What is the value of this complex line integral?

[burak100]

find the value,

$\int\limits_{0}^{2\Pi} e^{-\sin t} \sin\lbrace (\cos t ) - (n-1) t \rbrace dt$ ?

I have no idea...

 

[HallsofIvy]

Well, since you title this "complex line integral 2" but there is no complex line integral in the problem, did you consider converting it to one? Can you find complex function that reduces to that integrand on the unit circle in the complex plane?

 

[burak100]

integral

I can't find

$\int\limits_{0}^{2\Pi} e^{-\sin t} \sin\lbrace (\cos t ) - (n-1) t \rbrace dt$ ?

 

[jackmell]

burbak . . . remove that other one. Need to just try things and these things lead you to other things and sometimes they lead you to the solution. Tell you what, how about . . . I don't know, say e^z? What happens if I consider:

$$\oint_{|z|=1} e^z dz$$

and I let z=e^{it} and convert that all to sines and cosines? What's it look like? Close huh? One of the most important things I can tell you about succeeding in math is just get it close to start. See, that's it! Ok, say e^{iz}. What about that? What's that look like? Better? How about ze^{z}? Again, convert it all to sines and cosines. We makin' progress I think. How about z^2e^{iz}. Again, turn the crank. Then maybe e^{z}/z or e^{z}/(z^2). What's that look like? Now here's what to do. You try one or a few of theses and then report back what you found. That way it looks like you're trying and others will be motivated to help you further.

 

[phyzguy]

Interesting! From evaluating this integral numerically, it is clear that it is equal to zero if n is odd, and it is equal to:
$$\frac{2\pi (-1)^\frac{n+2}{2}}{(n-1)!}$$
if n is even. However, I don't see a way to show this. Does anyone?

 

[obafgkmrns]

The integral = 0 if n is an integer. Otherwise, you'll probably have to do it numerically. Gradshteyn & Ryzhik 3.936.2 is a near miss.

 

[phyzguy]

The integral = 0 if n is an integer. Otherwise, you'll probably have to do it numerically. Gradshteyn & Ryzhik 3.936.2 is a near miss.

It's only zero for odd integers. If you plot it, you'll see that it is clearly not zero for even integers.

 

[Redbelly98]

Moderator's note: merged two threads created by duplicate posts.