Power Series
We will cover following topics
Power Series
A power series centered at a complex number \(z_0\) is an expression of the form \(\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}\), 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, \(f(z)=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}\), where the series converges on any disk \(\vert z-z_{0} \vert < r\) contained in \(A\). Also, \(a_n\) can be calculated as follows:
\[a_{n}=\dfrac{f^{(n)}\left(z_{0}\right)}{n !}=\dfrac{1}{2 \pi i} \int_{\gamma} \dfrac{f(z)}{\left(z - z_{0}\right)^{n+1}} d z\]where \(\gamma\) 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 \(A:r_1 < \vert z-z_0 \vert < r_2\). Then, \(f(z)\) can be expressed as the below series:
\[f(z)=\sum_{n=1}^{\infty} \dfrac{b_{n}}{(z-z_{0})^{n}}+\sum_{n=0}^{\infty} a_{n}\left(z - z_{0}\right)^{n}\]where
\[a_{n} =\dfrac{1}{2 \pi i} \int_{\gamma} \dfrac{f(w)}{\left(w - z_{0}\right)^{n+1}} d w\] \[b_{n} =\dfrac{1}{2 \pi i} \int_{\gamma} f(w)\left(w - z_{0}\right)^{n-1} d w\]and
\(\gamma\) is any circle \(\vert w-z_0 \vert=r\) inside the annulus, i.e., \(r_1 < r < r_2\).
The series \(\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}\) converges to an analytic function for \(\vert z-z_0 \vert < r_2\) and the series \(\sum_{n=1}^{\infty} \dfrac{b_{n}}{\left(z - z_{0}\right)^{n}}\) converges to an analytic function for \(\vert z-z_0 \vert > r_1\).
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 \(f\left(\dfrac{1}{z}\right)\).
[2017, 10M]
2) For a function \(f: C \rightarrow C\) and \(n \geq 1\), let \(f^{(n)}\) denotes the \(n^{th}\) derivative of \(f\) and \(f^{(0)}=f\). Let \(f\) be an entire function such that for some \(n \geq 1\), \(f^{(n)}\left(\dfrac{1}{k}\right)=0\) 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 \(\vert z-a \vert < R\), prove that for \(0< r < R\), \(f^{\prime}(a)=\dfrac{1}{\pi r} \int_{0}^{2 \pi} \mathrm{P}(\theta) e^{-i \theta} d \theta\), where \(P(\theta)\) is the real part of \(f\left(a+r e^{i \theta}\right)\).
[2011, 15M]
5) Let \(f(z)\) be an entire function satisfying \(\vert f(z) \vert \leq \vert z \vert\) for some positive constant \(k\) and all \(z\). Show that \(f(z)=az^3\) for some constant \(a\).
[2008, 15M]
Laurent’s Series
1) Obtain the first terms of the Laurent series expansion of the function \(f(z)=\dfrac{1}{(e^z-1)}\) about the point \(z=0\) valid in the region \(0< \vert z \vert<2 \pi\).
[2010, 10M]
2) Find the Laurent’s series which represent the function \(\dfrac{1}{(1+z^2)(z+2)}\) when
(i) \(\vert z \vert <1\) (ii) \(1< \vert z \vert <2\) (iii) \(\vert z \vert >2\)
[2018, 15M]
3) Find all possible Taylor’s and Laurent’s series expansions of the function \(f(z)=\dfrac{2z - 3}{z^{2}-3 z+2}\) about the point \(z=0\).
[2015, 20M]
4) Expand in Laurent series the function \(f(z)=\dfrac{1}{z^{2}(z-1)}\) about \(z=0\) and \(z=1\).
[2014, 10M]
5) Expand the function \(f(z)=\dfrac{1}{(z+1)(z+3)}\) in Laurent series valid for:
i) \(\quad 1 < \vert z \vert < 3\)
ii) \(\vert z \vert>3\)
iii) \(0< \vert z+1 \vert < 2\)
iv) \(\vert z \vert < 1\)
[2012, 15M]
6) Find the Laurent series for the function \(f(z)=\dfrac{1}{1-z^{2}}\) with centre \(z=1\).
[2011, 15M]
7) Find the Laurent series of the function \(f(z)=\exp \left[\dfrac{\lambda}{2}\left(z-\dfrac{1}{z}\right)\right]\) as \(\sum_{n=-\infty}^{\infty} C_{n} z^{n}\) for \(0< \vert z \vert<\infty\), where \(C_{n}=\frac{1}{\pi} \int_{0}^{\pi} \cos (n \phi-\lambda \sin \phi) d \phi\), \(n=0\), \(\pm 1\), \(\pm 2\), \(\ldots\) with \(\lambda\) a given complex number and taking the unit circle \(C\) given by \(z=e^{i \phi}(-\pi \leq \phi \leq \pi)\) as contour in this region.
[2010, 15M]
8) Expand \(f(z)= \dfrac{1}{(z+1)(z+3)}\) in Laurent’s Series which is valid for:
i) \(1 < \vert z \vert < 3\),
ii) \(\vert z \vert < 3\) and
iii) \(\vert z \vert < 1\)
[2005, 30M]
9) Show that, when \(0< \vert z-1 \vert < 2\), the function \((z)=\dfrac{z}{(z-1)(z-3)}\) has the Laurent series expansion in powers of \((z-1)\) as \(\dfrac{-1}{2(z-1)}-3 \sum_{n=0}^{\infty} \dfrac{(z-1)^{n}}{2^{n+2}}\).
[2002, 15M]
10) Find the Laurent series for the function \(e^{1 / z}\) in \(0< z < \infty\). Using this expansion, show that \(\dfrac{1}{\pi} \int_{0}^{\pi} \exp (\cos \theta) \cos (\sin \theta-n \theta) d \theta=\dfrac{1}{n !}\) for \(n=1\), 2, 3, \(\ldots\)
[2001, 15M]