Computation of complex integral over circle

Question

I am beginning Mathematica user. Please show me the syntax on how to compute the following complex integral in Mathematica:

$\int_{|z-2i|=10} \frac{dz}{z(1-e^{-5z})}$

What I really want is a subroutine that takes $f$ - my function and $c$ - my circle and compute the following complex integral. Samples in documentation force me to alter $f$ every time. But calculation insets relies on residues, so I want to have clean syntax.

Thanks a lot for your help!

The documentation here has examples for Integral along a complex line, Along a piecewise linear contour in the complex plane and also Along a circular contour in the complex plane http://reference.wolfram.com/language/ref/Integrate.html — Lotus, May 18 '17 at 11:55
Hello. Thanks for the comment. I do not think your reference is practical. Imagine my that analytic express of $f$ has 30 instances of $z$. I do not want to covert every $z$ to parametric form. What I want is to have one unified function for different function $f$ and different contour $c$. — Hedgehog, May 18 '17 at 12:06
If you're actually computing residues, why not use Residue[]? — J. M.'s missing motivation, May 18 '17 at 12:20
J.M. I already used it, but I wanted to know to compute the integral having $f$ and $c$ with Mathematica in natural and elegant way. With residues one has to know something about what is he doing. I have 2 parameters - cicle and function, how do I symblicaly integrate? Since Mathematica can compute residues, it must be able to handle this task. — Hedgehog, May 18 '17 at 12:25
Possible duplicate of Paths integrals in the complex plane. Also this can be instructive How to calculate contour integrals with Mathematica? — Artes, May 18 '17 at 14:31

score 1 · Answer 1 · edited May 17 '21 at 16:57

1

Try this:

    z = 2*I + 10*E^(I*t);
    NIntegrate[10/(z + (1 - E^(-5*z))), {t, 0, 2 \[Pi]}]
(*   2.07045 - 0.0177716 I  *)

Have fun!

edited May 17 '21 at 16:57

Community

1

answered May 18 '17 at 11:56

Alexei Boulbitch

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Thanks for your reply. However, I do not need numeric value, I need symbolic one. – Hedgehog May 18 '17 at 12:00
Then try Integrate instead, but I doubt that it will work. – Alexei Boulbitch May 18 '17 at 12:14

Computation of complex integral over circle

1 Answers1