Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
The Taylor series of f …z† converges to f …z† only within a circular region around
the point z ˆ a, the circle of convergence; and it diverges everywhere outside this
circle.
Taylor series of elementary functions
Taylor series of analytic functions are quite similar to the familiar Taylor series of
real functions. Replacing the real variable in the latter series by a complex variable
we may `continue' real functions analytically to the complex domain. The
following is a list of Taylor series of elementary functions: in the case of multiplevalued
functions, the principal branch is used.
e z ˆ X1
nˆ0
sin z ˆ X1
nˆ0
cos z ˆ X1
nˆ0
sinh z ˆ X1
nˆ0
cosh z ˆ X1
nˆ0
ln…1 ‡ z† ˆX1
nˆ0
z n
n! ˆ 1 ‡ z ‡ z2
‡; jzj < 1;
2!
…1† n z 2n‡1
…2n ‡ 1†! ˆ z z3
3! ‡ z5
‡; jzj < 1;
5!
z 2n
…1† n
…2n†! ˆ 1 z2
2! ‡ z4
‡; jzj < 1;
4!
z 2n‡1
…2n ‡ 1†! ˆ z ‡ z3
3! ‡ z5
‡; jzj < 1;
5!
z 2n
…2n†! ˆ 1 ‡ z2
2! ‡ z4
‡; jzj < 1;
4!
…1† n‡1 z n
n
ˆ z z2
2 ‡ z3
‡; jzj < 1:
3
Example 6.20
Expand (1 z† 1 about a.
Solution:
1
1 z ˆ 1
…1 a†…z a† ˆ 1 1
1 a 1 …z a†=…1 a† ˆ 1
1 a
X 1
nˆ0
z a
n:
1 a
We have established two surprising properties of complex analytic functions:
(1) They have derivatives of all order.
(2) They can always be represented by Taylor series.
This is not true in general for real functions; there are real functions which have
derivatives of all orders but cannot be represented by a power series.
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