Calculating the residue of complicated expression

[VVS]

Homework Statement

Hi, I want to calculate the residue of this expression:

Homework Equations

I know that the residue of a function with a pole of k-th order is given by this:

The Attempt at a Solution

I know that the function has infinite number of poles at k*∏, for k=-∞ to k=+∞
I think one has to expand the sine function or something like that but I don't know how it simplifies things.

 

[Office_Shredder]

Well the obvious first question is what is the order of the poles so you can use your formula?

 

[Ray Vickson]

Homework Statement

Hi, I want to calculate the residue of this expression:
View attachment 59634

Homework Equations

I know that the residue of a function with a pole of k-th order is given by this:
View attachment 59635

The Attempt at a Solution

I know that the function has infinite number of poles at k*∏, for k=-∞ to k=+∞
I think one has to expand the sine function or something like that but I don't know how it simplifies things.

This function does not have just ONE residue, so it is wrong to talk about THE residue. It has residues at lots of different points. What, exactly, do you want?

 

[VVS]

I'd say it is of third order. Do I need to expand sin(z) around pi?

 

[VVS]

This function does not have just ONE residue, so it is wrong to talk about THE residue. It has residues at lots of different points. What, exactly, do you want?

I know the answer, but I can't work it out myself. We want an expression for the residues as a function of k, since we have poles at multiples k*pi.

 

[Ray Vickson]

I know the answer, but I can't work it out myself. We want an expression for the residues as a function of k, since we have poles at multiples k*pi.

Look first at the case k = 0. What do you think you should do? (You really should work this out for yourself; that is the only way to learn!)

Then look at k = ±1, ±2, etc.; these give different results from the k = 0 case. Do you see why?

 

[jackmell]

Homework Statement

Hi, I want to calculate the residue of this expression:
View attachment 59634

Homework Equations

I know that the residue of a function with a pole of k-th order is given by this:
View attachment 59635

The Attempt at a Solution

I know that the function has infinite number of poles at k*∏, for k=-∞ to k=+∞
I think one has to expand the sine function or something like that but I don't know how it simplifies things.

You got them k's in there representing two different things, one of the worst things you can do in math. The formula for the residue is in terms of the order of the pole given as k. Me, I'd change the pi thing to $n\pi$. Ok, got that straight. Now what is the order of the pole? How you know it's third order or k=3? Do that later. Let's assume it is for $n\neq 0$, then by your formula, we could write:

$$\text{Res}\left(f(z),n\pi\right)= \lim_{z\to n\pi} \frac{1}{2}\frac{d^2}{dz^2}\left\{(z-n\pi)^2 f(z)\right\},\quad n=\pm 1, \pm 2,\cdots$$

We can't take that limit?