1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications

0 ..

x

Fig. 9.11

= lim { r-f [_::i_ +g(x)] dx+jb [_::i_ +g(x)] dx}

E-+O+ 1 a X - C c+E X - C

= lim {Aln Ix -cl ]~-' + Aln Ix -cl i:+,

f-+O+

• r-fg(x)dx+jb g(x)dx}

la c+E

• b
  = lim {A[lnE-ln(c-a)+ln(b-c)-lnE]}+ f g(x)dx

E-+O+ la

=Alo --b-c + lb g(x)dx

c-a a

9.11 EVALUATION OF REAL IMPROPER INTEGRALS

BY USING THE RESIDUE THEOREM

685

The reader will recall that there are three types of improper real integrals:

1. Improper integrals of the first kind. For functions that are Riemann
   integrable over every finite interval [a, b] (in particular, for continuous
   functions) they are defined as follows:

(a) fa

00

J(x)dx = limb-+oof: J(x)dx if the limit exists

(b) J~ 00 J(x)dx = lima-+-oof: J(x)dx ifthe limit exists

(c) J~ 00 f(x)dx = J~ 00 f(x)dx + fc

00

f(x)dx if both limits exist

The terms convergent and divergent are applied to these integrals ac-

cording to whether thy corresponding limits do or do not exist (as finite

numbers). If in case ( c) the integral diverge:;; it may nevertheless be

convergent in the sense that

lim JR J(x)dx

R-+oo -R

does exist. If that occurs, the limiting value 1s called Cauchy's principal

value of the integral (or simply its principal value), and we write