Page 41 - Calculus with Complex Numbers

3 .7 Taylo r exp an si o ns
The M aclaurin exparlsion is a particular case of a more general exparlsion due to
Taylor (17 15) which represents flz) as a series in powers of z - c for any lixed c as
X
J'lz) - l'-qtzzztz - cln,
n=0
where the rlth coeflicient an is given by the formula
y(n) (c)
Jn = .
n !
We call this expansion the Taylor expansion of flz) at z = c, and we call the
coeflicient an the rlth Taylor coeflicient of flz) at z = c.
For example, suppose f (z) = 1/z and c = 1. We can calculate an as follows:

ao = ./(1) = 1.
f' (z) = - 1/z2 = - 1 at c = 1. Therefore tz1 = - 1.

=z 1 - (z - 1) + (z - 1)2 - (c - 1)3 + .-
as before.

The range of validity for this expansion is Iz - 11 < 1.
3.8 Laurent expansions

The Taylor exparlsion is a special case of a still more general expansion due to
Laurent (1843) which represents f (z) as the sum of a two-way power series
X
J'lz) - J--2 anlz - c)n