Questions tagged [contour-integration]
Questions on the evaluation of integrals along a locus in the complex plane.
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how can I use contour integration in complex plane to solve this problem?
enter image description here
$$\int_0^{\tfrac\pi2} \frac{\ln(\cos(x))}{x^2+\ln^2(\cos(x))} \, dx$$
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Generalizing contour integration to indefinite integrals [closed]
Is it possible to evaluate indefinite integrals using contour integration. For example take the integral of $\left(\dfrac{\ln x}x\right)^{2011}$. Is it possible to find its primitive solely using ...
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Contour integrals on unit circle.
When dealing with a contour along a unit circle, we can set $|z| = 1$ and $z(t) = e^{it}$ so that $\frac{d z}{dt} = ie^{it}$ with $t\in [0,2\pi]$.
Find the integral of:
$$\int_{\Gamma}(z^7+z^4) dz$$
...
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hard integral: $\sin(x)/\sqrt{1+2\sin(2x)}$ between $0$ and $\pi/2$
I found this integral in one of my math textbooks.
$$\int_0^{\pi/2}\frac{\sin(x)}{\sqrt{1+2\sin(2x)}}\mathrm{d}x$$
I tried to solve it using the substitution $u=\tan(\frac{x}{2})$ but it got me ...
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Computing the energy of the electrostatic field for a Gaussian distribution
I want to compute, as a function of $\sigma$,
the electrostatic energy of a Gaussian distribution of charges (omitting the constants):
$${1\over 2}\int \rho\ \Phi\ d{\bf y}
= {1\over 2}\int_{\mathbb ...
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Bounding the large semicircular contour in integrals like $\oint\frac{e^{iz}}{z}\, dz$ [duplicate]
Considering the large semicircular arc $\Gamma$ in the upper half plane of $\oint{\frac{e^{iz}}{z}\,dz}$ using the Estimation Lemma with $z=Re^{i\theta}, \theta\in[0,\pi]$ means :
$$\sup\left|\frac{e^{...
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an inequality on the area represented by a contour integral
This question was inspired by Carnot's great theorem on the efficiency of reversible thermodynamic cycles.
We have two continuous positive functions $T(u)$ and $S(u)$ of the parameter $u\in [0,1]$. ...
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Integrating the exponential of a fractional order complex polynomial
This post potentially has two questions related to the following integral:
$$
\int_{-\infty}^{\infty}e^{i(ax-bx^{s})}dx
$$
where $1<s\leq2,$ $a\in\mathbb{R}$ and $b>0$.
The first question: is ...
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$\int^{c+i\infty}_{c-i\infty}\frac1za^z$: how to use this contour?
"Show that $\int^{c+i\infty}_{c-i\infty}\frac1za^z$ is 0 for $a<1$, $\frac12$ for $a=1$, 1 for $a>1$.
For $a>1$, use the rectangle from $c-Bi$ to $-X+Ai$.
For $a<1$, use $X$ instead ...
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Evaluating the sum $\sum_{k=1}^\infty (-1)^k / k^2$ via a contour integral
I'm evaluating the sum:
\begin{align*}
\sum_{k=1}^\infty (-1)^k \frac{1}{k^2}
\end{align*}
I expressed the sum via a complex contour integration. But I'm not getting out the right answer.
My method: ...
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How is a Hankel contour different from a keyhole contour?
From what I'm guessing a keyhole contour is one that looks like this
and because it can be shown that the contribution from $C_R$ and $C_\epsilon$ vanishes as $R\to \infty$, a Hankel contour looks ...
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Saddle Point analysis of integral
I want to make saddle point approximation of the integral $$ I = \frac{1}{2 \pi i } \int_{\gamma - i \infty} ^{\gamma + i \infty} \frac{1}{z^6} e^{zt} dz $$ but as I see the function in the exponent i....
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Contour integration with $\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$
$$\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$$
The contour is shaped like this (image from this answer)
With $\epsilon$ being the radius of the smaller semi-circle. The integral over ...
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Integrals of $\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$
Consider a type of integrals
$$
\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1}
f(k)\text{d}k
$$
where $K=K(k),K^\prime=K(\sqrt{1-k^2})$ are complete elliptic integrals, and $k$ is an elliptic ...
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Integrate over curve specified by Dirac delta functions
I need to numerically evaluate an integral of the following form:
$$
I(\omega) = \int_0^\infty\int_0^\infty\int_0^\infty f(\omega_1,\omega_2,\omega_3,\omega)\delta(\omega_1+\omega_2-\omega_3-\omega)\...
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If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.
If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.
