# 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.

Note by Vishnu C
3 years, 2 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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?

- 3 years, 2 months ago

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

- 3 years, 2 months ago