Questions on the evaluation of integrals along a locus in the complex plane.

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Inverse Gamma function for integers (Hankel)

So I want to prove that for all integers $n \in \mathbb{Z}$ it holds that $$F(n):= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-n}e^{s} ds = \frac{1}{\Gamma(n)},$$ with $\gamma$ the 'Hankel'-contour: ...
8
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1answer
179 views

Physical interpretation of residues

What is physical interpretation of residues of poles (of any order) of a complex function? Poles represents the points where a complex function cease to be analytic and residues are calculated to ...
6
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4answers
5k views

How to measure the volume of rock?

I have a object which is similar to the shape of irregular rock like this I would like to find the volume of this. How to do it? If I have to find the volume, what are the things I would need. eg., ...
1
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0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
2
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3answers
63 views

A question about complex integration of $\frac{1}{p(z)}$

Let $p(z)$ be a polynomial of degree $n\ge 2$. Is it true that, there is a $R>0$ such that $$\int\limits_{|z|=R}{\frac{1}{p(z)}dz}=0?$$ My attempt is: there is a $R>0$ such that $|p(z)|\ge ...
1
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1answer
30 views

Integration of Bessel functions:Finding a suitable contour

I have below function to integrate; $$\int_{0}^{\infty} \frac{J_{0}(ax)x^3}{k^2-x^2} dx$$ here $a,k$ are constants. Since this is an odd function, I am not allowed to extend the limits from negative ...
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0answers
50 views

Contour integration from zero to infinity

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...
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0answers
18 views

Residue Theorem on an integral contains a Hankel function and a cosine function

I am trying to solve below integration; $$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. This is related to the question link. Below shows my approach to get ...
3
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0answers
75 views
+50

Asymptotic form of an integral to an power law decaying function

$$ f(x)=\frac{1}{2}+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right| $$ This function is not analytic at $x=1$. The plot is shown: The integral is: $$ I=\int_0^\infty g(x) \sin(2b rx) dx $$ where ...
2
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1answer
42 views

Residue theorem on even function integration

I need to integrate below function; $$\int_{-\infty}^{\infty} \frac{\sin(pR)}{R}\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. Since this is an even function of $p$, I tried applying the residue ...
4
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1answer
144 views

Integrating around a pole

I recently made the following observation: Assume that $f(z)$ has a Laurent expansion at $z=z_{0}$ of the form $$ f(z) = \sum_{k=-n}^{0} a_{2k-1} (z-z_{0})^{2k-1} + g(z) \, ,$$ where the function ...
2
votes
1answer
383 views

Integrating $\int \frac{e^{ipz}}{(\cos z)^{a}} \frac{dz}{z- \xi}$ on the complex plane

In a book I'm reading, the author states the following: Let us integrate $$ (i) \ \int \frac{e^{ipz}}{(\cos z)^{a}} \frac{dz}{z- \xi} \, , $$ $$ (ii) \ \int \frac{e^{ipz}}{(\sin z)^{a}} ...
3
votes
3answers
141 views

integral $\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt$

I want to compute this integral $$\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt$$ where $0<b \leq a$. I have this results $$I_1=-\frac{ab}{2\pi}\int_0^\pi ...
12
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3answers
327 views

Evaluating the integral $ \int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \ dx$

I'm curious about the proper way to evaluate $$ \int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \, dx = \text{Re} \int_{-\infty}^{\infty} \frac{e^{i(x- ...
6
votes
2answers
111 views

Is this a Morera´s Theorem Application?

Let $G \subset \mathbb C$ be a domain and $f: G \to \mathbb C$ a continuous function such that for any closed and rectifiable path $\gamma \subset G$, $$ \left| \oint_\gamma f(z)dz \right|\leq \left( ...
9
votes
6answers
726 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
5
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4answers
127 views

calculate $\int_{0}^{\pi} \frac{dx}{a+\sin^2(x)} $using complex analysis

where $a>1$ calculate $$\int_{0}^{\pi} \dfrac{dx}{a+\sin^2(x)}$$ I tried to use the regular $z=e^{ix}$ in $|z|=1$ contour. ($2\sin(x) = z-\frac1z)$, but it turned out not to work well because ...
1
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1answer
59 views

Application of Complex Variables

Considering the integral : $$\int_{-\infty}^{+\infty}\left(\dfrac{\sin\alpha z}{\alpha z}\right)^2 \dfrac{\pi}{\sin{\pi} z}dz,\quad \alpha \lt \dfrac{\pi}{2}$$ around a circle of large radius, prove ...
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0answers
38 views

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
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3answers
581 views

Integral $\int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

Inspired by the user @Integrals, I thought I'd find some nice integrals! Especially interesting are those involving $\log \pi$. From Borwein and Devlin's "The Computer as Crucible", pg. 58 - show that ...
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0answers
24 views

Contour integration of logarithm: $g(\omega) \log[1 - \chi(q,\omega)]$

I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced ...
1
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2answers
44 views

definite integral of form G(cos(x), sin(x)) by complex integration

Given: the following integral: $$ \int_0^{2\pi} \frac{\mathrm{sin}(3x)}{5-3\mathrm{cos}(x)}\,\mathrm{d}x = 0 $$ Prove it by using complex integration and the residue theorem. But I do something wrong ...
2
votes
2answers
47 views

Laplace Inverse

I want to find the laplace inverse of $$s^{-3/2}$$ the steps given in the solution manual are as follows: $$\frac{2}{\sqrt\pi}\frac{\sqrt\pi}{2s^{3/2}}=2\sqrt{\frac t\pi}$$ I know the first part ...
3
votes
1answer
52 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
0
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1answer
33 views

