COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

Complimentary software to accompany the textbook:

COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
Jones and Bartlett Publishers, Inc., 40 Tall Pine Drive, Sudbury, MA 01776
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CHAPTER 8 RESIDUE THEORY

Section 8.1 The Residue Theorem

The Cauchy-integral formulas in Section 6.5 are useful in evaluating contour integrals over a simple closed contour where the integrand has the form and is an analytic function. In this case, the singularity of the integrand is at worst a pole of order at . In this section we extend this result to integrals that have a finite number of isolated singularities and lie inside the contour . This new method can be used in cases where the integrand has an essential singularity at and is an important extension of the previous method.

Definition 8.1: Residue

Let have a nonremovable isolated singularity at the point . Then has
the Laurent series representation . The coefficient of is
called the residue of at and we use the notation Res[f , ] = .

Load Maple's "residue" procedure.
Make sure this is done only ONCE during a Maple session.

> readlib(residue):

Example 8.1, Page 307. Use Laurent series to find the residue at for the function .

> f:='f': s:='s': z:='z':
f := z -> exp(2/z):
s := series(f(z), z=infinity, 7):
`f(z) ` = f(z);
`f(z) ` = s;

The coefficient of is so the residue is .

Example 8.2, Page 308. Find the residue at for the function .

> g:='g': s:='s': z:='z':
g := z -> 3/(2*z + z^2 - z^3):
s := series(g(z), z=0, 5):
`g(z) ` = g(z);
`g(z) ` = s;

The coefficient of is so the residue is .
We compare this with Maple's residue procedure for computing residues.

> `g(z) ` = g(z);
`g(z) ` = s;
`Res[g,0] ` = residue(g(z), z=0);

Example 8.3, Page 308. Use residues to integrate around : .
From Example 8.1 . Thus the value of the integral is .

> f:='f': F:='F': s:='s': z:='z':
f := z -> exp(2/z):
`F(z) ` = f(z);
s := series(f(z), z=infinity, 5):
res := 2:
`Res[F,0] ` = res;
print(int(F(z),z=C..``) = `2*Pi*I*Res[f,0])`);
print(int(F(z),z=C..``) = 2*Pi*I*res);

Theorem 8.1 (Cauchy's Residue Theorem)

Let be a simply connected domain and let be a simple closed positively oriented contour that lies in .

If is analytic inside and on , except at the points , , ..., that lie inside , then

= .

End of Section 8.1.