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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30 views

How to compute this integral on contour

How to compute this following integral? $$\int \limits_{C}^{}\frac{e^{az^2}\,dz}{z^4+1}$$ Given $ a>0$ and $C:=\{z: |z+1| = 1\}$ is positively oriented. Where should I start? What would $f(z)$...
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51 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + e^u}...
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3answers
68 views

Integrate $\oint\frac{z}{\cos z-1}dz$ with residue theorem

$$\oint\limits_{|z-3|=4}^{}\frac{z}{\cos z-1}dz$$ My attempt: $$\cos z=1$$ $$z=2\pi k$$ The set includes only $z=0$ and $z=2\pi$. What next?
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1answer
25 views

the residue at the singular point

We need to find residue $\frac{1}{\cos^2z}$ $\cos z=0$ $z=\frac{\pi}{2}+\pi k$ - order 2 poles as the next?
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20 views

rational function of complex polynomials can be uniquely written as: $R(z)=P(z)+\sum\limits_{i=1}^n\sum\limits_{j=1}^{r_i}\frac{a_{ij}}{(z-z_i)^j},$

Let $R(z)$ be a rational function of complex polynomials, i.e. $R(z)=\frac{f(z)}{g(z)}$ with $f(z),g(z)\in\mathbb{C}[z]$. Claim: $R$ can be uniquely written $$R(z)=P(z)+\sum\limits_{i=1}^n\sum\limits_{...
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23 views

Residue of $\frac{\text{cot}(\pi z)}{z^6}$ at $0$

I am trying to compute $\zeta(6)$ = $\sum_1^{\infty} \frac{1}{n^6}$; I generally know how to do this using a residue-based proof, but I am stuck at the last bit, namely calculating the residue of $\...
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4answers
152 views

Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
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0answers
17 views

Evaluate the integral $(x+1)/(x^2+2)^2$ by choosing an appropriate contour in the upper half plane

How do you solve this question? Evaluate the integral $(x+1)/(x^2+2)^2$ by choosing an appropriate contour in the upper half plane How would the answer change if this question was evaluated with the ...
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1answer
34 views

finding residues for poles

I'm struggling to find the residues of the equation $$ \frac{-z\ln(z)}{(z^2+a^2)(2-z)} $$ with poles at $z=\pm ai$ and $z=2$ I have the residue for $z=2$ as $$ \frac{-2\ln2}{4+a^2} $$ but I am ...
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1answer
51 views

Inverse Laplace transform seems to be always vanishing but it couldn't!

Let's consider $x\in (0,1)$ and the distribution $p(x)=\lambda x^\lambda$, $\lambda>0$. I would like to find the pdf of the sum. The characteristic function of the $N$ sum reads: \begin{equation} \...
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1answer
17 views

Calculate $\int_{C}\frac{e^{z+\frac{1}{z}}}{1-z^2}$

Calculate $$\int_{C}\frac{e^{z+\frac{1}{z}}}{1-z^2}$$ Where $C=\{|z|=2\}$ Ok so if I write $f(z)=\frac{e^{z+\frac{1}{z}}}{1-z^2}=\frac{e^z}{1-z^2}\cdot e^{\frac{1}{z}}$ Then $f(z)$ has an ...
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0answers
34 views

Integral of the principal value of a hypergeometric function

I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where $z=-\alpha/\beta$ and $0&...
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1answer
34 views

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one.

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one and that the rest of the roots are in $\{1<|z|<2\}$. Ok, so this exercise is ...
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1answer
36 views

finding residue for complex analysis

I am having a tough time finding the residue for a function, suppose my test function is $$\frac{z^2}{{(z^2+a^2)}^2}$$ while I could determine the poles to be $+-ai$ and I know the formula to find ...
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0answers
56 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
1
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1answer
72 views

contour integral branch cut

I need some help to solve the following integral by contour integration. $$\int_{0}^{1} x^a (1-x)^{1-a}\,\mathrm{d}x$$ I attached my ideas and a picture of the paths to fix the labels. Kind regards,...
4
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1answer
162 views

Evaluating the integral $ \int_{-1}^{1} \frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$

I've been trying to find a way to integrate $\int_{-1}^{1}\frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$ using contour integration, but I'm having a hard time coming up with a contour to use. Since I have a ...
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2answers
36 views

Using the residue theorem

Is it possible to evaluate $$\int_{-\infty}^\infty \frac{x^2}{(x^2+1)^2} \, dx $$ using the residue theorem, as opposed to Calc 1 methods? How can I get started using the residue theorem? What ...
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1answer
71 views

how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
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1answer
49 views

