Residue Calc: $\cot^n(z)$ at $z=0$

MHB
Thread starter polygamma
Start date Feb 29, 2016
Tags

Calculation Residue

In summary, the residue of $\cot^n(z)$ at $z=0$ is 1 if n is even, and 0 if n is odd. This can be found using the formula $\text{Res}(f,c) = \frac{1}{2\pi i} \int_C f(z)dz$, where C is a small circle around $z=0$. The significance of the residue is its use in evaluating complex integrals and solving differential equations. It can never be negative and is equal to the order of the pole at $z=0$ for $\cot(z)$.

Feb 29, 2016

polygamma

Show that the residue of $\cot^{n}(z)$ at $z=0$ is $\sin \left( \frac{n \pi}{2}\right)$, $n \in \mathbb{N}$.

Mathematics news on Phys.org

Mar 8, 2016

polygamma

Hint:

Integrate $\cot^{n} (z)$ around a rectangular contour with vertices at $- \frac{\pi}{2} - iR$, $ \frac{\pi}{2} - i R$, $\frac{\pi}{2} + iR$, and $- \frac{\pi}{2} + iR$. Then let $ R \to \infty$.

Related to Residue Calc: $\cot^n(z)$ at $z=0$

What is the residue of $\cot^n(z)$ at $z=0$?

The residue of $\cot^n(z)$ at $z=0$ is 1 if n is even, and 0 if n is odd. This can be proven using the Laurent series expansion of $\cot(z)$ around $z=0$.

How do you find the residue of $\cot^n(z)$ at $z=0$?

To find the residue of $\cot^n(z)$ at $z=0$, you can use the formula $\text{Res}(f,c) = \frac{1}{2\pi i} \int_C f(z)dz$, where C is a small circle around $z=0$. You can then use the Laurent series expansion of $\cot(z)$ and the Cauchy Integral Formula to evaluate the integral.

What is the significance of the residue of $\cot^n(z)$ at $z=0$?

The residue of $\cot^n(z)$ at $z=0$ is important in evaluating complex integrals involving $\cot^n(z)$. It can also be used in solving differential equations and in other areas of mathematics and physics.

Can the residue of $\cot^n(z)$ at $z=0$ be negative?

No, the residue of $\cot^n(z)$ at $z=0$ can never be negative. This is because the residue is defined as the coefficient of the $\frac{1}{z}$ term in the Laurent series expansion, and this coefficient is always a non-negative integer.

What is the relationship between the residue of $\cot^n(z)$ at $z=0$ and the poles of $\cot(z)$?

The residue of $\cot^n(z)$ at $z=0$ is equal to the order of the pole at $z=0$ for $\cot(z)$. This means that if $z=0$ is a simple pole for $\cot(z)$, then the residue of $\cot^n(z)$ at $z=0$ is 1. If $z=0$ is a pole of higher order, the residue will be a higher power of $n$.