Calculate complex integral

Let $C$ be a circle $\gamma=\partial B (0,2)$ oriented positively. I have to calculate $$\int_\gamma \frac{-\cos(1/z)}{\sin(1/z)z^2}dz$$ My attempt: Notice that $\sin(1/z)$ is meromorphic inside ...
3
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1answer
89 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
3
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0answers
126 views

An interesting identity involving powers of $\pi$ and alternating zeta series

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
3
votes
1answer
75 views

Steepest descent method with movable maximum

Suppose we want to find the asymptotic behavior as $n \rightarrow \infty$ of the integral $$\int_C \frac{dz}{z} \frac{e^z}{z^n}=\int_C \frac{dz}{z} \exp(z-n \ln z)$$ where $C$ is some contour in the ...
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0answers
38 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
2
votes
1answer
63 views

$\int_0^{\infty} \exp(i(t-\alpha)^2) dt$

It's fairly straight forward to show that $$ \int_0^{\infty} \exp(it^2) dt = \frac{\sqrt{\pi}}{2}\exp\left(i\frac{\pi}{4}\right) $$ via complex contour integration over a contour shaped like a piece ...
6
votes
6answers
276 views

Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
3
votes
2answers
88 views

Definite integral (in the complex plane?)

I want to prove that $$\int_{0}^{\infty} \frac{dx}{1+x^b} = \frac{\pi}{b \sin(\pi/b)} \ ,$$ where $b\in (1,\infty)$. I thought about doing it in the complex plane since the integrand is a ...
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0answers
30 views

Series involving Laguerre polynomials

Given the series \begin{align} S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a) \end{align} where $L_{m}(x)$ is the Laguerre polynomial. By using \begin{align} L_{n}(z) = ...
2
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1answer
53 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
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1answer
69 views

What enclosure should I choose to evaluate the next integral using residues?

I have to solve the next integral: $$\int_{-\infty}^{\infty} e^{ibx}(e^{ia/x}-1)dx$$ where $a,b$ are real parameters. I can use Jordan´s Theorem to show that as $f(z)=e^{ibz}g(z)$ where $g(z)=(e^{ ia ...
3
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0answers
50 views

Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
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2answers
49 views

Contour Integration of a Complex function

In the context of contour integration: For positive real values of $\alpha$ the following integral is $$ I(\alpha)=\int_{-\infty}^{\infty}\frac{e^{it\alpha}}{1+t^2}dt=\frac{\pi}{e^{\alpha}} $$ Why ...
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0answers
68 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
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0answers
50 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
2
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2answers
65 views

evaluate $\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx $

This is supposed to be a very easy integral, however I cannot get around. Evaluate: $$\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx$$ What I did is: $$\int_{0}^{2\pi}\frac{dx}{\cos x + \sin x ...
2
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1answer
26 views

Complex contour integral of fraction of polynomials

Let $n \in \mathbb{N}_0$ and set $p(z) = z^n + a_1 z^{n-1} + \cdots$ and $q(z) = z^{n+1} + b_1 z^{n} + \cdots$ to be two monic complex polynomials with no common zeros. I want to prove that ...
0
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1answer
29 views

Contour Integration of Simple Closed Contour

Let C be any simple closed contour inside the annulus 4 < |z| < 6. Show that there holds: $$ \int_C \frac{dz}{z^2+1} = 0$$ To begin: I know that there are poles at $\pm i$ and that the ...
2
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3answers
60 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. ...
1
vote
2answers
21 views

Locations of singularities of a function with respect to given contours

Show that $\int_{C_1}f=\int_{C_2}f$, where $C_1:|z|=1$, $C_2:|z|=2$, and $f(z)=\frac{2z+1}{\sin z}$. Hint: Locate the singularities of $f$ in each case and indicate their location with respect ...
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1answer
34 views

Complex Integrations and Contours

Show that $$\int_C \frac{2 z^2-5}{(z^2+1)(z^2+4)} dz \le \frac{\pi R (2 R^2+5)}{(R^2-1)(R^2-4)} $$ Let $C$ be the upper half of the circle $z=R$ for any $R>2$. Do I need to actually find the ...
0
votes
0answers
53 views

Prove: $\int_C \frac{dz}{z^2+1} =0$ on the annulus $6\lt |z| \lt 8$

Let $D$ be the annulus $6\lt |z| \lt 8$ and let $C$ be any simple closed contour inside $D$. Show that there holds: $$\int_C \frac{dz}{z^2+1} =0$$ This has two singular points, $z=\pm i$, these are ...
4
votes
1answer
39 views

$\int_{\gamma}f(z)\log\left(\frac{z+1}{z-1}\right)dz = 2\pi i\int_{x=-1}^{x=1}f(x)dx$ on an ellipse

I am a self-studier and this is a problem from a course I've been doing. I would appreciate help showing: $$\int_{\gamma}f(z)\log\left(\frac{z+1}{z-1}\right)dz = 2\pi i\int_{x=-1}^{x=1}f(x)dx$$ ...
0
votes
2answers
49 views

Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
2
votes
2answers
41 views

Calculating residue $\int_C \frac{8-z}{z(4-z)}dz$

I want to calculate the following: $$\int_C \frac{8-z}{z(4-z)}dz$$ $C$ is a circle of radius $7$, centered at the origin,negative oriented. I want to do this via finding the residues at $z=0,4$. I ...
3
votes
4answers
93 views

Integral using contour integration [duplicate]

Here is the integral I want to evaluate: $$\int_{0}^{2\pi} \frac{dx}{a+b \cos x }, \quad a>b >0$$ Apparently there are limitations as to what values $a, b$ are supposed to take but let us not ...