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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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1answer
24 views

Residue of pole at higher power in the denominator

I need to calculate the residue of $\frac{1}{z^{2017}}$ My thought process would be to use the derivative formula for a pole of higher order for the pole at $0$ of order $2017$ but I can’t be ...
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0answers
26 views

Proving $j$-invariant is surjective

I'm trying to solve exercise $1.1.9$ from Diamond's A First Course in Modular Forms, in which we must prove the $j$-invariant $j:\mathcal{H}\to\mathbb{C}$ with $j(\tau)=1728\frac{g_2(\tau)^3}{\Delta(\...
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2answers
36 views

Residue of $1/(e^z+1)$

How do I find the residues of $\frac{1}{1+e^{z}}$. I have calculated that the root of $1+e^{z}=0$ and the answer is $z=i\pi(2k+1)$, but the problem is that I get stucked at $$Re(f,i\pi)=\lim_{z\...
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0answers
48 views

Integral through complex methods $\int_{0}^{2\pi}\log \sin^{2}2\theta d\theta=4\int_{0}^{\pi}\log \sin\theta d\theta=-4\pi \log 2$

Using residue theorem show that: $$\int_{0}^{2\pi}\log \sin^{2}2\theta d\theta=4\int_{0}^{\pi}\log \sin\theta d\theta=-4\pi \log 2$$ I cannot show even the first identity let alone the second one. ...
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2answers
58 views

Calculus of Residue

I finding difficulties calculating : $res(f,0)$. with $f(z)=\frac{1}{z^2sinz}$ I thought of the method : defining $g(z)=z^3f(z)$, since $0$ is a pole of order $3$. then : $res(f,0)=\frac{1}{2!}g^{(...
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1answer
34 views

Finding $\mathcal{L}^{-1}_t\left(\frac{e^{|x|\sqrt{\frac{s}{k}}}}{\sqrt{ks}}\right)$ Using the General Inversion Formula

I am trying to classify the singularities in the function $$\mathcal{L}^{-1}_t\left(\frac{e^{|x|\sqrt{\frac{s}{k}}}}{\sqrt{ks}}\right), \ k>0$$ as the final part of solving a PDE. To use the ...
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1answer
22 views

Residues of $\frac{e^{iwx}}{kw^2+s}$

I am trying to find the residues of the simple poles in $$f(w)= \frac{e^{iwx}}{kw^2+s}.$$ The simples poles are at points $$w=\pm i\sqrt{\frac{s}{k}}.$$ In general, I have used the formula $$\text{...
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1answer
33 views

Computation of integral $\int_{\rho} \frac{dz}{(z-a)(z-b)}$ [duplicate]

Let $a,b$ be complex number and $|a| < r < |b|$, compute. $\int_{\rho} \frac{dz}{(z-a)(z-b)}$ where $\rho$ is the circle with radius $r$ and the usual orientation. I've tried the common path ...
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0answers
23 views

Res($\zeta(s+2) \Gamma(s)\cos(\frac{\pi s}{2})x^{-s}$|$s=-1$)?

$S=\frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} \zeta(s+2) \Gamma(s)\cos(\frac{\pi s}{2})x^{-s} ds$ I understand there are poles at $s=-1,0,-2$ My answer is $S = \frac{\pi^2}{6} - \frac{x\pi}{2} + \...
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1answer
41 views

Compute $\oint_{\gamma}f(z)dz$, where $f(z)=\frac {ze^{\pi z}}{z^4-16}+ze^{\frac \pi z}$ where $\gamma:9x^2+y^2=9$

Compute $\oint_{\gamma}f(z)dz$, where $f(z)=\frac {ze^{\pi z}}{z^4-16}+ze^{\frac \pi z}$ where $\gamma:9x^2+y^2=9$ Using the residue theorem. I don't know how to start this, first I thought about ...
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2answers
72 views

Calculate $\oint_{\gamma}\frac {\sin z}{z(z-2i)}dz$ on $|z| =3$ in trigonometric sense and on the inverse trigonometric sense.

