Mathematical Tools for Physics

14—Complex Variables 450

14.38 Evaluate ∫∞

0

dx

1

a^3 +x^3

Ans: 2 π

√

3 / 9 a^2

14.39 Go back to problem3.45and find the branch points of the inverse sine function.

14.40 What is the Laurent series expansion of 1 /(1+z^2 )for small|z|? Again, for large|z|? What is the domain
of convergence in each case?

14.41 Examine the power series

∑∞

0 z

n!. What is its behavior as you move out from the origin along a radius

at a rational angle? That is,z=reiπp/qforpandqintegers. This result is called a natural boundary.

14.42 Evaluate the integral Eq. ( 7 ) for the casek < 0. Combine this with the result in Eq. ( 12 ) and determine
if the overall function is even or odd ink(or neither).