I came across this problem today:

\(Verify\quad for\quad u(x,y)=e^{ x }sin(y)\quad the\quad mean\\ value\quad theorem\quad for\quad harmonic\quad functions\\ on\quad a\quad circle\quad C\quad of\quad radius\quad r=1,\quad with\quad its\\ centre\quad at\quad z=2+2i.\)

I tried to simplify it but I got stuck at the integral of \(cosh(e^{i\theta})\). So my question is : how do I integrate \(cosh(e^{i\theta})\)?

I know that it is somehow related to \(Chi(e^{i\theta})\), but I don't know how.

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TopNewestAfter some simplification I was able to verify, by integration, that it is true for the given function. But the question still stands: How is it related to \(Chi(e^{i\theta})\)? I was able to solve the case where the function had limits from 0 to 2*pi, i.e, I had to use some properties of definite integrals to simplify it. But is it possible to evaluate it with a general limit?

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@Sandeep Bhardwaj Sir, @Raghav Vaidyanathan @Shashwat Shukla @Pranjal Jain @Abhishek Sinha Sir\[\] Please help him. Thanks a lot!\[\] @vishnu c

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