Mathematical Methods for Physicists: A concise introduction - Site Map

COMPLEX INTEGRATION
But R AF f …z†dz ˆR FA
f …z†dz, therefore this becomes
Z
Z
f …z†dzy ‡ f …z†dzy ˆ 0
or
I
C
ABDEA
FGHF
I I
f …z†dz ˆ f …z†dz ‡
C 1
f …z†dz ˆ 0;
C 2
…6:26†
where both C 1 and C 2 are traversed in the positive direction (in the sense that an
observer walking on the boundary always has the region R on his left). Note that
curves C 1 and C 2 are in opposite directions.
If we reverse the direction of C 2 (now C 2 is also counterclockwise, that is, both
C 1 and C 2 are in the same direction.), we have
I I
I
I
f …z†dz f …z†dz ˆ 0 or f …z†dz ˆ f …z†dz:
C 1 C 2 C 2 C 1
Because of Cauchy's theorem, an integration contour can be moved across any
region of the complex plane over which the integrand is analytic without changing
the value of the integral. It cannot be moved across a hole (the shaded area) or a
singularity (the dot), but it can be made to collapse around one, as shown in Fig.
6.9. As a result, an integration contour C enclosing n holes or singularities can be
replaced by n separated closed contours C i , each enclosing a hole or a singularity:
I
I
f …z†dz ˆ Xn
f …z†dz
C
C i
kˆ1
which is a generalization of Eq. (6.26) to multiply-connected regions.
There is a converse of the Cauchy's theorem, known as Morera's theorem. We
now state it without proof:
Morera's theorem:
If f(z) is continuous in a simply-connected region R and the Cauchy's theorem is
valid around every simple closed curve C in R, then f …z† is analytic in R.
Figure 6.9.
Collapsing a contour around a hole and a singularity.
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