Singularities/Residues/ Applications 707~
b~r+
b~;q
- A2 R x
b~Fig. 9.20which shows thatSimilarly,lim J za f ( z) dz = 0
R->oor+
Thus if we let r -t 0, then R -t oo in (9.11-31), we see that J 000 xa f(x) dx
exists, and ·
[
000xaf(x)dx=^2 7ri. ~Resz.af(z)
Jo 1 - e^2 7ria L.J z=bk
k=l
-7ria m
= -~e L Res za f(z)
sin a7r k=l z=bk(9.11-32)Formula (9.11-32) applies, in particular, to the case of a rational function
J(x) = P(x)/Q(x) provided that
O<a+l<v-μ
where v denotes the degree of Q and μ the degree of P, and z = 0 is not
a pole of P / Q ( P and Q being prime to each other).
Remark The function g(a) = f 0
00
xaf(x)dx = J 0
00
xa-^1 [xf(x)]dx is
called the Mellin transform of xf(x).