(0)(z + 1) + C = H z,
C = H z − ψ(0)(z + 1),C = γ.
limDz→+∞ H z = γ.
f (z ) a ∈ C {∞(0)} f (z ) =F 1(z )+F 2(z ) a F 1(z ) F 2(z ) c0
D
limz→a(α)±
f (z ) =D
limz→a(α)±
F 1(z ) + c0.
a ∈ C limDz→a(α)± f (z ) = limDz→a(α)±(F 1(z )+F 2(z )) = limDz→a(α)±(−1
k=−n ck·(z −a)k+∞
k=0 ck·(z −a)k) = limDz→a(α)±
−1k=−n ck·(z −a)k+limz→a
∞k=0 ck·(z −a)k = limDz→a(α)± F 1(z )+
c0 = limDz→a(α)±
nk=1 c−k · (z −a)−k + c0 = limDr→0(0)±
nk=1 c−k ·(a + r−ke−iαk−a)−k +
c0 = n
k=1 c−k · e−iαk limDr→0(0)± r−k + c0 = n
k=1 c−k · e−iαk(∓ ∓∞∓1 r
−k−2dr) + c0 =nk=1 c−k · e−iαk (±1)
k
k+1 + c0 =
−1k=−n ck · eiαk (±1)
k
2(−k+1) + c0. a = ∞ limDz→±∞ f (z ) =
nk=1 ck · eiαk (±1)
k
2(k+1) + c0
a ∈ C lim
Dz→a(α) f (z ) =
12(lim
Dz→∞(α)+ f (z )+lim
Dz→∞(α)− f (z )) =
12(−1k=−n ck ·eiαk (+1)
k
−k+1 + c0 +−1k=−n ck · eiαk (−1)
k
−k+1 + c0) =−1
k=−n ck · eiαk 1+(−1)k
2(−k+1) + c0. a = ∞ limDz→∞(α) f (z ) =
nk=1 ck · eiαk 1+(−1)
k
2(k+1) + c0
f (z ) z = a ∈ C {∞} c0 f (z ) a
D
limz→a(α)
f (z ) = c0.
ζ (z ) z = 1 z = 1 1
z−1 + γ − γ 1(z − 1) + 12γ 2(z − 1)2 − 16γ 3(z − 1)3 + 124γ 4(z − 1)4 + O((z − 1)5).
limDz→1(0) ζ (z ) = limDz→1(0)(F 1(z ) + c0) = lim
Dz→1(0)(
1z−1 + γ ) = 0 + γ = γ.
ψ(0)(z) = Γ(z)Γ(z) Γ(x) =
+∞0 tx−1e−tdt
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f (x) a ∈ C {∞} f (x) =F 1(x) + F 2(x) a c0
D
limz→a
f (z ) = c0.
α ∈ [0, 2π) a ∈ C limDz
→a f (z ) =
12π
· 2π
0 limDz→a(α) f (z )dα =
12π
· 2π
0 −1k=−n ck121+(−1)k
−k+1
eiαkdα + c0 =12π
·−1k=−n ck 12 1+(−1)k−k+1 2π0 eiαkdα + c0 = 0 + c0 = c0. a = ∞
ζ (z ) z = 1 z = 1 1
z−1 + γ − γ 1(z − 1) + 12γ 2(z − 1)2 − 16γ 3(z − 1)3 + 124γ 4(z − 1)4 + O((z − 1)5).
limDz→1 ζ (z ) = c0 = γ.
1 − 1 + 1 − 1 + ... + (−1)n−1 + ... = 12
.
sn = 12 · ((−1)n+1 + 1) = 12 · (−1)n+1 + 12 s = 12 .
1 + 2 + 4 + 8 + ... + 2n−1 + ... = −1.
sn = 2n − 1 s = −1.
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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1 − 2 + 4 − 8 + ... + (−2)n
−1
+ ... =
1
3 .
sn = −13 · ((−2)n − 1) = −13 · (−2)n + 13 s = 13 .
1 + a + a2 + a3 + ... + an−1 + ... = 1
1 − a.
sn = 1a−1 · (an − 1) = 1a−1 · an − 1a−1 s = 11−a .
