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
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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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
37 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
63 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 ...
1
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1answer
50 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
150 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 ...
2
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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=...
3
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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
32 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
53 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 ...
2
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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
25 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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70 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
29 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
101 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}^...
2
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1answer
256 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
201 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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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 ...
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2answers
45 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,...
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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
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3answers
56 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 $...
1
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1answer
90 views

Residue of 1/(z-sin(z)) at z=0

I am to find the residue of f(z)=1/(z-sin(z)) at z=0. I am confused as to which method to use. Your help will be greatly appreciated! Thanks!
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35 views

Is this integral automatically zero?

If I integrate $\int e^{iz}\,dz$ for z complex, along the positive real line, then is the imaginary part of the integral $i\int \sin(x)\,dx$ automatically equal to zero (integration only along the ...
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0answers
25 views

Inverse Laplace transform of the form $F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$

I am trying to solve the inverse Laplace transform of the form \begin{equation} F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}} \end{equation} where, $a$ and $b$ are known constants, $m$, $n$, ...
2
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1answer
83 views

Use contour integration to compute the Fourier transform,

The problem statement is: Use contour integration to determine the Fourier transform, $\large \hat f(ξ)=∫_{-\infty}^{\infty}f(x)e^{−iξx}dx$, of $\large f(x)=\frac{1}{2−2x−x^2}$. Some issues that I ...
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38 views

In P.V. contour integration in complex analysis, using a wedge vs. a keyhole contour

When is it clearly better -- perhaps even necessary -- to use a keyhole contour, instead of a wedge contour? The wedge contour minimizes computation of residues, as we can choose it so that it ...
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2answers
76 views

Compute $\int _0^\infty\frac{x \sin x}{1+x^2}dx$ with the residue theorem

Compute $\int _0^\infty\frac{x \sin x}{1+x^2}dx$ with the residue theorem Ok so I have done a couple of these but I'm stuck on this one. I want to use $$ \int_0^\infty \frac{ze^{iz}}{1+z^2}dz $$ ...
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0answers
51 views

Contour integral over a pole: how to choose sign?

My function is $f(z)$ is analytic in Re$(s)>1$, defined by meromorphic continuation elsewhere. It has a simple pole only at $s=0$. I want to integrate $$\int_{-i\infty}^{i\infty}f(z)dz$$ knowing ...
2
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2answers
99 views

How can I convert this tricky complex number into a real number?

The problem statement is: $$∫_0^{\infty}\frac{x^α}{x^3+1}dx$$ for α in the range −1<α<2. $$\huge \frac{2\pi i}{1-e^{\frac{i2\pi (\alpha+1)}{3}}} \frac {e^{\frac{i \pi \alpha}{3}}} { 3e^{\frac{...
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25 views

On the right half-plane, what is an upper bound for $\frac{1}{\log(z+2)}$?

I am trying to estimate some factors in my integrand in complex integration, and I think the upper bound for $\frac {1}{log(z+2)}$ on the semicircle in the right half plane is just $\frac {1}{log(2)}$....
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2answers
28 views

Find the residue of $(z^2-1)\cos\frac {1}{z-1}$ at $z=1$.

Question: Find the residue of $(z^2-1)\cos\frac{1}{z-1} $ at $z=1$. Attempt: I tried to expand the series of $\cos\frac{1}{z-1}$ about $z=1$ and multiply through by $z^2-1$ but I couldn't isolate ...
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0answers
55 views

How can I prove $\int^{\infty}_{0} \frac{\ln(x)}{(x+a)^2+b^2} dx = \frac{\ln(\sqrt{a^2+b^2})}{b} \arctan(\frac{b}{a})$

I consider the following: $$f(z) = \frac{\operatorname{Ln} (z)}{(z+a)^2+b^2}$$ And the contour: Then, $f(z)$ has a simple pole at $z= -a+ib$ But after developing , I didn't get that result. How ...
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1answer
50 views

How to prove the formula for the residue of $f$ at a pole of order $m$?

Let $f$ holomorphic on $z_0$. I saw this awesome formula on a book : the residual of $f$ on $z_0$ is given by $$\text{Res}_{z_0}(f)=\frac{1}{(m-1)!}\frac{\mathrm d^m}{\mathrm dz^{m-1}}(z-z_0)^mf(z)$$ ...
0
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1answer
31 views

Estimations for integrals(complex) in the upper half circle

So I'm trying to solve some real valued improper integrals with the residue theorem and I have some questions about the curve-contour in the upper half circle. When I want to show that this integral ...
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0answers
44 views

Does a function with an exponential growth condition necessarily have infinitely many zeros?

This is part (2) of a question that I am working on. In part(1), I have constructed an entire function $f:=\cosh(\sqrt{z})$ that grows like $$\lim_{r \to \infty} \frac {\log M(r)}{\sqrt{r}}=1$$ ...
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0answers
17 views

How well does $L_{n,f}$ approximate $f$?

My Try: I did part a,b and c. Having trouble with d. Followed the hint and got $\displaystyle f(z)-RHS=\frac{1}{2\pi i}\Bigg[\int_\Gamma \frac{f(\zeta)}{(\zeta - z)}\Bigg(1-\frac{w(z)}{w(\zeta)}\Bigg)...
1
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1answer
105 views

How to compute this contour integral with a modulus sign in the integrand,

Evaluate the integral $$∫_{∣z∣=ρ} \frac {1}{|z−a|^{2}}|dz|$$ where ρ is a positive number, a is a complex number, and |a|<ρ. I welcome any hints on how to get started on this problem. The ...
0
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1answer
37 views

Taking inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ using residues, something's wrong

I am trying to compute the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ using residues So $$\mathcal{L}^{-1}\dfrac{1}{(s^2+1)^2} = res(\dfrac{e^{st}}{(s^2+1)^2}, i) + res(\dfrac{e^{st}}{(s^2+...
1
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2answers
113 views

Residue theorem for Multi-valued functions

Im stuck with this problem show that: $$\int_0^\infty{\frac{x^a}{(x^2+1)^2}dx} = \frac{\pi (1-a)}{4cos(a \pi /2)}, \, -1<a<3, \, a \neq 1$$ I have the solution for it and everything but i don'...
1
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1answer
74 views

Continuous extension of $\int_\mathbb{R} dt\, e^{-t^2}/(t-z)$ from $\operatorname{Im} z < 0$ onto $\mathbb R$

I am asked to show that the continuous extension of $$ F(z) = \int_{-\infty}^{\infty} dt\, \frac{e^{-t^2}}{t-z}, \quad \operatorname{Im} z < 0 $$ onto $\mathbb R$ is given by $$ \int_{-\infty}^{\...