Page 775 - Handbook Of Integral Equations

Expansion into series in powers of x as x → 0:

∞ k+1 2k–1
(–1) x
Si(x)= .
(2k – 1) (2k – 1)!
k=1
Asymptotic expansion as x →∞:
M–1 m N–1 m
(–1) (2m)! (–1) (2m – 1)!
–2M–1 –2N
si(x)= – cos x + O |x| + sin x + O |x| ,
x 2m+1 x 2m
m=0 m=1
where M, N =1, 2, ...
Integral cosine
Definition:
∞
cos t cos t – 1
x
Ci(x)= – dt = C +ln x + dt, C = 0.5572 ...
t t
x 0
Expansion into series in powers of x as x → 0:
∞ k 2k
(–1) x
Ci(x)= C +ln x + .
2k (2k)!
k=1
Asymptotic expansion as x →∞:
M–1 m N–1 m
(–1) (2m – 1)! (–1) (2m)!
–2M –2N–1
Ci(x) = cos x + O |x| + sin x + O |x| ,
x 2m x 2m+1
m=1 m=0
where M, N =1, 2, ...
Fresnel integrals
Definitions: √
1 x sin t 2 x 2
S(x)= √ √ dt = sin t , dt,
2π 0 t π 0
√

1 x cos t 2 x 2
C(x)= √ √ dt = cos t dt.
2π 0 t π 0
Expansion into series in powers of x as x → 0:
∞ k 2k+1
2 (–1) x
S(x)= x ,
π (4k +3) (2k + 1)!
k=0
∞ k 2k
2 (–1) x
C(x)= x .
π (4k +1) (2k)!
k=0
Asymptotic expansion as x →∞:
1 cos x sin x
S(x)= – √ P(x) – √ Q(x),
2 2πx 2πx
1 sin x cos x
C(x)= + √ P(x) – √ Q(x),
2 2πx 2πx
1 ⋅ 3 1 ⋅ 3 ⋅ 5 ⋅ 7 1 1 ⋅ 3 ⋅ 5
P(x)=1 – + – ··· , Q(x)= – + ··· .
(2x) 2 (2x) 4 2x (2x) 3

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 760