Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
(b) ln‰…1 ‡ z†=…1 z†Š ˆ ln…1 ‡ z†ln…1 z†. Next, replacing z by z in
Taylor's expansion for ln…1 ‡ z†, we have
Then by subtraction, we obtain
ln…1 z† ˆz z2
2 z3
3 z4
4 :
ln 1 ‡ z
1 z ˆ 2 z ‡ z3
3 ‡ z5
5 ‡ !
ˆ X1
nˆ0
2z 2n‡1
2n ‡ 1 :
Laurent series
In many applications it is necessary to expand a function f …z† around points
where or in the neighborhood of which the function is not analytic. The Taylor
series is not applicable in such cases. A new type of series known as the Laurent
series is required. The following is a representation which is valid in an annular
ring bounded by two concentric circles of C 1 and C 2 such that f …z† is single-valued
and analytic in the annulus and at each point of C 1 and C 2 , see Fig. 6.14. The
function f …z† may have singular points outside C 1 and inside C 2 . Hermann
Laurent (1841±1908, French mathematician) proved that, at any point in the
annular ring bounded by the circles, f …z† can be represented by the series
f …z† ˆ X1
nˆ1
a n …z a† n
…6:34†
where
a n ˆ 1 I
2i C
f …w†dw
…w a†
n‡1;
n ˆ 0; 1; 2; ...; …6:35†
Figure 6.14.
Laurent theorem.
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