Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Notice that the Eq. (6.31) is just a Taylor series at the origin of a function with
f n …0† ˆa n n!. Every choice we make for the in®nite variables a n de®nes a new
function with its own set of derivatives at the origin. Of course we can go beyond
the origin, and expand a function in a Taylor series centered at z ˆ z 0 . Thus in the
complex analysis there is a Taylor expansion for every analytic function. This is
the question addressed by Taylor's theorem (named after the English mathematician
Brook Taylor, 1685±1731):
If f(z) is analytic throughout a region R bounded by a simple
closed curve C, and if z and a are both interior to C, then f(z)
can be expanded in a Taylor series centered at z ˆ a for
jz aj < R:
f …z† ˆf …a†‡f 0 …a†…z a†‡f 00 …z a†2
…a† ‡
2!
‡ f n …z a†n1
…a† ‡ R
n!
n ; …6:32†
where the remainder R n is given by
I
R n …z† ˆ…z a† n 1
2i C
f …w†dw
…w a† n …w z† :
Proof:
To prove this, we ®rst rewrite Cauchy's integral formula as
f …z† ˆ 1 I
f …w†dw
2i C w z ˆ 1 I
2i C
f …w†
w a
1
1 …z a†=…w a†
dw:
…6:33†
For later use we note that since w is on C while z is inside C,
z a
< 1:
w a
From the geometric progression
we obtain the relation
1 ‡ q ‡ q 2 ‡ ‡q n ˆ 1 qn‡1
1 q ˆ 1
1 q qn‡1
1 q
1
1 q ˆ 1 ‡ q ‡‡qn ‡ qn‡1
1 q :
270