Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Thus on letting z ! a
that is,
d m1
lim
z!a dz m1 ‰…z a†m f …z†Š ˆ …m 1†!a 1 ;
( )
Res f …z† ˆ 1
zˆa …m 1†! lim d m1
z!a dz m1 ‰ …z a†m f …z† Š : …6:40†
Of course, in the case of a rational function f …z† the residues can also be
determined from the representation of f …z† in terms of partial fractions.
The residue theorem
So far we have employed the residue method to evaluate contour integrals whose
integrands have only a single singularity inside the contour of integration. Now
consider a simple closed curve C containing in its interior a number of isolated
singularities of a function f …z†. If around each singular point we draw a circle so
small that it encloses no other singular points (Fig. 6.15), these small circles,
together with the curve C, form the boundary of a multiply-connected region in
which f …z† is everywhere analytic and to which Cauchy's theorem can therefore be
applied. This gives
1
2i
I
C
I
I
f …z†dz ‡ f …z†dz ‡‡
C 1
f …z†dz
C m
ˆ 0:
If we reverse the direction of integration around each of the circles and change the
sign of each integral to compensate, this can be written
I
1
f …z†dz ˆ 1 I
f …z†dz ‡ 1 I
f …z†dz ‡‡ 1 I
f …z†dz;
2i C 2i C 1
2i C 2
2i C m
Figure 6.15.
Residue theorem.
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