Mathematical Methods for Physicists: A concise introduction - Site Map

SERIES REPRESENTATIONS OF ANALYTIC FUNCTIONS
Since r 1 = < 1, the last expression approaches zero as n !1. Hence
lim n!1 R n2 ˆ 0 and we have
f …z† ˆ 1 I
" f …w†dw
2i C 1
w a ‡ 1 I #
f …w†dw
2i C 1 …w a† 2 …z a†
"
‡ 1 I #
f …w†dw
2i …w a† 3 …z a† 2 ‡
C 1
1
‡ 1 I
f …w†dw
2i C 2
z a ‡ 1 I
…w a† f …w†dw
2i C 2
1
…z a† 2 ‡:
Since f …z† is analytic throughout the region between C 1 and C 2 , the paths of
integration C 1 and C 2 can be replaced by any other curve C within this region
and enclosing C 2 . And the resulting integrals are precisely the coecients a n given
by Eq. (6.35). This proves the Laurent theorem.
It should be noted that the coecients of the positive powers (z a) in the
Laurent expansion, while identical in form with the integrals of Eq. (6.28), cannot
be replaced by the derivative expressions
f n …a†
n!
as they were in the derivation of Taylor series, since f …z† is not analytic throughout
the entire interior of C 2 (or C), and hence Cauchy's generalized integral
formula cannot be applied.
In many instances the Laurent expansion of a function is not found through the
use of the formula (6.34), but rather by algebraic manipulations suggested by the
nature of the function. In particular, in dealing with quotients of polynomials it is
often advantageous to express them in terms of partial fractions and then expand
the various denominators in series of the appropriate form through the use of the
binomial expansion, which we assume the reader is familiar with:
…s ‡ t† n ˆ s n ‡ ns n1 t
n…n 1†
s n2 t 2 ‡
2!
n…n 1†…n 2†
s n3 t 3 ‡:
3!
This expansion is valid for all values of n if jsj > jtj: If jsj jtj the expansion is
valid only if n is a non-negative integer.
That such procedures are correct follows from the fact that the Laurent expansion
of a function over a given annular ring is unique. That is, if an expansion of the
Laurent type is found by any process, it must be the Laurent expansion.
Example 6.23
Find the Laurent expansion of the function f …z† ˆ…7z 2†=‰…z ‡ 1†z…z 2†Š in
the annulus 1 < jz ‡ 1j < 3.
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