Determine the singularities and evaluate residuesby brydustin Tags: complexanalysis, mathematical physics 

#1
Sep1112, 11:11 PM

P: 209

1. The problem statement, all variables and given/known data
f(z) = \frac{z*exp(+i*z)}{z^2+a^2} 2. Relevant equations Res(f,z_0) = lim_z>z_0 (1/(m1)!) d^{m1}/dz^{m1} {(zz_o)^m f(z)} 3. The attempt at a solution I have no clue how to do this because I don't know how to determine the order of the pole for a function of this form. For example, I could easily do this for a function like (1/(z^2+a^2)) 



#2
Sep1212, 12:20 AM

P: 202





#3
Sep1212, 01:22 AM

P: 209

Okay... . I'm now specifically asking how to compute it for the essential singularity, when z=∞ what is the residue?




#4
Sep1212, 01:33 AM

P: 202

Determine the singularities and evaluate residues
I thought you needed to determine the singularities and their residues? I don't know if you've learned this yet, but if you have a function f(z)=g(z)/h(z) where h(z0)=0 and g(z0)!= 0 then the order of the pole at z0 is equal to the order of the zero at h(z0). The order of the zero can be found by looking at the order of the first nonvanishing coefficient in the series expansion of h(z) around z0. In this case, we have h(z)=z^2+a^2 so for example h(ia) = 0. If we try to expand in a series h(z)=c0 +c1(zia)+c2(zia)^2+... then c0=0 because h(ia) = 0 and c1=dh(z)/dz evaluated at z=ia which gives c1=2ia. That's nonvanishing so the order of the zero at h(ia) is 1. Hence there's a simple pole.
Then the residue at z=ia is given by g(ia)/h'(ia). I could be wrong because it's been a while, but good luck. 


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