I'm busy with complex analysis and once again i'm having some trouble. Any help would be appreciated.I apologize in advance for the lengthiness of the question I'm just really uncertain about this and would like to understand where my mistakes are.Thanks.
The question is:
Locate and classify all the singularities of the function [sinz/(z^2-pi^2)]. Then compute the residues at each of these singularities.
The only way I know how to do this is using Laurent expansion. I know that the function has isolated singularities at z=pi and z=-pi. I also know the expansion for sinz is sinz=z-z^3/3!+z^5/5!-z^7/7!+.... now for the singularity z=pi we want all terms expressed in terms of (z-pi) so sin(z-pi+pi)= -sin(z-pi)= -[(z-pi)-(z-pi)^3/3!+(z-pi)^5/5!-....] and finding the laurent series for 1/(z^2-pi^2)=1/(z-pi)(z+pi) with 0<lz-pil<2pi we end up with 1/(z-pi)(z+pi) = 1/2pi(z-pi)-1/(2pi)^2+(z-pi)/(2pi)^3-...… we then multiply the series together looking for terms with (z-pi) in the denominator which requires multiplying the first term of 1/(z-pi)(z+pi) by -[(z-pi)-(z-pi)^3/3!+(z-pi)^5/5!-....] and we see that no term will have (z-pi) in the denominator hence this singularity is a removable singularity and the residue=0. The same reasoning applies to z=-pi and we get the same answer except with (z+pi) replacing (z-pi).
I'd like to know if this is correct and if not where I went wrong? If it is correct I'd like help with the following question:
Give an example of a function with pole of order 5 at (1-2i), an essential singularity at 1, and a removable singularity at 0.Justify all assertions.
for a function with pole of order 5 at (1-2i) we need a function with the form: a/(1-2i)+b/(1-2i^2)+...+e/(1-2i)^5 so can we choose a to e arbitrarily then add together to get a single term with (1-2i)^5 as the denominator? for an essential singularity at 1 we need a funtion with the form: a/(z-1)+b/(z-1)^2+..... with infinite terms so can we just choose exp^[1/(z-1)]? and for a removable singularity at 0 we need a function such that no term has any power of z in the denominator so like (1-cosz)/z^2? and then do you just add the three functions together to get 1 function with all three singularities or what?
Thanks in advance for all help.