1
vote
3answers
59 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
2
votes
1answer
47 views

Residue of a 1-form in a Riemann Surface does not depend of the chart

Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by $$ ...
1
vote
2answers
568 views

finding Laurent series for $\dfrac{1}{z(z-2)^3}$

I am trying to get the Laurent series for $\dfrac{1}{z(z-2)^3}$. I know there are poles at $z = 0$ and $z=2$, and so I am looking for expansions about the singularities. Using $\dfrac{1}{1-z} = ...
1
vote
2answers
151 views

Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?

How do I use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...