Compute using residuals the integral of the following function over the positively oriented circle |z|=3, f(z)=(e^-z)/(z^2)

Rubi Garner 2022-10-21 Answered
Compute using residuals the integral of the following function over the positively oriented circle | z | = 3
f   ( z ) = e z z 2
My solution: The only singular point of f in | z | 3 is z = 0 (double pole) and its remainder is therefore
Res z = 0 f ( z ) = lim z 0 1 ( 2 1 ) ! ( e z z 2 z 2 ) = lim z 0 e z = 1
Consequently, | z = 3 | f ( z ) = 2 π i Res z = 0 f ( z ) = 2 π i .
this right?
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Answers (1)

RamPatWeese2w
Answered 2022-10-22 Author has 15 answers
That is correct, although it is simpler, in order to compute the residue, to note that
e z z 2 = 1 z + 1 2 ! z 2 1 3 ! z 3 + z 2 = z 2 z 1 + 1 2 ! 1 3 ! z +
and that therefore, by definition,
res z = 0 ( e z z 2 ) = 1.
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