§ 3. Some Applications of Cauchy’s Method
83
(iii) Hankel’s integral. For x > 0 and a = i, (8) can also be written as
2πi/Γ (s) =
e
2πix(y−i)
[2πix(y − i)]
s 2πixdy, Re(s) > 1 .
Setting 2πix(y−i) = z, we get an integral computed over the vertical Re(z) =
2πx = a > 0. As dz = 2πixdy, we finally get that
2πi/Γ (s) =
Re(z)=a>0
e
z z
−s dz , Re(s) > 1 .
(10.10)
Having said that, and the function z
−s being defined on U = C − R − by
z
−s = |z|
−s exp [−is Arg(z)] with | Arg(z)| < π ,
(10.11)
integral (10) is about a holomorphic function on U . We show that a formula
holding for all s ∈ C can be obtained by deforming the integration contour.
This is not the case of (10) since the integral diverges for Re(s) ≤ 1. We use
C
G
B
A
D
F
E
0
a
Fig. 10.12.
the above path, along which the integral is zero, and set r and R to be the
radii of the two circular arcs. By (11), |z
−s
| | |z|
− Re(s) on U = C − R − and
in particular for small or large |z|. Since, moreover,
|e
z
| = e
Re(z)
≤ e
a