# Tagged Questions

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

41 views

### What is the Residue of this function?

What is $$\operatorname*{Res}_{z=i \pi}{\frac{e^{(1-a)z}}{\left (1+e^{z} \right )^n}}$$ where $0<a<1$ and $n$ is an integer??
30 views

### Express ${1\over 2\pi i}\int_{\partial \Omega}{g(z){f'(z)\over f(z)}}dz$ with poles and zeros of $f$

The question states, in fact, show ${1\over 2\pi i}\int_{\partial \Omega}{g(z){f'(z)\over f(z)}}dz=\sum_{a_k}g(a_k)-\sum_{b_k}g(b_k)$, where $a_k$ are the zeros of $f$ in $\Omega$ and $b_k$ are the ...
37 views

### Evaluating the residue of $(1 - e^{-z})^n$ at $z = 0$ with $n \in \mathbb{Z}$

For $n \in \mathbb{Z}$, I want to find a way to determine the residue of the function $f(z) := (1 - e^{-z})^n$ at $z = 0$. I must admit that I haven't come too far yet. In case that $n ≥ 0$, $f$ ...
271 views

### What is the integral of 1/(z-i) over the unit circle?

At present there is a simple pole on the closed contour, so the Residue Theorem appears to be inapplicable. But I want to claim that we can enlarge this circle to make sure that it encloses the ...
42 views

### Integrating secans over the imaginary axis using the residue theorem

I am trying to integrate $\sec(z)$ over the whole imaginary axis using the residue theorem. i.e., I want to calculate the integral $$\int_{\Gamma} \frac{dz}{\cos{z}}$$ where $\Gamma$ is the (open) ...
74 views

### Limit of a difference of integrals that both look almost identical,

Let $\gamma (t) = t+i(e^t-1)$ for $-1\le t \le 1$. find $$\lim_{\epsilon \to 0^+} \left[\int_{\gamma} \frac{\sin(z)}{(z-i\epsilon)^2} dz - \int_{\gamma} \frac{\sin(z)}{(z+i\epsilon)^2} dz\right]$$ ...
44 views

### Multivariate/multidimensional residues

My very specific question: Given $(z_1,z_2) \in \mathbb{C}^2$ and $$I = \frac{1}{z_1 z_2 +1}$$ a) Where are the poles of $I$? b) What are the residues of $I$? Note: $z_1$ and $z_2$ are not in ...
681 views

### Need a hint for this integral

I'm trying to evaluate the following integral $$\int_0^{\infty} \frac{1}{x^{\frac{3}{2}}+1}\,dx.$$ This is an old complex analysis exam question, so I plan to use the residue theorem. How can I ...
35 views

44 views

### Find the residues of $\frac {z^2} {(z^4+1)^2}$

I know that $f(z)=\frac{z^2}{(z^4+1)^2}$ has four poles at -1 .$z_1= e^{i\frac{\pi}{4}}$, $z_2= e^{i\frac{3\pi}{4}}$, $z_3= e^{i\frac{5\pi}{4}}$,$z_4= e^{i\frac{7\pi}{4}}$.But how do I find the ...
147 views

72 views

### Inverse Laplace transform of an exponential function

What is the inverse Laplace transform of $$\frac{e^{\frac{-2}{s}}}{s}$$ I have seen an answer using Maclaurin series expansion of this function. This function is not analytic at $0$, so, is such ...
60 views

### how to integrate the definite integral using residue theorem? [duplicate]

How to evaluate $\int_{0}^{\infty}\dfrac{1}{x^a+1}dx$, where $a>1$. I don't know where to start since $x^a+1$ could have infinitely many roots, then it is impossible(?) to evaluate its residues. ...