- #1

MatinSAR

- 597

- 182

- Homework Statement
- Calculate laurent series of following function.

- Relevant Equations
- series expansion formulas.

1. ##f(z)=\dfrac {\sin z}{z- \pi}## at ##z=\pi## : $$ \dfrac {\sin z}{z- \pi}=\dfrac {\sin(\pi +z- \pi)}{z- \pi}=\dfrac {- \sin(z- \pi)}{z- \pi}=\dfrac {-1}{z- \pi} \sum_{n=0}^\infty \dfrac {(-1)^n (z- \pi)^{2n}}{(2n+1)!}$$My answer has extra ##\dfrac {-1}{z- \pi} ## according to a calculator. Am I wrong?

2. Find the residue of ##f(z)=\dfrac {\sin z} {z^4}## at ##z=0## :

I wrote laurent series ot this. It doesn't have ##1/z##. I think the residue is ##0##. Is it it true?

3. Calculate the residue of the ##f(z)=z^2 \sin \dfrac {1}{z+1}## at its singular points.

I'm not sure what should I do. Should I expand both ##z^2## and ##\sin \dfrac {1}{z+1}## at ##z=-1##?

Any help would be appreciated.

2. Find the residue of ##f(z)=\dfrac {\sin z} {z^4}## at ##z=0## :

I wrote laurent series ot this. It doesn't have ##1/z##. I think the residue is ##0##. Is it it true?

3. Calculate the residue of the ##f(z)=z^2 \sin \dfrac {1}{z+1}## at its singular points.

I'm not sure what should I do. Should I expand both ##z^2## and ##\sin \dfrac {1}{z+1}## at ##z=-1##?

Any help would be appreciated.