Find the residues of the function

[DanniHuang]

Homework Statement

Find the residues of the function

Homework Equations

f(z)=\frac{1}{sin(z)} at z=0, \frac{∏}{2}, ∏

The Attempt at a Solution

Since the function has a simple pole at z=0

I used: Res(f,0)=lim_{z->0}(z-0)\cdot\frac{1}{sin(z)}=1. This means the residue of the function at z=0 is 1.

And Res(f,\frac{∏}{2})=lim_{z->∏/2}(z-\frac{∏}{2})\cdot\frac{1}{sin(z)}=0. This means the residue of the function at z=\frac{∏}{2} is 0.

However I think this method can not be applied on solving z=∏. How can I work it out?

 

[vela]

Try expanding sin z in a series about $z=\pi$. Or, better yet, use $\sin z = \sin[(z-\pi)+\pi]$. The basic idea is to get everything in terms of $z-\pi$.

 

[DanniHuang]

Try expanding sin z in a series about $z=\pi$. Or, better yet, use $\sin z = \sin[(z-\pi)+\pi]$. The basic idea is to get everything in terms of $z-\pi$.

If I turn everything into z-∏, do we need to use the same method as I used to work out z=0 and z=\frac{∏}{2}?
But how? Turn the limit to z-∏→0?
Because I used computer to work out the answer of z=∏ which is -1. But I still cannot get this answer by myself.

 

[vela]

You find the residue by evaluating the limit
$$\lim_{z \to \pi}\ (z-\pi)\frac{1}{\sin z}.$$ Rewriting the limit in terms of $z-\pi$ is simply to make the evaluation easier. I just realized you could simply apply L'Hopital's rule to do that and avoid unnecessary complications. If you had a more complicated function, however, writing things in terms of $z-\pi$ is often less work than using L'Hopital's rule.

But say you did it with using the trig identity anyway. You should end up with
$$\lim_{z\to\pi} \frac{z-\pi}{-\sin(z-\pi)}.$$ You might already recognize that limit, but it not, use the substitution $w=z-\pi$ to turn it into one you should recognize.