Mathematics of Physics and Engineering

Residue Integration^223

where C(zo) is a simple, closed, piece-wise smooth curve enclosing the point

zo and oriented counterclockwise; see Exercises 4.2.13 and 4.2.17. Term-

by-term integration of (4.4.10) over the curve C(ZQ) then results in

<p f(z)dz = 2iric- 1 =2iri Ties f(z). (4.4.12)
JC{z 0 ) z=z°
Note also that we get (4.4.12) after setting k = — 1 in formula (4.4.3) for
the coefficients Cfc (remember that, for our integration purposes, the curve
C(ZQ) can be replaced with a circle centered at ZQ). Consequently, c_i is the
only coefficient in the Laurent series expansion contributing to the integral
over a closed curve around the singular point.
The more general result is as follows.
Theorem 4.4.2 Let G be a simply connected domain. Assume that the
function f = f(z) is analytic at all points in G except finitely many points
z\,..., zn. If a simple, closed, piece-wise smooth curve C in G encloses the
points z\,..., zn and is oriented counterclockwise, then

I f(z)dz = 2m V Res f(z). (4.4.13)

Jc fiz=Zk

EXERCISE 4.4.7. (a)B Prove the above theorem. Hint: surround each zk

with a small circle that stays inside the domain bounded by C, then apply Theorem

4-2-4, page 198, in the (not simply connected) domain bounded by C and the n

circles around z\,..., z„. (b)c Explain why both the Cauchy Integral Formula

(4-2.11), page 199, and formula (4-2.16), page 201, are particular cases of

(4-4-13). Hint: for (4-2.16), write the Taylor expansion and integrate term-by-

term.

The idea of the residue integration is to find the residues of a function

without computing any integrals, and then use the above theorem to evalu-

ate the integrals of the function over different closed curves. Let us mention

that if the curve does not enclose any singular points of the function, then

the integral along the curve is zero by the Integral Theorem of Cauchy.

If the curve passes through a singular point, then, in general, the integral

along such a curve is not defined (remember that, to define the integral, we

require the function to be continuous at all points on the curve).

COMPUTING THE RESIDUES. The residue is a certain coefficient in the

Laurent series expansion at an isolated singular point, and there are three

types of isolated singular points: removable singularity, pole, and essential