Mathematical Methods for Physicists: A concise introduction - Site Map

SERIES REPRESENTATIONS OF ANALYTIC FUNCTIONS
By setting q ˆ…z a†=…w a† we ®nd
1
1 ‰…z a†=…w a†Š ˆ 1 ‡ z a
w a ‡
‡
z a 2‡‡
z a
w a w a
‰…z a†=…w a†Šn‡1
:
…w z†=…w a†
We insert this into Eq. (6.33). Since z and a are constant, we may take the powers
of (z a) out from under the integral sign, and then Eq. (6.33) takes the form
f …z† ˆ 1 I
f …w†dw
2i C w a ‡ z a I
I
f …w†dw a†n f …w†dw
‡‡…z
2i
2
C …w a† 2i C …w a† n‡1 ‡ R n…z†:
Using Eq. (6.28), we may write this expansion in the form
where
f …z† ˆf …a†‡ z a f 0 …a†‡
1!
…z a†2
f 00 …a†‡‡
2!
I
R n …z† ˆ…z a† n 1
2i C
f …w†dw
…w a† n …w z† :
n
…z a†n
f n …a†‡R
n!
n …z†;
Clearly, the expansion will converge and represent f …z† if and only if
lim n!1 R n …z† ˆ0. This is easy to prove. Note that w is on C while z is inside
C, sowehavejw zj > 0. Now f …z† is analytic inside C and on C, so it follows
that the absolute value of f …w†=…w z† is bounded, say,
f …w†
w z < M
for all w on C. Let r be the radius of C, then jw aj ˆr for all w on C, and C has
the length 2r. Hence we obtain
I
jz ajn
f …w†dw
jR n j ˆ 2
C …w a† n …w z† < jz
ajn
M 1 2 r n 2r
ˆ Mr z a
n ! 0 as n !1:
r
Thus
f …z† ˆf …a†‡ z a
1!
f 0 …z a†2
…a†‡ f 00 …z a†n
…a†‡‡ f n …a†
2!
n!
is a valid representation of f …z† at all points in the interior of any circle with its
center at a and within which f …z† is analytic. This is called the Taylor series of f …z†
with center at a. And the particular case where a ˆ 0 is called the Maclaurin series
of f …z† [Colin Maclaurin 1698±1746, Scots mathematician].
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