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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0
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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, ...
1
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1answer
32 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
votes
1answer
96 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 ...
2
votes
1answer
207 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 ...
4
votes
3answers
194 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
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 ...
2
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2answers
54 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 ...
1
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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
55 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
vote
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
75 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!
2
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0answers
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 ...
1
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0answers
24 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
votes
1answer
79 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 ...
0
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0answers
34 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 ...
0
votes
2answers
75 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 $$ \int_0^\infty \frac{ze^{iz}}{1+z^2}dz $$ ...
1
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0answers
48 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
votes
2answers
97 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}}} { ...
0
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0answers
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
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2answers
27 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 ...
0
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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 ...
0
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1answer
47 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
votes
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 ...
0
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0answers
43 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 $$\lim_{r \to \infty} \frac {\log M(r)}{\sqrt{r}}=1$$ ...
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 - ...
1
vote
1answer
101 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
votes
1answer
36 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) + ...
1
vote
2answers
111 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 ...
1
vote
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 $$ ...
0
votes
1answer
67 views

how to check if a singularity is isolated?

I have a function $1/(\sin(1/z))$ and I must show if the singularities are isolated or not. Is taking the limit of the number a little to the right and a little to the left enough? If not, how can ...
0
votes
2answers
48 views

Compute integral with residue theorem

I have the following integral $$\int_0^{\infty} \frac{\sin^2x}{x^2}\mathrm{d}x$$ And a hint: integrate $\frac{e^{2iz}-1-2iz}{z^2}$ within a semi circle. But this function residue zero (what I ...
0
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0answers
13 views

How to calculate residues of 1/(GaussianMixture)?

If I numerically calculate the complex roots of a Gaussian Mixture... $$g(z)\equiv\sum_{k=1}^{n}\frac{1}{\sqrt{2\pi B_{k}}}e^{-\frac{\left(z-x_{k}\right)^{2}}{2B_{k}}} $$ how do I calculate the ...
0
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0answers
33 views

When can we solve integrals by residue method without explicitly finding the poles?

An issue that prompted a Question earlier got me thinking about a more general class of problem. $$\int_{-\infty}^{\infty}\frac{f(z)}{g(z)}dz$$ I know I've seen some integral problems where it was ...
4
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0answers
478 views

How to solve an integral with a Gaussian Mixture denominator?

I am trying to solve this integral: $$t(v)\equiv\sum_{k=1}^{n}\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{w_{k}N(x-x_{k},B_{k})N(x-x_{j},v)}{\sum_{m=1}^{n}w_{m}N(x-x_{m},B_{m}+v)}dx $$ where ...
0
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1answer
39 views

Integrate with residue

I have this integral $$\int_0^{\infty}\frac{x^2}{(x^2+1)(x^2+4)}dx$$ and i want to use residues to compute it, how do i do that when the limits are $0$ to $\infty$? First of i define $$f(z):= ...
1
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1answer
93 views

Residue theorem for infinitely many singularities

The residue theorem is a standard result in complex analysis, I state it below so we are on the same page: note that $\overline{\mathbb{C}}$ is the extended complex plane (ie. $\simeq$ Riemann sphere) ...
1
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1answer
38 views

How to calculate this integral via residues.

I get into trouble in evaluating this integral: $$ C(a)=\frac{1}{i\beta}\int_\Gamma \cot\frac{\pi z}{\beta}\frac{1}{\sin^2\frac{z}{2}}dz $$ where the contour $\Gamma$ consists of two vertical lines, ...
0
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0answers
32 views

Pole and residue of the following function at infinity

I am confused about one particular problem regarding complex infinities. Suppose i have EXP[-z^2] with z being the complex number. Clearly it has poles at z=+i(infinity) and -i(infinity). . How to ...
0
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2answers
39 views

Finding the residue of $\frac{1}{z(z^2+4)(z+2i)}$ at $z=-2i$

I've literally tried every technique I know of and they all lead to explosions of the kind $1/0$. Generally speaking the residue at $c$ for a function can be calculated as: ...
1
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1answer
79 views

Definite Integral: $\int_0^{2 \pi} \frac{d\phi}{z + b \cos(\phi)}$

During my work, I stumbled upon this definite integral $$\int_0^{2 \pi} \frac{d\phi}{z + b \cos(\phi)} = \mathrm{sgn}(\Re(z))\frac{2\pi}{\sqrt{z^2-b^2}} \qquad z \in \mathbb{C}$$ which result I ...
6
votes
2answers
210 views

The meaning of the Imaginary value of the Residue while Evaluating a Real Improper Integral

When evaluating the improper integral $$\int_{0}^{\infty}\frac{x^{3}\sin\left(2x\right)}{\left(x^{2}+1\right)^{2}}\,dx$$ (which is an even function, so half of the $(-\infty,\infty)$ integral), I used ...
1
vote
1answer
108 views

Calculating the residue of a complex funciton with ln(z) at z=0

How can I calculate this residue: $$Res\left(\frac {z\ln(z)}{(z^2 +1)^3} , 0\right) $$ if it's possible at all. I know $0$ is a branch point for $\ln(z)$ and therefore isn't a pole, but when i plug ...
2
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1answer
51 views

How to solve this equation with implicit sum

I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& ...
2
votes
1answer
93 views

My complex integral cancels at the end; how can I modify the integrand to prevent this?

$$\int_0^\infty \frac{x^a}{x^2 + b^2}$$ for $-1< a < 1$ and b>0 -- these constraints help with estimating the integral on the big circle and small circle of a keyhole contour that I chose to ...
3
votes
4answers
163 views

Can I use an upper semi-circle to integrate this function?

I'm trying to integrate $$\int_{-\infty}^{\infty} \frac{e^{iz}}{e^z + e^{-z}}dz$$ Do I have have to integrate this over a box, or can I use my first guess at a contour and use an upper semi-circle ...
2
votes
0answers
137 views

Contour integral mystery: why is the answer different from Maple/Matlab?

The mystery is that here is a fairly standard contour integral which can be done by the residue theorem. Yet when I tried to evaluate it using numerical softwares like Maple or Matlab, the answer is ...
1
vote
3answers
211 views

What is a residue?

I've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on what ...
1
vote
1answer
96 views

Inverse Laplacetransform of rational function with multiple pole

I have to calculate the inverse Laplacetransorm of this function using Residue calculus $$ \frac{s^4 + 6s^3 - 10s^2 + 1}{s^5} $$ but I can't find any Residue rule that would solve this. Can you show ...
2
votes
1answer
76 views

Finding the residues of poles

Consider the equation $\mathcal{F}(\lambda)=0\ \ \ \forall\ \lambda = \lambda_{n},\ n \in \mathbb{N}$. I understand that the expression $\frac{d}{d\lambda}\ ...