complex analysis

[Anonymous No.6100442]

Good afternoon /sci/

does any of you have a good ressource for learning about poles, singularities, etc because i'm having a hard time with it. Not a wikipedia kind, but something with examples etc maybe from a textbook?

thank you

[Anonymous No.6100457]

I found the book we use to explain 'em pretty well
>Fundamentals of complex analysis - Saff, Snider

What are you having problems with?

[Anonymous No.6100473]

I think this is absolutely wonderful
http://rutherglen.science.mq.edu.au/wchen/lnicafolder/lnica.html
The presentation is very pedestrian and, at least the first 10 or so chapters, are fully rigorous. You can find notes on more courses on his homepage too. http://rutherglen.science.mq.edu.au/wchen/

[Anonymous No.6100509]

>>6100457
>> 6100473

thx guys i will look it. This is a typical question on midterms :
find singularities and its type. if it is a pole give its order and justify. (pic attached)

[Anonymous No.6100537]

>>6100509
I'd recommend sitting down with a few easy functions you know the singularities of, doing a series expansion and having a look at that. You get a pretty clear picture that way.

In any case, a point z_0 is a pole of order m to f(z) if f can be rewritten as

<span class="math">f(z) = \frac{g(z)}{(z-z_0)^m}[/spoiler]

<span class="math">g(z_0) != 0[/spoiler] and g(z) is analytic at z_0


In your example z = i is a pole of order 5, think of the rest of the function as g(z). Likewise it has a pole of order 2 at z = 0.

For the cosine I believe a series expansion is the only way to go about it. and the square root I honestly couldn't say, we didn't cover it at all.

[Anonymous No.6100542]

>>6100537
z = -i*

[Anonymous No.6100831]

I have the countour integral of e^(iz)/z^3 around |z|=2. I have a pole of order 2 at z=0 right (by doing the limit lim z->0 z^2*e^(iz)/z^3) and the residue is i-1. However the answer is -pi*i. What did i do wrong?

[Anonymous No.6100847]

>>6100831
Cauchy's integral formulas.

[Anonymous No.6100909]

>>6100831
you fucked up the residue

[Anonymous No.6102132]

>>6100442
Ah, I see your picture fails to include Jacob Barnett! What kind of mistake is that?