0
votes
2answers
87 views

Partial fractions for $\pi \cot(\pi z)$

I want to derive $$\pi \cot(\pi z) = \sum_{-\infty}^{\infty}\frac{1}{z-n} + \frac{1}{n}$$ without taking derivatives. I know through Mittag Leffler that $$\pi \cot(\pi z) = g(z) ...
1
vote
3answers
52 views

Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
3
votes
0answers
75 views

the partial fraction of ${\pi}\cot{\pi}z$ from the partial fraction of $\frac{\pi^{2}}{\sin^{2}{\pi}z}$

I want to deduce the equation $${\pi}\cot{\pi}z=\frac{1}{z}+ \sum_{n=1}^{\infty} \frac{2z}{z^{2}-n^{2}}$$ where the convergence is uniform on compact subsets of $\mathbb{C}-\mathbb{Z}$ from the ...
1
vote
0answers
53 views

How do you integrate the “logarithmic part”?

This entry in Wikipedia states the following theorem (http://en.wikipedia.org/wiki/Partial_fraction_decomposition#Application_to_symbolic_integration) Let $f$ and $g$ be nonzero polynomials over a ...
6
votes
4answers
183 views

Expressing $\sum_{n=-\infty}^\infty\dfrac{1}{z^3-n^3}$ in closed form

I want to express $$\sum_{n=-\infty}^\infty\dfrac{1}{z^3-n^3}$$ in closed form. I know that $$\pi z\cot(\pi z)=1+2z^2\sum_{n=1}^\infty\dfrac{1}{z^2-n^2}$$ which looks close, but I don’t know how to ...
2
votes
2answers
212 views

Partial Fractions over complex variables.

We proved the following theorem in my Complex Variables class the other day. Let $R(z)=\frac{P(z)}{Q(z)}$ be a rational function with $deg(P)<deg(Q)$ and let the roots of $Q(z)$ be $z_{i}$. Then ...
0
votes
3answers
4k views

How to find a Laurent Series for this function

How do I give a Laurent Series on various ranges of $|z|$? I need to find the Laurent series expansion for $$f(z)=\frac{1}{z(z-1)(z-2)}$$ for the following ranges of $|z|$: $0<|z|<1$ ...
0
votes
1answer
136 views

Find the 3 laurent series of $f(z)=\frac{3}{(1-z)(z+2)}$.

Find the Laurent series of $$f(z)=\frac{3}{(1-z)(z+2)}$$ centered at $0$ with the three domains $|z|<1$, $1<|z|<2$, $2<|z|$. I know you use partial fractions and I got $\frac{1}{1-z} + ...
1
vote
2answers
1k views

Power series by partial fractions

I am trying to find a power series centered at the origin for the function $f(z) = \frac {1}{1-z-2z^2}$ by first using partial fractions to express $f(z)$ as a sum of two simple rational functions. If ...
0
votes
1answer
88 views

How do I write $\frac{1}{z^n+1}$ as partial fractions in general?

I'm trying to integrate $\frac{1}{z^7+1}$ around the circle of radius 2, and was wondering how I express this in terms of partial fractions, hopefully as linear factors so as to apply the Cauchy ...
3
votes
1answer
326 views

Partial fraction of $\sec(z)$ from $\frac{\pi}{\sin (\pi z)}$

Given $$\frac{\pi}{\sin (\pi z)}=\sum_{n=-\infty}^\infty (-1)^n \frac{1}{z-n},$$ is there a fast way to get $$\sec(z)=\sum_{n=1}^\infty \frac{(-1)^n(2n-1)\pi}{z^2-(n-1/2)^2\pi^2}?$$ I've tried ...
0
votes
3answers
567 views

What is the complex partial fraction decomposition of $\frac{1}{z^2 - z + 1}$

I need this result to compute a residue. I haven't been successful so far. What I have tried: I have tried decomposing $\frac{1}{z^2 - z + 1} = \frac{A + Bi}{z - \omega} + \frac{C + Di}{z + ...
4
votes
2answers
296 views

Show the existence of a complex differentiable function defined outside $|z|=4$ with derivative $\frac{z}{(z-1)(z-2)(z-3)}$

My attempt I wrote the given function as a sum of rational functions (via partial fraction decomposition), namely $$ \frac{z}{(z-1)(z-2)(z-3)} = \frac{1/2}{z-1} + \frac{-2}{z-2} + \frac{3/2}{z-3}. ...