Purchase Solution

Singular Point : Pole and Residue

Not what you're looking for?

Ask Custom Question

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 see attachment for functions}

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

Attachments
Purchase this Solution

Solution Summary

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

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 ...

Purchase this Solution


Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Probability Quiz

Some questions on probability

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.