A First Course in Complex Analysis · Problem 5

Isolated Singularities and the Residue Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · power-series

Problem

Find the terms through third order and the radius of convergence of the power series for each of the following functions, centered at

z_0

. (Do not find the general form for the coefficients.) (a)

f(z) = \dfrac{1}{1+z^2}

z_0 = 1

(b)

f(z) = \dfrac{1}{\exp(z)+1}

z_0 = 0

(c)

f(z) = (1+z)^{\frac{1}{2}}

z_0 = 0

(d)

f(z) = \exp(z^2)

z_0 = i