# Singularities of a complex function

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1. Dec 2, 2015

### Mattbringssoda

1. The problem statement, all variables and given/known data

Find and classify all singularities for (e-z) / [(z3) ((z2) + 1)]

2. Relevant equations

3. The attempt at a solution

This is my first attempt at these questions and have only been given very basic examples, but here's my best go:

I see we have singularities at 0 and i.

The 0 corresponds with z3, so upon inspection it's a third order pole.

To determine the order for the i singularity, I multiply the function by (z - i) and use L'Hospital's rule

Lim (z -> i ) of [ (z-i) (e-z) ] / [ (z3) ((z2) + 1) ]

= Lim (z -> i) of [ (z-i) (-e-z) + (e-z) ] / [ (5z4) + 3z2 ]

= (e^-i) / 2

Which is a finite number, and since I used the first order term (1-i), this is indeed a first order pole, according to what I've been taught.

I'm worried that I'm misunderstanding how to use L'Hospital here and was hoping I could get a second set of eyes from someone familiar with these problems.

Thanks!

2. Dec 2, 2015

### Svein

You missed one. Remember that (z2+1)=(z+i)(z-i).
You should slow down a bit - multiply by (z-i) is correct, but then you have (z-i) in both nominator and denominator...

3. Dec 3, 2015

### HallsofIvy

Staff Emeritus
Writing $\frac{e^{-z}}{z^3(z^2+ 1)}$ as $\frac{e^{-z}}{z^2(z- i)(z+i)}$ it should be immediately obvious that z= 0 is pole of order three and that z= i and z= -i are poles of order one.

4. Dec 4, 2015

### Mattbringssoda

Thank you two very much; it all stemmed down to a simple oversight that I wasn't catching from the (z^2 + 1) term. Thanks again!