# Questions tagged [residue-calculus]

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

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### Residue of a removable singularity at inifinity

Exercise: Find all the singularities of $$\frac{z^3e^{\frac{1}{z^2}}}{(z^2+4)^2},$$ classify them, and find each residue. I found that $+2i, \ -2i$ are poles of order two. I was able to calculate ...
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### Use residues to verify the integral $\int_0^{\infty} \frac{x^{1/2}}{(x+1)^2} dx$ [duplicate]

I found this question on page $285$ of the book COMPLEX ANALYSIS FOR MATHEMATICS AND ENGINEERING by J, Mathews. I tried to draw the contour and calculate the contour integral and each path integral,...
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### Finding mistake in contour integral. $f(z)=\frac{\exp{(-1+i)z}}{z \cdot z^{1/2}}$

I'm trying to calculate the following real integral: $$I=2\int_{0}^{\infty}\frac{e^{-x^2}\sin(x^2)}{x^2}\mathop{\mathrm{d}x} = \int_{0}^{\infty}\frac{e^{-t}\sin{t}}{t^{3/2}}\mathop{\mathrm{d}t}.$$ I ...
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### $\int_{0}^{\infty} \frac{x^{m-1}}{x^n+1} dx$ [duplicate]

Show that $\large\int_{0}^{\infty} \frac{x^{m-1}}{x^n+1} dx= \frac{\frac{\pi}{n}}{\sin(\frac{m}{n}\pi)}$; $n>m$ Given $z \in \mathbb{C}; z^n\neq-1$, I can define $f(z)= \frac{z^{m-1}}{z^n+1}$ and ...
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### Taylor-Laurent series expansions

I'm having some issues finding how to series expand some complex functions that my professor gave past years in exams. For example, in this exercise, it is asked to find the first two terms of the ...
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### Residue's theorem- analytic continuation

Let $U$ be an open set and $A \subset U$ a finite set ; let $\gamma$ be a simple loop and null homotopic in $U$ such that $tr(\gamma) \cap A= \varnothing$. Let $f: U -A \to \mathbb{C}$ be an ...
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### Calculus of residues; $\frac{f'}{f}$

Let $U \subseteq \mathbb{C}$ be an open set, $z_0 \in U$ and $f: U-\{z_0\} \to \mathbb{C}^*$ an analytic function such that $\DeclareMathOperator{\ord}{ord} \ord(z_0,f) \in \mathbb{Z}^*$. Then $z_0$ ...
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I am trying to calculate the residue of $ze^{\frac{1}{z}}$, here's what I got: We have a singularity at $z=0$. We know that $e^w=\sum_{n=0}^\infty \frac{w^n}{n!}$ so $e^{\frac{1}{z}}=\sum_{n=0}^\infty ... • 99 0 votes 0 answers 40 views ### Trying to calculate$\int_{\vert z \vert = 2} \frac z{\cos z}dz$[duplicate] Trying to calculate $$\int\limits_{\vert z \vert = 2} \frac{z}{\cos z}dz$$ but running into a lot of issues. I decided to try and use $$\int\limits_{\vert z \vert = 2} \frac{z}{\cos{z}}dz = 2 \pi i \... 3 votes 2 answers 56 views ### Issue with evaluating the residue at a double pole I need to find the residue of the function f(z)=\frac{A(z)}{B{{(z)}^{2}}C(z)} at a zero {{\bar{z}}_{B}} of B(z). The functions A(z), B(z), and C(z) are not elementary polynomials, and have ... 2 votes 1 answer 79 views ### Residue of inverse function using Lagrange inversion Let v(z) be a infinite power series of terms z^k where k>0 with coefficients v_k and a analytic function except at z=0 and V(z) be the inverse of v(z),then show that Res_0(V(z)^{-k})=k\... 4 votes 1 answer 128 views ### How to calculate this definite-Integral? Is it possible to use residue theorem? I'm seeking assistance with the following integral:$$ \int_{0}^{\infty}\frac{1}{\left(1 + cx\right)^{a}\, \left(1 + dx\right)^{b}\,}\,{\rm d}x\quad\mbox{where}\quad \left\{\begin{array}{rcl} {\... • 85 0 votes 3 answers 74 views ### Q. If$C$is the circle$|z|=1$taken with positive orientation, evaluate$\displaystyle\oint_C\dfrac{e^{\sin z}}{z^4}dz$[closed]$\displaystyle\oint_C\dfrac{e^{\sin z}}{z^4}dz$my answer is coming out to be zero I did it using the residue theorem, the residue is coming out to be zero at the point$z=0$is this the correct way? ... 2 votes 2 answers 102 views ### Find$\int_0^\pi \frac{8 \, d \theta}{5+2 \cos \theta}$Find$\int_0^\pi \frac{8 \, d \theta}{5+2 \cos \theta}$Let$z = e^{i \theta}$. Then$dz = \frac{d \theta}{iz}\begin{align} \begin{split} 8 \int_0^\pi \frac{\, d \theta}{5+2 \cos \theta}... • 2,758 0 votes 1 answer 51 views ### evaluate\int_{|z-e|=2} \frac{1}{(z-1)\log z}dz$evaluate$\int_{|z-e|=2} \frac{1}{(z-1)\log z}dz$we know that$\log(z) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(z-1)^n}{n}$so$\int_{|z-e|=2} \frac{1}{(z-1)\log z}dz = \int_{|z-e|=2} \frac{1}{(z-1)(\...
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I'm trying to compute the following integral $$\mathcal{PV}\int_0^{+\infty}dy\frac{y}{\sqrt{1+y^2a^2}{(y-b)}}$$ with $a$ and $b$ positive. I have tried to compute an ...
I encountered the integral: $$\oint_{|z|=1} \frac{f(z) \ dz}{\sqrt{(z-a)(z-b)}} \ \ \ \ \text{with} \ \ \ |a|,|b| < 1$$ So that the branch points are inside the contour. I am not adding the ...