$\oint_{\gamma}\frac {\sin z}{z(z-2i)}dz$ on $|z| =3$ in trigonometric sense and on the inverse trigonometric sense. On the trigonometric sense, we can apply the Residue theorem, we have : $z_1 = 0$ ...
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2answers
18 views

Closed-form expression for residue of a pole of order 2

Suppose $f(z)/g(z)$ has a pole of order 1 at $c$. Then its residue at $c$ is $f(c)/g'(c)$. I want a formula like this of a quotient for a pole of order 2. I know it's $\lim_{z\to c} \frac{d}{dz}[(z-c)^...
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2answers
43 views

$\oint_{\gamma}\frac 1{z^2+1}dz$ on different curves

$\oint_{\gamma}\frac 1{z^2+1}dz$ on the curves: $\gamma_1: |z + i| = 1$, $\gamma_2:|z - i| = 1$, $\gamma_3: |z| = \frac 12$, $\gamma_4: |z - i| = \frac 32$. I would like to use the residue theorem and ...
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1answer
46 views

Integral $\oint_{\gamma}\frac {\cos z}{z}dz$ on 2 curves

$\oint_{\gamma}\frac {\cos z}{z}dz$ on the curve $\gamma = \gamma_1\colon|z| =1$ and $\gamma = \gamma_2\colon|z|=3$. I calculated on $|z| = 1$ but I don't see why it would be different if I would ...
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3answers
81 views

Show $\frac{f(z)}{\sin(\pi z)} = \sum_{-\infty}^{\infty} \frac{(-1)^nf(n)}{(z-n)}$

Let $f(z)$ be entire function satisfying $|f(x+iy)| \leq Ce^{a|y|}$ for $C > 0$ and $a \in (-\pi, \pi).$ Show $\frac{f(z)}{\sin(\pi z)} = \sum_{-\infty}^{\infty} \frac{(-1)^nf(n)}{(z-n)}$ All I ...
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0answers
38 views

Find $\oint\exp((z-1)^{-1})(z+3)^{-1}\,dz$

Compute $\oint\exp((z-1)^{-1})(z+3)^{-1}\,dz$ I have to calculate this. On the contour : $|z+3| = 7$. First I want to use the residues formula for this, so I calculated the residue at the simple pole ...
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2answers
37 views

Compute $\oint\frac{e^{2z}+\sin z}{(z^2+1)^3}dz$, over the curve C

Compute $I = \oint\frac{e^{2z}+\sin z}{(z^2+1)^3}dz$, over the curve $C:|z+i|=r, r\neq2$ So what I understood from my classes I have to find what it's poles are and then apply the residual theorem ...
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2answers
30 views

Residue theorem /Integral

I want to calculate the following integral using residue theorem: $$\int_{-\infty}^{\infty} \frac{x^2}{x^4+1} $$ When I conisder the singularities, I get: $ \text{Rez}(f, z_k)=\frac{1}{4z_k}$ with $...
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3answers
102 views

Show that $\text{Res}_{z = \infty}\left(f(z)\log\left(\frac{z-a}{z-b}\right)\right) = - \int_{a}^{b}f(z)\,dz$

I would like to show that $\text{Res}_{z = \infty}\left(f(z)\log\frac{z-a}{z-b} \right)= - \int_{a}^{b}f(z)\,dz$, where $f(z)$ is an entire function and for $\log\left(\frac{z-a}{z-b}\right)$ we take ...
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0answers
50 views

Contour integral of square root on its Riemann surface

Consider a branch of the square root function $f(z)=z^{1/2}, z\in\mathbb{C}$ with $Im \thinspace {f(z)}>=0$, i.e.                        ...
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2answers
38 views

What type of singularity is $z=\infty$ for $f(z)=\frac{1}{(sin(1/z))}$?

Consider the function $$f(z)=\frac{1}{(sin(1/z))}$$ At $z=\infty$ does $f$ have an isolated singularity or not? Or is $z=\infty$ a regular point? $f(1/t)=1/(sin(t))$ has simple poles in $t=k \pi$, ...
2
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3answers
93 views

Real integral using residue theorem - why doesn't this work?

Consider the following definite real integral: $$I = \int_{0}^\infty dx \frac{e^{-ix} - e^{ix}}{x}$$ Using the $\text{Si}(x)$ function, I can solve it easily, $$I = -2i \int_{0}^\infty dx \frac{e^{-...
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2answers
104 views

Solving the improper integral

The question is asking to evaluate the following integral: $$\int_{0}^{\infty} \frac{x^{1/3}}{x^2+7x+6}dx$$ I am required to use complex analysis methods to solve this integral but I cannot seem to ...
2
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3answers
47 views

Compute the integral $ \int_{Re(s)=2}\frac{x^{-s}}{s^3}ds\quad\text{for real}\ x>0. $

