Does f(z) have a singularity at infinity and how can its residue be obtained?

[abode_x]

does f(z)=\frac{ze^{iz}}{z^2+a^2} have a singularity at infinity?

if so, how do i get the residue there?

 

[quasar987]

Does f(w):=f(1/z) has a singularity at w=0 is what you must ask yourself.

Btw - you're really not in the right forum.

 

[abode_x]

okay so i transform z -> 1/w then take lim w-> 0... if it blows up then i do have a singularity... how do i get lim w->0 of exp(i/w) ?

well first, i think i need l'hopitals (for the whole function). then, can i use the fact that when taking a limit it can be approached along any line on the Z-plane? i.e. use the path along i-axis ?

i think the conclusion will be that it blows up. three follow up questions. 1. how do i get the residue at infinity? 2. what is the conclusion in a case wherein the limit does not exist? 3. Is the singularity at infinity and/or its residue useful? (i mean i know the finite singularities are useful in integration, does this arise in some physical theory?)

i am very sorry for posting in the wrong forum.. thanks for all the help