1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 711

Next, in view of condition (5) there is an R 1 > r such that x ;:=: R 1

implies that

A

xc+l

But J: (A/xc+l) dx converges as R--+ oo. Hence (9.11-41) converges also

at the \ipper limit. Therefore, if we let r --+ o+ in (9.11-40) and then

R --+ oo, we get

+ i7r Res za f(z)

z=p

When f(z) has n simple poles p 8 ( s = 1, 2, ... , n) on the positive real axis,

we obtain, similarly,

m n

= 27ri L Res za f(z) + i7r L Res za f(z) (9.11-42)

k=l z=bk s=l z=ps

By equating real and imaginary parts in (9.11-42), the values of

and

can be obtained. In particular, (9.11-42) applies to a rational function
f(x) = P(x)/Q(x) provided that

O<a+c+l=v-μ

and z = 0 is not a pole of P / Q. As before, μ = degree of P and v = degree
of Q.

Example To evaluate (PV) f 000 xa dx/(b-x), -1 <a< 0, b > 0. In this

case f(z) = 1/(b - z) has a simple pole at z = b > 0, and za+l f(z) =

za+l /(b - z) tends t~ 0 as jzj --+ 0. If we choose c = -a> 0, we get

Also, we have

--- = -- -+1

I

za+c+l I I z I

b-z z-b

za

Res --= -ba

z=b b - z

as lzl--+ oo