Power Series
We will cover following topics
Power Series
A power series centered at a complex number z0 is an expression of the form ∑∞n=0an(z−z0)n, where an 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 ∞] is called an entire function or integral function. The functions ez, sinz, cosz 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 ∞, must be a constant.
Taylor’s Series
Taylor’s Theorem:Suppose f(z) is an analytic function in a region A and let z0∈A, then, f(z)=∑∞n=0an(z−z0)n, where the series converges on any disk |z−z0|<r contained in A. Also, an can be calculated as follows:
an=f(n)(z0)n!=12πi∫γf(z)(z−z0)n+1dzwhere γ is any simple closed curve in A around z0, with its interior entirely in A.
The above series is called the Taylor’s series representing f around z0.
Laurent’s Series
Suppose f(z) is analytic on the annulus A:r1<|z−z0|<r2. Then, f(z) can be expressed as the below series:
f(z)=∞∑n=1bn(z−z0)n+∞∑n=0an(z−z0)nwhere
an=12πi∫γf(w)(w−z0)n+1dwand
γ is any circle |w−z0|=r inside the annulus, i.e., r1<r<r2.
The series ∑∞n=0an(z−z0)n converges to an analytic function for |z−z0|<r2 and the series ∑∞n=1bn(z−z0)n converges to an analytic function for |z−z0|>r1.
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(1z).
[2017, 10M]
2) For a function f:C→C and n≥1, let f(n) denotes the nth derivative of f and f(0)=f. Let f be an entire function such that for some n≥1, f(n)(1k)=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)=1πr∫2π0P(θ)e−iθdθ, where P(θ) is the real part of f(a+reiθ).
[2011, 15M]
5) Let f(z) be an entire function satisfying |f(z)|≤|z| for some positive constant k and all z. Show that f(z)=az3 for some constant a.
[2008, 15M]
Laurent’s Series
1) Obtain the first terms of the Laurent series expansion of the function f(z)=1(ez−1) about the point z=0 valid in the region 0<|z|<2π.
[2010, 10M]
2) Find the Laurent’s series which represent the function 1(1+z2)(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)=2z−3z2−3z+2 about the point z=0.
[2015, 20M]
4) Expand in Laurent series the function f(z)=1z2(z−1) about z=0 and z=1.
[2014, 10M]
5) Expand the function f(z)=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)=11−z2 with centre z=1.
[2011, 15M]
7) Find the Laurent series of the function f(z)=exp[λ2(z−1z)] as ∑∞n=−∞Cnzn for 0<|z|<∞, where Cn=1π∫π0cos(nϕ−λsinϕ)dϕ, n=0, ±1, ±2, … with λ a given complex number and taking the unit circle C given by z=eiϕ(−π≤ϕ≤π) as contour in this region.
[2010, 15M]
8) Expand f(z)=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)=z(z−1)(z−3) has the Laurent series expansion in powers of (z−1) as −12(z−1)−3∑∞n=0(z−1)n2n+2.
[2002, 15M]
10) Find the Laurent series for the function e1/z in 0<z<∞. Using this expansion, show that 1π∫π0exp(cosθ)cos(sinθ−nθ)dθ=1n! for n=1, 2, 3, …
[2001, 15M]