Explore BrainMass

# Singular Point : Pole and Residue

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

2. Show that the singular point of each of the following functions is a pole. Determine the order m of that pole and the corresponding residue B.

Please specify the terms that you use if necessary and clearly explain each step of your solution.

https://brainmass.com/math/graphs-and-functions/singular-point-pole-residue-36385

#### Solution Preview

In general a pole (z0) of f(z) is of order m, if for the first whole number m the following limit exists:

lim (z-z0)^m*f(z)
z-->z0

Naturally if this limit exists, z0 is a pole.

(a)
The singularity is at z=0, so z0=0. Now we want to find m such that:

lim z^m*(1-cosh z)/z^3
z--->0

exists. m=0:

lim (1-cosh z)/z^3 ---> lim -sinh z/(3z^2) ---> lim -cosh z/(6z) ---> infinity
z--->0 z--->0 z--->0

so the order is not 0. Now m=1:

lim z(1-cosh z)/z^3 = lim (1-cosh z)/z^2 ---> lim -sinh ...

#### Solution Summary

Poles and residues are investigated. The functions for singular points are given.

\$2.49