Use the residue theorem to compute the integral $$ \int_{Re(s)=2}\frac{x^{-s}}{s^3}ds\quad\text{for real}\ x>0, $$ where the contour is oriented upwards. (Hint: treat the cases of $ x<1 $ ...
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0answers
57 views

Applying the residue theorem to $\log^2(|1+hz|)$

I have to calculate an integral of the following form:$$\oint_{|z|=1} \log^2\left(\frac{|1+hz|^2}{(1-y2)^2}\right)\left(\frac{1}{z-r^{-1}}\right)\,\mathrm dz,$$ where $h,r$ and $y2$ are constants with ...
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1answer
53 views

Trigonometric residue integral

The trigonometric complex integral is $$ \int_0^{2\pi} \frac 1{(2+\sin \theta)^2} d\theta. $$ My answer is below, and it is correct because I verified this numerically on WolframAlpha: https://www....
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1answer
39 views

Finding residues of $ 1/(\exp{z^2}-1)$

i was asked to find the poles and the residues of $f(z) = \frac{1}{\exp{(z^2)}-1}$ So i found the poles at $z=0$ and $z = \sqrt{2 n \pi i}, n \in \mathbb{N}$ I wanted to find the residues, but it'...
2
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1answer
22 views

Complex integral depending on the chosen path?

Let's say $f:\mathbb C\backslash\{0\}\to\mathbb C$ is holomorphic and $\text{Res}(f,0)=1$. Now if I look for example at the integral $$\oint_{|z|=2}f(z)dz$$ I get confused by the following: The path ...
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1answer
19 views

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$? where res means the residue of the function? I know $\zeta(s)$ has a pole at $s=1$ but i can't ...
2
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2answers
55 views

Taking Residues of Infinity of Square Roots.

I am looking for worked out exercises of real valued integrals with square roots where ideas of residues at infinity are used. I was hoping that from $$\int_0^1 \sqrt{x} \thinspace dx $$ I could ...
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0answers
28 views

Residue theorem and a two-dimensional integral: not working?

Consider the following integral: \begin{align} \iint_{\mathbb{R}^2}d t\,dT\, \frac{e^{-i(t-T)}e^{-t^2}e^{-{T}^2}}{(t-T-i\epsilon)^2}\,. \end{align} The $i\epsilon$ prescription simply tells me that if ...
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1answer
40 views

Inverse laplace transform of $\hat{f} (s) =\frac{1}{s(\cosh{\sqrt{2s}}-1)}$

Doing some computations on a unknown but desirable function $f:[0,\infty) \to \Bbb R $ has lead me to that its Laplace transform can be known: $$\hat{f} (s) = \frac{1}{s(\cosh{(\sqrt{2s})}-1)}, \quad ...
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3answers
108 views

Complex Contour Integral Evaluation.

I am having troubles evaluating this integral: $$\int_{-\infty}^{\infty} \frac{\cos x-1}{x^2(x^2+4)}dx$$ Well, actually, I'm quite sured about the numerical part: the value of this integral is likely ...
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1answer
54 views

Integral calculation: residue theorem or it is useless?

I would like to calculate the following integral: $$\int_{0}^{1}dx\frac{x(1-x)^2}{(1-x)^2+ax},$$ where $a$ is a parameter which defined as the integral converges. I would like to try use residue ...
1
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1answer
29 views

Compute the integral $I=\int_\gamma f(z)\, dz$, where $f(z)=\frac{e^{z^2}}{z^2-6z}$ and $\gamma=\{z:|z-2|=3\}$.

Compute the integral $I=\int_\gamma f(z)\,dz,\ $ where $f(z)=\frac{e^{z^2}}{z^2-6z}$ and $\gamma=\{z:|z-2|=3\}$. I thought of using $$f(z_0)=\frac{1}{2\pi i}\int\frac{f(z_0)}{z-z_0}\,dz$$ Rewriting $...
1
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1answer
32 views

How does the fact that every integral around a toy contour vanishes imply that the function is holomorphic?

I know that if a function is holomorphic in the enclosed domain, then it follows that the integral around the contour vanishes. However, my question is rather, how does the other direction follow? If ...
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1answer
64 views

Expressing $G_{m,m+1}^{m+1,0}\left(x\middle| \begin{array}{c}1,\cdots,1 \\0,0,\cdots,0\\\end{array}\right)$ as a power series.