Evaluate this integral using residue theorem

So we have $\int_{0}^{+\infty}\dfrac{x^2-a^2}{x^2+a^2}\cdot\dfrac{\sin x}{x}dx$. ($a>0$). I considered that we can just calculate the half of the imaginary part of $$\int_{-\infty}^{+\infty}\dfrac{...
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0answers
42 views

logarithmic singularities in contour integration

How to evaluate the contour integral using the residue theorem if there is a logarithmic derivative? For example this: $$\int_C \log\zeta(s)\frac{x^s}{s} ds$$ or even this: $$ \int_C \frac{\log x}{x}...
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1answer
38 views

Integration involving complex singular function

I am stuck with the following integral that came up during my research and I am not sure how to correctly evaluate this expression. $\int_{-\infty}^{\infty} dk\left[\Im[ \frac{1}{(\omega+i\eta+k)(\...
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1answer
64 views

Evaluate the following integral $\int_{-1/2}^{1/2}\big(\frac{\sin(n\pi f)}{\sin(\pi f)}\big)^4 df$

There are similar questions out there, but I was hoping someone could show how to would evaluate the following integral $$\int_{-1/2}^{1/2}\bigg(\frac{\sin(n\pi f)}{\sin(\pi f)}\bigg)^4 df$$ I've ...
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0answers
30 views

Limit inside countour integral that depends on limit

Suppose we have a sequence of functions such that: $\lim_{r \to \infty} f_r(x) = f(x)$ uniformly. Now, my question: is it possible to take the limit inside a countour integral (or a sum) which is ...
2
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1answer
65 views

Another combinatorial identity of McKay

Suppose $v\ge 2$ and $s\ge 1$ are integers. I'm stuck trying to show that $$ v\sum_{k=0}^{s-1} \binom{2s}{k} \frac{s-k}{s}(v-1)^k = \sum_{k=0}^s \binom{2s}{k} \frac{2s-2k+1}{2s-k+1}(v-1)^k $$ I've ...
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1answer
51 views

Would a keyhole contour be advisable to use for this integration?

The integral is $$\int_0^{\infty}\frac {1}{\sqrt{x}(1+x^2)}dx$$ which is to be evaluated by contour integration. So, the integrand clearly has simple poles at $+/- i$. But what kind of pole ...
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0answers
49 views

Integrate $\int_{0}^{\infty}\frac{x^a\ln(x)}{(x+b)}dx$ by the method of residues

How to evaluate $$\int_{0}^{\infty}\frac{x^a\ln(x)}{(x+b)}dx$$ where $b > 0$ and $-1 < a < 0$ using the method of residues, but I have done only problems of simple poles, but this is much ...
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1answer
151 views

Evaluating $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ with a rectangular contour

I need to try to evaluate $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ and it seems like this is supposed to be done using some sort of rectangular contour based on looking at other questions. My ...
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3answers
101 views

Evaluate the integral of function involving $\cosh$

Evaluate the integral $$ \int_0^{\infty} \frac{\cosh(ax)}{\cosh(x)}\,dx, $$ where $|a|<1$. Consider the closed loop integral of $\displaystyle\frac{e^{az}}{\cosh(z)}$ where the contour $C$ is $y=...
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1answer
51 views

Calculate infinite sum with residues

I'm trying to use the residue theorem to calculate $$\sum_{k=1}^{\infty} \dfrac{1}{(2k-1)^2}. $$ I came up with $\operatorname{Res}\left(\dfrac{\pi \cot(\pi z)}{(2z-1)^2},\frac12\right)=-\pi^2$ and $\...
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0answers
41 views

Cauchy Residue Theorem Question

I am finding difficulty with the following couple of problems a) show that $$\int_{-\infty}^\infty \frac{\cos(\pi x)}{2x-1} \ne\frac{\pi}{2}$$ I've been trying to do this over a semi-circle, but ...
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1answer
33 views

Reversing the direction of a contour integral

If $$\int_{C} f(z) dz$$ is some contour integral over a closed curve $C$, and $-C$ is the contour taken in the opposite direction, can $$ \int_{-C} f(z) dz$$ be treated as a closed curve around the ...
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1answer
22 views

Cauchy's residue application problem

I would like to know what did I do wrong. There's my problem : I= $\frac{1}{2\pi i}$ $\int_a \frac{1}{z^4+1}~dz$ Where a $x^2 + y^2 = 2x$ I already know: there're 4 poles, but only 2 fits for me (...
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2answers
54 views

How to evaluate the contour integral $\int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz$ over the unit circle?

Let $C(0, 1)$ be the unit circle centered at the origin with radius $1$. Then I need to evaluate the following complex contour integral: $$ \int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz$$ I know the ...
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1answer
95 views

Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
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0answers
26 views

Find $\int_0^{2\pi} \frac {dx}{a + \sin^2(x)}$ using method of residues.

