We will cover following topics

Power Series

A power series centered at a complex number $z_{0}$ is an expression of the form $\sum_{n = 0}^{\infty} a_{n} {(z - z_{0})}^{n}$ , where $a_{n}$ can be complex numbers.

Singularities

Singularity of a function of the complex variable $z$ is a point at which the function is not analytic.

Removable Singularity: It is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.

Entire Function

A function that is analytic everywhere in the finite plane [i.e., everywhere except at $\infty$ ] is called an entire function or integral function. The functions $e^{z}$ , $\sin z$ , $\cos z$ are entire functions.

An entire function can be represented by a Taylor series that has an infinite radius of convergence. Conversely, if a power series has an infinite radius of convergence, it represents an entire function.

Note that by Liouville’s theorem, a function which is analytic everywhere, including $\infty$ , must be a constant.

Taylor’s Series

Taylor’s Theorem:Suppose $f (z)$ is an analytic function in a region $A$ and let $z_{0} \in A$ , then, $f (z) = \sum_{n = 0}^{\infty} a_{n} {(z - z_{0})}^{n}$ , where the series converges on any disk $| z - z_{0} | < r$ contained in $A$ . Also, $a_{n}$ can be calculated as follows:

a_{n} = \frac{f^{(n)} (z_{0})}{n!} = \frac{1}{2 π i} \int_{γ} \frac{f (z)}{{(z - z_{0})}^{n + 1}} d z

where $γ$ is any simple closed curve in $A$ around $z_{0}$ , with its interior entirely in $A$ .

The above series is called the Taylor’s series representing $f$ around $z_{0}$ .

Laurent’s Series

Suppose $f (z)$ is analytic on the annulus $A : r_{1} < | z - z_{0} | < r_{2}$ . Then, $f (z)$ can be expressed as the below series:

f (z) = \sum_{n = 1}^{\infty} \frac{b_{n}}{(z - z_{0})^{n}} + \sum_{n = 0}^{\infty} a_{n} {(z - z_{0})}^{n}

where

a_{n} = \frac{1}{2 π i} \int_{γ} \frac{f (w)}{{(w - z_{0})}^{n + 1}} d w

b_{n} = \frac{1}{2 π i} \int_{γ} f (w) {(w - z_{0})}^{n - 1} d w

and

$γ$ is any circle $| w - z_{0} | = r$ inside the annulus, i.e., $r_{1} < r < r_{2}$ .

The series $\sum_{n = 0}^{\infty} a_{n} {(z - z_{0})}^{n}$ converges to an analytic function for $| z - z_{0} | < r_{2}$ and the series $\sum_{n = 1}^{\infty} \frac{b_{n}}{{(z - z_{0})}^{n}}$ converges to an analytic function for $| z - z_{0} | > r_{1}$ .

Together, both the series converge on the annulus $A$ where $f$ is analytic.

PYQs

Taylor’s Series

1) Determine all entire functions $f (z)$ such that 0 is a removable singularity of $f (\frac{1}{z})$ .

[2017, 10M]

2) For a function $f : C \to C$ and $n \geq 1$ , let $f^{(n)}$ denotes the $n^{t h}$ derivative of $f$ and $f^{(0)} = f$ . Let $f$ be an entire function such that for some $n \geq 1$ , $f^{(n)} (\frac{1}{k}) = 0$ for all $k = 1$ , 2, 3, show that $f$ is a polynomial.

[2017, 15M]

3) Prove that every power series represents an analytic function inside its circle of convergence.

[2016, 20M]

4) If the function $f (z)$ is analytic and one valued in $| z - a | < R$ , prove that for $0 < r < R$ , $f^{'} (a) = \frac{1}{π r} \int_{0}^{2 π} P (θ) e^{- i θ} d θ$ , where $P (θ)$ is the real part of $f (a + r e^{i θ})$ .

[2011, 15M]

5) Let $f (z)$ be an entire function satisfying $| f (z) | \leq | z |$ for some positive constant $k$ and all $z$ . Show that $f (z) = a z^{3}$ for some constant $a$ .

[2008, 15M]

Laurent’s Series

1) Obtain the first terms of the Laurent series expansion of the function $f (z) = \frac{1}{(e^{z} - 1)}$ about the point $z = 0$ valid in the region $0 < | z | < 2 π$ .

[2010, 10M]

2) Find the Laurent’s series which represent the function $\frac{1}{(1 + z^{2}) (z + 2)}$ when

(i) $| z | < 1$ (ii) $1 < | z | < 2$ (iii) $| z | > 2$

[2018, 15M]

3) Find all possible Taylor’s and Laurent’s series expansions of the function $f (z) = \frac{2 z - 3}{z^{2} - 3 z + 2}$ about the point $z = 0$ .

[2015, 20M]

4) Expand in Laurent series the function $f (z) = \frac{1}{z^{2} (z - 1)}$ about $z = 0$ and $z = 1$ .

[2014, 10M]

5) Expand the function $f (z) = \frac{1}{(z + 1) (z + 3)}$ in Laurent series valid for:
i) $1 < | z | < 3$
ii) $| z | > 3$
iii) $0 < | z + 1 | < 2$
iv) $| z | < 1$

[2012, 15M]

6) Find the Laurent series for the function $f (z) = \frac{1}{1 - z^{2}}$ with centre $z = 1$ .

[2011, 15M]

7) Find the Laurent series of the function $f (z) = \exp [\frac{λ}{2} (z - \frac{1}{z})]$ as $\sum_{n = - \infty}^{\infty} C_{n} z^{n}$ for $0 < | z | < \infty$ , where $C_{n} = \frac{1}{π} \int_{0}^{π} \cos (n ϕ - λ \sin ϕ) d ϕ$ , $n = 0$ , $\pm 1$ , $\pm 2$ , $\dots$ with $λ$ a given complex number and taking the unit circle $C$ given by $z = e^{i ϕ} (- π \leq ϕ \leq π)$ as contour in this region.

[2010, 15M]

8) Expand $f (z) = \frac{1}{(z + 1) (z + 3)}$ in Laurent’s Series which is valid for:
i) $1 < | z | < 3$ ,
ii) $| z | < 3$ and
iii) $| z | < 1$

[2005, 30M]

9) Show that, when $0 < | z - 1 | < 2$ , the function $(z) = \frac{z}{(z - 1) (z - 3)}$ has the Laurent series expansion in powers of $(z - 1)$ as $\frac{- 1}{2 (z - 1)} - 3 \sum_{n = 0}^{\infty} \frac{(z - 1)^{n}}{2^{n + 2}}$ .

[2002, 15M]

10) Find the Laurent series for the function $e^{1 / z}$ in $0 < z < \infty$ . Using this expansion, show that $\frac{1}{π} \int_{0}^{π} \exp (\cos θ) \cos (\sin θ - n θ) d θ = \frac{1}{n!}$ for $n = 1$ , 2, 3, $\dots$

[2001, 15M]

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