Answer to Question #93683 in Complex Analysis for Edward

Question #93683
Determine the residue of e^zt/(z-2)^3
1
Expert's answer
2019-09-03T13:29:05-0400

The function"f(z)=e^zt\/(z-2)^3" has only two special points: 2, "\\infin" (f has a singularity at z = 2, z="\\infin" only). Therefore "Res(2,f)=-Res(\\infin,f)".

"Res(2,f)=c_{-1} -\\text{ Laurent's coefficient}."

"e^z=e^2+\\frac{e^2}{1!}(z-2)+\\frac{e^2}{2!}(z-2)^2+..."

Therefore Laurent's series for f is

"f(z)=t(\u2026 +\\frac{e^2}{2!}(z-2)^{-1}+...)".

Then

"Res(2,f)=t\\frac{e^2}{2}, Res(\\infin,f)=-t\\frac{e^2}{2}."

Finally, if z is a non singularity point for f, then "Res(z,f)=0."


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