Taylor's and laurent's series

TAYLOR'S AND LAURENT'S SERIES

1. Taylor's Series

A function f(z) analytic inside a circle C with centre at a, can be expanded in the series

which is convergent at every point inside C.

Let y be a circle having z in its interior and concentric with the circle C and lying inside C, so that f(z) is analytic inside and on y.

Let ω be any point on y.

We have |z - ɑ | < | ω - ɑ|

Note 1. This is Taylor's series about z = a.

Note 2. Taking a = 0, Taylor's series reduces to

which is known as Maclaurin's series.

Note 3. There is no negative powers of (z - ɑ).

2. Laurent's Series.

Let C₁, C₂ be two concentric circles |z - ɑ| = R₁ and |z - ɑ | = R₂, where R₂ < R₁. Let f (z) be analytic on C₁ and C₂ and in the annular region. R between them. Then, for any point z in R,

Let z be any point in the annulus R. Let y be a small circle with centre z and radius p lying entirely in R.

Taking ω as the current variable,

f(ω) / ω – z  is analytic in the domain bounded by C₁, C₂ and y.

Hence, for ω ∈ C₁

 and this series coverges absolutely and uniformly for z ∈R and ω ∈ C₁

Hence, term by term integration is valid and we have

and the series on the right side

converges absoultely and uniformly for z ∈ R, ω ∈ C₂.

Hence, term by term integration is valid and, we have

where an, bn are given by (2) and (3).

3. Some important Results:

I. Taylor's series

1. Taylor's series about z = ɑ is

2. Taylor's series about z = 0 is

II. Laurent's series

Note :

1. If f (z) is analytic inside C₂, then the Laurent's series residues to the Taylor's series of f (z) with centre a, since in this case all the co-efficients of negative powers in Laurent's are zeros.

2. The part  consisting of positive integral powers of (z - ɑ) is called the analytic part of the Laurent's series, while  consisting of negative integral powers of (z - ɑ) is called the principal part of the Laurent's series.

3. As the Taylor's and Laurent's expansions in the given region are unique, they are not usually found by the theorems given, but by the other simpler methods such as use of binomial series.