How do I integrate this?

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.

#CalculusDoubts #ContourIntegrals

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

User 123 - 6 years, 1 month ago

After 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?

vishnu c - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$