Still having trouble solving integrals via residues

[saybrook1]

Homework Statement

Hi guys, I don't quite understand how to solve closed integrals "around a given circle." I was given $$\oint\frac{dz}{sinz}$$ around $$|z-6|=4$$ and said that the integral is equal to zero because the singularity $$n\pi i$$ is not within the circle. Is this correct? Also I need to solve $$\oint \frac{sinz}{sinh^{2}z}dz$$ around $$|z|=3$$. In this case do I just need to solve for the residues of the singularities which are inside of this circle? Or am I going about this completely wrong? Any help would be greatly appreciated.

Homework Equations

All are given above.

The Attempt at a Solution

I have attempted to solve the second integral by expanding sinhz and looking for the residues of this function within the given circle. I'm not sure if I've done that right or not. Thanks in advance for any help or a point in the right direction; if anyone knows where I could find some similar examples that would also be awesome.

 

[saybrook1]

It seems like the second integral will also be zero since the singularity is n*pi*i which is also out of the circle for any value of n other than zero? What am I missing here?

 

[vela]

Homework Statement

Hi guys, I don't quite understand how to solve closed integrals "around a given circle." I was given $$\oint\frac{dz}{\sin z}$$ around $$|z-6|=4$$ and said that the integral is equal to zero because the singularity $$n\pi i$$ is not within the circle. Is this correct?

Why do you think the singularities are on the imaginary axis?

Also I need to solve $$\oint \frac{\sin z}{\sinh^{2}z}dz$$ around $$|z|=3$$. In this case do I just need to solve for the residues of the singularities which are inside of this circle? Or am I going about this completely wrong? Any help would be greatly appreciated.

Yes, you only care about the poles inside of the contour.

 

[saybrook1]

Why do you think the singularities are on the imaginary axis?


Yes, you only care about the poles inside of the contour.

Okay right, so the singularities in the first case are actually n*pi and there are three of them within that circle. Thanks a bunch, I've got that problem down now. For the second problem the singularities are also n*pi but they are second order. So I can either do kind of a nasty derivative or try and expand sinz in powers of (1-n*pi)? Any idea about how I would go about this expansion in order to find the necessary residues? Thanks a ton!

 

[saybrook1]

Okay right, so the singularities in the first case are actually n*pi and there are three of them within that circle. Thanks a bunch, I've got that problem down now. For the second problem the singularities are also n*pi but they are second order. So I can either do kind of a nasty derivative or try and expand sinz in powers of (1-n*pi)? Any idea about how I would go about this expansion in order to find the necessary residues? Thanks a ton!

Also for some reason, wolfram tells me that the poles of 1/(sinz) are 2*n*pi and 2*n*pi +pi. And for 1/(sinh^2(z)) it gives poles at 2*n*pi*i and 2*n*pi*i +i*pi. Are these correct? Sorry for all of the questions and thanks again.

 

[vela]

You need to be able to figure out where the poles are. What's the condition for the integrand to have a pole at some point? When is it met for this particular integrand?

 

[saybrook1]

The denominator must equal zero at that point.

 

[saybrook1]

I'm just confused because wolfram tells me that the zero's of sinhz are n*pi*i. I'm not sure as to why when I ask for the poles it adds a factor of 2 and a phase. Shouldn't that be irrelevant? The factor of 2 still makes it go to zero. I think I will just stick with saying the poles for 1/sinhz are at n*pi*i and the poles for 1/sinz are at n*pi.