1 − 2 + 3 − 4 + ... + (−n)n−1 + ... = 14
.
sn = −14 · ((−1)n(2n + 1) − 1) = −14 · (−1)n(2n + 1) + 14 s = 14 .
1 + 2 + 3 + 4 + ... + n + ... = −1
12 .
sn = n(n+1)
2 s = limDn→+∞
n(n+1)2 =
0−1
n(n+1)2 dn = − 112 .
1 + 1 + 1 + 1 + ... + n0 + ... = −1
2 .
sn = n s = limDn→
+∞
n = 0−1 ndn = −12
.
2 + 3 + 4 + 5 + ... + (n + 1) + ... = −7
12 .
sn = n(n+3)
2 s = limDn→+∞
n(n+3)2
= 0−1
n(n+3)2
dn = − 712
.
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0 + 1 + 2 + 3 + ... + (n − 1) + ... = 5
12 .
sn = n(n−1)
2 s = limDn→+∞
n(n−1)2 =
0−1
n(n−1)2 dn =
512 .
ln 1 + ln 2 + ln 3 + ln 4 + ... + ln(n) + ... = 1
2 ln(2π).
sn = ln((1)n) (1)n = Γ(1+n)
Γ(1)
s(n) = sn n = ∞ ln(2−
arg(z+1)+π2π
cscarg(z+1)+π
2π (π(z +1))+arg(z+1)+π2π (iπz + iπ2 +O((1z )7))+arg(z)+π2π (2iπz +
iπ + O((1z
)7)) + ((− ln(1z
) − 1)z + 12
(ln(2π) − ln(1z
)) + 112z
− 1360z3
+ 11260z5
+ O((1z
)6))) s = 12 ln(2π).
ln 1 − ln2 + ln 3 − ln4 + ... + (−1)n−1 ln(n) + ... = −12
ln(1
2π).
sn = −12(−1)n ln(2)+(−1)n ln(Γ(n+12 )) − (−1)n ln(Γ(n+22 )) − 12 ln(2π) +ln(2) s = −12 ln(2π) + ln(2) = − 12 ln( 12π).
cos θ + cos 2θ + cos 3θ + cos 4θ + ... + cos(nθ) + ... = −12
.
sn = 12 cot( θ2) sin(nθ) + 12 cos(nθ) − 12 s = −12 .
sin θ + sin 2θ + sin 3θ + sin 4θ + ... + sin(nθ) + ... = 1
2 cot
θ
2.
(x)n = Γ(x+n)
Γ(x) = x(x + 1) · · · (x + n − 1)
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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sn = −12 cot(θ2)cos(nθ) + 12 sin(nθ) + 12 cot( θ2) s = 12 cot θ2 .
cos θ − cos2θ + cos 3θ − cos4θ + ... + (−1)n−1 cos(nθ) + ... = 12
.
sn = 12(cos((θ + π)(n + 1)) + tan
θ2 · sin((θ + π)(n + 1)) + 1), s = 12 .
sin θ − sin2θ + sin 3θ − sin4θ + ... + (−1)n−1 sin(nθ) + ... = 12
tan θ
2.
sn = 12(− tan θ2 cos((θ + π)n) − sin((θ + π)n) + tan θ2) s = 12 tan θ2 .
12k − 22k + 32k − 42k + ... + (−1)n−1n2k + ... = 0(k = 1, 2, 3,...).
sn = 22k(−1)n+1ζ (−2k, n+12 ) + 22k(−1)nζ (−2k, n+22 ) − 22k+1ζ (−2k) +
ζ (−2k) s = (1 − 22k+2)ζ (−2k) = 0.
12k+1−22k+1+32k+1−42k+1+ ... +(−1)n−1n2k+1+ ... = 22k+2 − 12k + 2
B2k+2(k = 0, 1, 2,...).
sn = 22k+1(−1)n+1ζ (−2k − 1, n+1
2 ) + 22k+1(−1)nζ (−2k − 1, n+2
2 ) −
22k+2ζ (−2k − 1) + ζ (−2k − 1) s = (1 − 22k+2)ζ (−2k − 1) = 22k+2−12k+2 B2k+2.