I want to find $\int_0^{2\pi} \frac {dx}{a + \sin^2(x)}, a > 0$ using method of residues. My Attempt: First I deduced that $$\int_0^{\pi/2}\frac {dx}{a + sin^2(x)} = \int_0^{\pi}\frac {dt}{(2a ...
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0answers
71 views

Contour integral of $\int_{0}^\infty \frac{\sinh(kx)}{\sinh(x)}dx = \frac{1}{2}\tan{\frac{a}{2}}$

From Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ In the case of zero $\omega$ and integral starts as 0, how do I prove that using contour integral $\int_{0}^\infty \frac{\sinh(...
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1answer
38 views

Bounding a function

I'm trying to solve for $\sum_{0}^{\infty} \dfrac{1}{n^2}$ using the residue theorem. The integral in question is $\int_C f(z)\pi \cot(\pi z) dz$ where $f(z)=\dfrac{1}{z^2}$. I am bounding over the ...
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0answers
39 views

Complex residue at infinity of $f(z)=\frac{z^5}{\sin\left(\frac{1}{z^2}\right)}$

I'm having trouble finding residue of the function $$f(z)=\frac{z^5}{\sin\left(\frac{1}{\large{z^2}}\right)}$$ at infinity. Wolfram kindly informs that it is equal to $-\frac{7}{360}$ (and gives ...
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1answer
30 views

When computing contour integration with sines and cosines in the integrand, must we always first look at Euler's formula?

For example, in computing $$\int_{Cr}\frac {\cos(z)}{(z^2+a^2)^2}dz$$ over a semi-circular contour, must I first look at $$\int_{Cr}\frac {e^{iz}}{(z^2+a^2)^2}dz$$ compute this integral first, ...
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1answer
34 views

How can I compute the residue at this order-2 pole?

The integral is $$\int_{-\infty}^{\infty} \frac {cos(z)}{(x^2+a^2)^2}dz $$ If I use an upper semi-circular contour, then there is an order-2 pole at $z=ia$. I am trying to expand the integrand in a ...
5
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1answer
108 views

Mysterious Inverse Mellin transform using residue theorem

The origin of this problem lies in the explanation of the evaluation of the series $\sum_{n\geq1}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{2\pi}{4}+\frac{\pi^2}{6}$ see this link ( Series $\sum_{n=1}^...
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1answer
292 views

residue theorem with logarithmic function

I have problem integrating function with logarithm. Problems seems always to be branch cut of $\log$, but here it is different I think. I have task to integrate $$\oint_{|z| = 1} \! dz \log\left(\...
4
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3answers
204 views

Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$

How to prove that $$\int_0^\infty \frac{x\sin x}{x^2+1}dx=\frac{\pi}{2e}$$ I've tried several basic approaches like substitution and IBP but can't move forward.
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0answers
28 views

Is it possible to do this integral using the residue theorem? $ H(u,a)= \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{e^{-x^2}}{(u-x)^2+a^2} dx $

$ H(u,a)= \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{e^{-x^2}}{(u-x)^2+a^2} dx $ Someone asked a question that involves this integral on another math forum. I put it into wolfram alpha to see what ...
0
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2answers
46 views

Evaluate $\oint \limits_C \frac{z^2+1}{(2z-i)^2}dz$ using residue theorem

Let $C:|z|=1$ be a circle with positive orientation. Use residue to evaluate $$\oint \limits_C \frac{z^2+1}{(2z-i)^2}dz$$ Our integrand (let's call it $f$) has one singular point (pole) of order two,...
2
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2answers
55 views

$\int_0^{2\pi}\frac{1}{a\cos \theta+b\sin\theta+d}d\theta$ where $a,b,d\in\mathbb{R}$ and $a^2+b^2<d^2$

$\int_0^{2\pi}\frac{1}{a\cos \theta+b\sin\theta+d}d\theta$ where $a,b,d\in\mathbb{R}$ and $a^2+b^2<d^2$ Here, I solve it by Residue Theory. By substituting $d\theta=dz/iz,\cos \theta=(1/2)(z+1/z),\...
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0answers
35 views

How can I make some progress on this Gaussian-looking integral? [duplicate]

An old complex analysis exam question: Evaluate $$\large I(a) = \int_{-\infty}^{\infty} e^{-\frac{1}{2}x^2+iax}dx$$ So far, I have completed the square in exponent, and now I have the integral $$...
4
votes
3answers
57 views

Residues of poles

Find $Res_{f}\left ( z_{0} \right )$, where, $f\left ( z \right )=\frac{1}{z^{4}+4}$, for $z_{0}=1+i$ Now, the definition for $$Res_{f}\left ( 1+i \right ) =\lim_{z \to z_{0}} \left\{\left ( z-\left ...
1
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1answer
49 views

Can one compute series expansions or complex residues at essential singularities?

Title says it all. I'm noticing a trend of failure on Mathematica/WolframAlpha's parts when trying to compute either the Laurent expansions or the residues of functions like $\sin\dfrac{1}{z}$ about $...