# Complex analysis

## Homework Statement

Hey guys.
So, I need to calculate this integral, I uploaded what I tried to do in the pic.
But according to them, this is not the right answer, according to them, the right answer is the one I marked in red at the bottom.
Any idea where this Sin came from?

Thanks.

## The Attempt at a Solution

#### Attachments

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lurflurf
Homework Helper
One way sin may appear is from combining exponetials
sin(z)=[exp(iz)-exp(-iz)]/(2i)

Try this
integrate around a pie slice
one edge
z=t t in (0,R)
one edge
z=t*exp(pi*i/n) t in (0,R)
the rounded edge
z=R*exp(pi*i*t/n) t in (0,1)
let R become large
express the integral of the pie slice in terms of I then as 2pi*i*Res(f,exp(pi*i/(2n))

can you explain to me how you got that term to drop out to 0, when you integrated round the contour from R to -R? did you find a bound for the integrand?

can you explain to me how you got that term to drop out to 0, when you integrated round the contour from R to -R? did you find a bound for the integrand?
When R goes to infinity, the integral of the semi circle goes to zero.

yes. but don't you need to find a bound, say M, such that the integrand is less than or equal to M at all points on the semi circle and then say as R goes to infinity the bound goes to 0.
my question was, what did you use to bound the intergal?

yes. but don't you need to find a bound, say M, such that the integrand is less than or equal to M at all points on the semi circle and then say as R goes to infinity the bound goes to 0.
my question was, what did you use to bound the intergal?
Oh, I thought it's right for any semi circle integral so I didn't try to prove it.
Now I'm not sure.

lurflurf
Homework Helper
can you explain to me how you got that term to drop out to 0, when you integrated round the contour from R to -R? did you find a bound for the integrand?
So for the curved edge
recall n>=3
|integral(curved edge)|<|R^4/(R^(2n)+1)|(2pi/n)*R<|R^5/R^(2n)|C<=C/R->0
The same bound holds for a semicircle (times n)

***It is easier to do a wedge than a semi circle as one avoids the sum of n residues only one appears.***

Last edited:
ok. how can you assume $n \geq 3$

lurflurf
Homework Helper
ok. how can you assume $n \geq 3$
It is given in the problem statement.