How to find the Laurent series about #z=0# and therefore the residue at #z=0# of #f(z) = 1/(z^4 sin(pi z))#, where #f(z)# is a complex valued function?
So I just had a complex analysis exam, went pretty well but I was agonising over this question for quite a few pages of scribble, if someone could put me out of my misery and show me how it should be done that'd be great.
So I just had a complex analysis exam, went pretty well but I was agonising over this question for quite a few pages of scribble, if someone could put me out of my misery and show me how it should be done that'd be great.
1 Answer
# f(z)=1/(piz^5) + (pi)/(6z^3) + (7pi^3)/(360z) + (31pi^5z)/15120 + O(z^3)#
# Res{f(z)}_(z=0) = (7pi^3)/(360) #
Explanation:
Typically in an exam you would need to derive the power series for
# csc x = 1/x + x/6 + (87x^3)/360 + (31x^5)/15120 + ...#
And so the Laurent series for
# f(z)=1/z^4 \ csc(piz)#
# \ \ \ \ \ \ \ =1/z^4 {1/(piz) + (piz)/6 + (7(piz)^3)/360 + (31(piz)^5)/15120 + O(z^7)}#
# \ \ \ \ \ \ \ =1/(piz^5) + (pi)/(6z^3) + (7pi^3)/(360z) + (31pi^5z)/15120 + O(z^3)#
So then;
# Res{f(z)}_(z=0) = (7pi^3)/(360) #