1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

has a Fourier expansion

+co

F(z) = L Ane+2.,..inz

n=-oo

valid in the strip -b < Im z < b, where

An= 1: J(x)e-2.,..inx dx

and derive the Poisson summation formula

+co +co jco

L J(n) = L J(x)e-2.,..inx dx

n=-oo n=-oo -oo

8.20 The Eulerian Integrals. The Gamma and Beta Functions

Definition 8.6 The integrals

613

r(z) = 1co e-ttz-l dt and B(z, () = 1

1

e-^1 (1-t)C-^1 dt (8.20-1)

where tis a real variable, tz-l = e<z-l)lnt, (1 - t)C-^1 = e<C-l)ln(l-t),
were discussed by L. Euler [11] for z and ( real. The first integral con-

verges for Re z > 0 and defines in this half-plane an analytic function r( z)

called the gamma Junction. The second integral converges for Re z > 0

and Re ( > 0 and defines over the Cartesian product of those half-planes

a function B(z, () analytic in each of the variables z and (, called the
beta function. There is a close connection between the beta and gamma
functions which we shall discuss later.

Theorem 8.46 The integral P(z) = J 0 co e-ttz-l dt converges absolutely
for Re z > 0. On this half-plane the function r( z) defined by the integral
is analytic, and

Proof We may write

P(z) = 1co e-ttz-^1 dt + 1co e-ttz-^1 dt = P(z) + Q(z)

In the first integral let t = 1/r. Then we have

P(z) = 1co e-lfrr-z-l dr

(8.20-2)

(8.20-3)