I have this family of MeijerG functions: $$ G_{m,m+1}^{m+1,0}\left(x\left| \begin{array}{c} 1,\cdots,1 \\ 0,0,\cdots,0 \\ \end{array} \right.\right) $$ which I'd like to express in terms of a power ...
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3answers
51 views

calcuating residue of a complex function

I need to calculate the residue of the function $\frac{(z^6+1)^2}{(z^5)(z^2-2)(z^2-\frac{1}{2})}$ at $z$=0. z=0 is a pole of order 5 so I tried using the general formula to calculate the residue but ...
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3answers
37 views

Verification of Complex Integral $\int \frac{e^{i\pi z}}{2z^2-5z+2}dz$

What is the value of the greatest integer less than or equal to the value of integral $$\int_C\frac{e^{i\pi z}}{2z^2-5z+2}dz$$ where $C$ be the curve $\cos t+i\sin t, t\in[0,2\pi]$. I think the ...
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1answer
44 views

Contour integral involving moduli in the integrand

Long-time lurker, first time questioner. Here is the problem: Let $ a,b \in \mathbb{C} $, with $|b|<1$. Calculate the integral $$ \frac{1}{2\pi i}\int_{|z|=1} \frac{|z-a|^2}{|z-b|^2} \frac{dz}{z}...
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1answer
34 views

contour integration of $I=\int^{\infty}_{-\infty}\frac{\cos x}{x+i}$

Evaluate:$$I=\int^{\infty}_{-\infty}\frac{\cos xdx}{x+i}$$ which means that $$f(z) = \frac{e^{iz}}{z+i}$$ with the simple pole $z=-i$ and since $z=-i$ i'd have to integrate it using the bottom half ...
0
votes
1answer
27 views

finding the residue ? Tough question

Find the residue at each of the isolated singularities of the following function on $\mathbb{C}$? $1) \frac{z}{z^2+3z +3}$ $ii) \frac{1}{(z^3 +1)(z+1)^2}$ My attempt : for $1) z^2 +3z +...
2
votes
1answer
50 views

Complex integral help: $\oint_C \frac{\sin(z)}{z(z-\pi/4)} dz$

I'm attempting to evaluate the following complex integral: $$\oint_C \frac{\sin(z)}{z(z-\pi/4)} dz, $$ where $C$ is a circle of radius $\pi$ centred on the origin. I have calculated the residues of ...
0
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0answers
47 views

Eigenvector normalization, poles and residue

I am trying to understand an eigenvector normalization procedure described in an article [1](appendix B). The problem involves a complex valued matrix $\mathbf{Z}$, of which each elements is a ...
0
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1answer
41 views

Compute the integral using complex analysis method

I want to compute this integral $$\int_{C(2i,5)} \frac{z}{e^z-1}dz$$ I was trying to trying to use residue theorem but I could not find residue of this function.
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0answers
63 views

Holomorphic coordinates on Riemann surfaces

I have a big problem understanding the meaning of holomorphic coordinates on Riemann surfaces, especially in relation to 1-forms. Holomorphic coordinates on a Riemann surface $X$ is an open set $U \...
2
votes
3answers
127 views

evaluate $\int_{|z|=1} \frac{1}{e^z -1-2z}dz$ by using Cauchy's residue theorem

evalulate the integral by using Cauchy's residue theorem $$\int_{|z|=1} \frac{1}{e^z -1-2z}dz$$ MY attempt : $ f(z) =\frac{1}{e^z -1-2z}$, now put $z= 1$ we get $f(z)=-1$ so $$\int_{|z|=1} \frac{...
2
votes
1answer
31 views

Residue-Calculus: Why does this equation hold?

$\DeclareMathOperator{\Res}{Res}$ $\DeclareMathOperator{\e}{e}$ We did the following equations in our lecture: Calculate $x(t)$ with the help of the residue-calculus: $X(s)=\frac{s^2}{(s+1)^3}$ $...
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0answers
23 views

solving integral with real exponent and real pole with residue theorem

I'm trying to solve this integral: $ \int_{-\infty}^{\infty} \frac{x^3 e^{- \alpha x^2}}{\beta - x} dx$ It looks similar to a complex integral with a pole but notice a few subtleties: The exponent ...
1
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2answers
35 views

Finding the residue of a function at $z_0$?

I encountered a particular question that led me to question the definition that I was given for a residue, after reviewing the literature I simply want to confirm that my understanding is correct. ...