
1514 SPECIAL FUNCTIONS AND THEIR PROPERTIES
Asymptotic expansion as x → 1:
li(x) = C + ln |ln x|+
∞
X
k=1
ln
k
x
k! k
.
Relation to the exponential integral:
li x = Ei(ln x), x < 1;
li(e
x
) = Ei(x), x < 0.
30.3 Sine Integral and Cosine Integral. Fresnel Integrals
30.3.1 Sine Integral
◮ Integral representations. Properties.
Definition:
Si(x) =
Z
x
0
sin t
t
dt, si(x) = −
Z
∞
x
sin t
t
dt = Si(x) −
π
2
.
Specific values:
Si(0) = 0, Si(∞) =
π
2
, si(∞) = 0.
Properties:
Si(−x) = −Si(x), si(x) + si(−x) = −π, lim
x→−∞
si(x) = −π.
◮ Expansions as x → 0 and x → ∞.
Expansion into series in powers of x as x → 0:
Si(x) =
∞
X
k=1
(−1)
k+1
x
2k− 1
(2k − 1) (2k − 1)!
.
Asymptotic expansion as x → ∞:
si(x) = − cos ...