Page 407 - Schaum's Outline of Theory and Problems of Advanced Calculus

398 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

þ
dz 0 if n 6¼ 1
n ¼ 2 i if n ¼ 1 ð12Þ
C ðz aÞ
(see Problem 16.13), it follows that

þ
f ðzÞ dz ¼ 2 ia 1 ð13Þ
C
i.e., the integral of f ðzÞ around a closed path enclosing a single pole of f ðzÞ is 2 i times the residue at the
pole.
More generally, we have the following important theorem.

Theorem. If f ðzÞ is analytic within and on the boundary C of a region r except at a finite number of
poles a; b; c; ... within r, having residues a 1 ; b 1 ; c 1 ; ... ; respectively, then

þ
f ðzÞ dz ¼ 2 iða 1 þ b 1 þ c 1 þ Þ ð14Þ
C
i.e., the integral of f ðzÞ is 2 i times the sum of the residues of f ðzÞ at the poles enclosed by C. Cauchy’s
theorem and integral formulas are special cases of this result, which we call the residue theorem.

EVALUATION OF DEFINITE INTEGRALS
The evaluation of various definite integrals can often be achieved by using the residue theorem
together with a suitable function f ðzÞ and a suitable path or contour C, the choice of which may reuqire
great ingenuity. The following types are most common in practice.
ð
1
1. FðxÞ dx; FðxÞ is an even function.
0 þ
Consider FðzÞ dz along a contour C consisting of the line along the x-axis from R to
C
þR and the semicircle above the x-axis having this line as diameter. Then let R !1. See
Problems 16.29 and 16.30.
ð 2
2. Gðsin ; cos Þ d , G is a rational function of sin and cos .
0 1 1
i
i
Let z ¼ e . Then sin ¼ z z ; cos ¼ z þ z and dz ¼ ie d or d ¼ dz=iz. The
2i 2
þ
given integral is equivalent to FðzÞ dz, where C is the unit circle with center at the origin. See
C
Problems 16.31 and 16.32.

ð
cos mx
1
3. dx; FðxÞ is a rational function.
sin mx
FðxÞ
1
þ
Here we consider FðzÞe imz dz where C is the same contour as that in Type 1. See
C
Problem 16.34.
4. Miscellaneous integrals involving particular contours. See Problems 16.35 and 16.38. In
particular, Problem 16.38 illustrates a choice of path for an integration about a branch point.