1k + 2k + 3k + 4k + ... + nk + ... = − Bk+1k + 1
(k = 1, 2, 3,...).
sn = 1k+1
km=0
k+1m
Bmn
k+1−m B1 =
12 .
s = limDn→+∞( 1k+1
km=0
k+1m
Bmn
k+1−m) = 1k+1
km=0
k+1m
Bm lim
Dn→+∞ n
k+1−m =1
k+1
km=0
k+1m
Bm
0−1 n
k+1−mdn = − 1k+1
km=0
k+1m
Bm
(−1)k+2−mk+2−m = − 1k+1 · Bk+1.
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B1
= 12
B1
= −
1
2
Bk+1 = 1 −k
m=0
k + 1
m
Bm
k + 2 − m,
Bk+1 = −k
m=0
k + 1
m
Bm
k + 2 − m.
1−s + 2−s + 3−s + 4−s + ... + n−s + ... = ζ (s)(Re(s) < 1).
sn = H (s)n = ζ (s) − ζ (s, n + 1) S = ζ (s)
1 + (−1) + 0 + 1 + (−1) + 0 + ... = 23
.
sn = 23 + n3 − n3 s = 23 .
1 + (−1) + 0 + 1 + (−1) + 0 + ... = 13
.
sn = −−23 + n3 + −13 + n3 s = 13 .
x ∈ R\Z x = −12 + x +∞
k=1sin(2kπx)
k
π
1 − 1! + 2! − 3! + 4! − ... + (−1)n−1(n − 1)! + ... = 0, 596347...
sn = −e((−1)nE n+1(1)Γ(n+1)+Ei(−1)) s = −eEi(−1) = 0, 596347...
E n(x) = +∞1
e−xtdttn
Ei(x) = − +∞−x
e−tdtt
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1 + 1! + 2! + 3! + 4! + ... + (n−
1)! + ... = 0, 697175...
sn = (−1)nn!!(−n − 1)+!(−2) + 1 s =!(−2) + 1 = 0, 697175...
1 + 1
2 +
1
3 +
1
4 +
1
5 + ... +
1
n + ... = γ = 0.57721566...
sn = ln(n) + γ + εn εn ∼ 12n s = γ.
sn = H n = ψ(0)(n + 1) + γ s = γ.
+∞0
sin xdxD = 1,
+∞0
ln x sin xdxD = −γ,
+∞0
√ x tanh(
√ x)dxD = −3
4ζ (3).
+∞0
sin xdxD = 1.
!n = Γ(n+1,−1)e
Γ(a, x) = +∞x
ta−1e−tdt
8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…
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+∞0 sin xdx
D = − cos x|+∞0 = −(limDx→+∞(cos x + 0) − cos 0) = −(0 −1) = 1.
+∞0
ln x sin xdxD = −γ.
+∞0
ln x sin xdxD = (Ci(x)−ln x cos x)|+∞0 = limDx→+∞(Ci(x)−ln x cos x)−limDx→0(Ci(x) − ln x cos x) = 0 − γ = −γ. Ci(x)− ln x cos x ∞ cos x(ln 1
x+O((1
x)7))+cos x(−( 1
x)2+ 6
x4− 120
x6 +O(( 1
x)7))+sin x( 1
x− 2
x3 + 24
x5 +O(( 1
x)7))+
O(( 1x
)9)−
iπ1
2 − arg(x)
π + 0.
Ci(x)− ln x cos x 0 γ + 14x
2(2ln x − 1) + 196x4(1 − 4 ln x) + x6(6lnx−1)
4320 + O(x7).
+∞0
√ x tanh(
√ x)dxD = −3
4ζ (3).
+∞0
√ xh(
√ x)dxD = (−2√ xLi2(−e−2
√ x)−Li3(−e−2
√ x)+2x
32
3 +2x ln(e−2
√ x+
1))|+
∞0 = limD
x→+∞(−2√ xLi2(−e−2√ x
) − Li3(−e−2√ x
) + 2x
32
3 + 2x ln(e−2√ x
+1) +0) −3ζ (3)4
= 0 − 3ζ (3)4
= −3ζ (3)4
.
Ci(x) = − +∞x
cos tdtt
Lin(z) ≡+∞
k=1zk
kn