Let f(z)=\frac{e^{z}}{z^{5}} and C be the circle |z|=10 in the counter-clockwise

Answered question

2022-01-17

Let
f(z)=ezz5 and C be the circle |z|=10 in the counter-clockwise direction. What is 12πiCf(z)dz?

Answer & Explanation

nick1337

nick1337

Expert2022-01-17Added 777 answers

Step 1
As another approach to Jan van Delden’s solution, we can use the Generalized Cauchy Integral Formula. Since ez is an entire function (and thus analytic inside and on C ), we obtain
12πiCezz4+1dz=1(51)!d51dz51ez|z=0=124

star233

star233

Skilled2022-01-17Added 403 answers

Step 1
The Laurent series of f(z), valid for z0 equals:
f(Z)=k=01k!zk5
The residue at z=0 is the coefficient belonging to the term z-1; we thus need to pick k=4 and find:
Res(f(z),z=0)=14!=124
and
|z|=10f(z)dz=2πi×124=πi12
since z=0 is the single singular point inside the given countour.

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