Hint is to use integration by part but I can't find a good reference for ...
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Pole-skirting during contour integration when evaluating a Green's function
I was reading the scattering section of Griffiths QM, the part where he derives the integral form of the Schrödinger equation, where he evaluates the following integral for the Green's function of the ...
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$\int_{-\infty}^{\infty}{\frac{1}{x^n+1}\, dx}$ by residues
For $f(x)=\int_{-\infty}^{\infty}{\frac{1}{x^n+1}\, dx}$ I tried using a semi-circle $\gamma$ in the upper half plane. So the integral over the whole contour $C$ is :
$$\oint_{C}=\int_{-R}^{R}{\frac{1}...
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any suggestions for solving this integral involving a complex exponential of trigonometric functions?
I'm looking to find a closed-form (or series) solution for the definite integral
$$\int^{\pi/2}_0 \sin(x) \cos(x) \sin(a*\cos(x)) \exp(i*b*\sin(x)) dx$$
where $a$ and $b$ are real constants and $i$ is ...
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how to choose the correct contour for the given integrals
can anybody tell me how to consider a correct contour for these integrals
$$ \int_{0}^{\infty} \frac{1}{x^\alpha} e^{-ikx} dx $$
$$ \int_{-\infty}^{0} \frac{1}{x^\alpha} e^{-ikx} dx $$
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Need help with a tricky complex integral
$$\int_{|z|=3} \frac{1}{z^2+2},dz$$
I got this on a quiz today, and I proceeded as follows.
I substituted $3e^{iθ}$ for z to get
$$\int_{-π}^{0} \frac{i3e^{iθ}}{(3e^{iθ})^2+2},dz+\int_{0}^{π} \frac{...
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Riemann-Zeta Function at non-integer points
I was wondering if there is a technique for evaluating the Riemann-zeta function at non-integer points please.
I am aware of the technique from complex analysis that when $s \in \mathbb{N}$, we can ...
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Connection between Integration by substitution and Line integral
I have read the following statement about Integration by substitution on wikipedia (https://en.wikipedia.org/wiki/Integration_by_substitution):
Let $U$ be an open set in $\mathbb{R}^{n}$ and $\varphi:...
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$f(z)=\frac{\ln(z)}{(1+z^2)^2}$ contour integral / estimation
Let $f(z)=\frac{\ln(z)}{(1+z^2)^2}$, whereas $\ln(z)$ is chosen such that $-\pi/2 < \arg(z) < 3 \pi/2$ so the branch cut is along the negative imaginary axis.
I now have an intented semicircular ...
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Mellin transform of sin(x) .
Proof of the identity $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$ for $\alpha\in (0,2)$.
In the above post, in the 2nd answer (given by mark ...
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Evaluate $\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$ using contour integration with branch cuts
I want to calculate the integral
$$\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$$
for $\beta>0$ with the residue theorem.
I tried to follow the steps in ...
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$\int_a^b \frac{(x-a)^p(b-x)^{1-p}}{\text{P}(x)} dx$ using contour integration and residue theorem
So a while ago I came across these types of integrals that has the form of
$$
I = \int_a^b (x-a)^p(b-x)^{1-p}dx
$$
Where $a$, $b$ and $p$ are real numbers, $a<b$ and $0<p<1$. And those types ...
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Closed line integral
It sounds like a stupid question. However, can closed line-integrals be defined for one-dimensional real valued functions? To add a little substance to the question. In two dimensions it makes total ...
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How to use Quarter-circle contour?
So I want to know how to evaluate $$\int_0^\infty f(x)dx$$for integrable $f(x)$ using the quarter circle contour, assuming that $f(x)$ has no singularities that aren't poles in that region. So, I know ...
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Can one evaluate Serret's integral using contour integration?
$$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$ This is the integral, and if possible could someone tell me whether we could solve any such type of problem via contour integration.
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Upper bound for modulus of a complex integral
Let $\gamma$ be a close piecewise smooth contour consisting of the three straight lines from $1$ to $2$, from $2$ to $1+i$ and from $1+i$ to $1$.
I have to show that
$$\left|\int_{\gamma}\frac{1}{\...
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Evaluating $\int_{-\infty}^{\infty}\frac{\ln\left(\frac{1}{2}+x+x^{2}\right)}{1+x^{2}}dx$
(Motivation) This is an integral I made up for fun. WolframAlpha doesn't seem to come up with a closed form for it and I'm surprised there doesn't seem to be a duplicate after using Approach0, but I ...
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Another Mellin transform question
Take $1\leq H\leq X$. Let $f(u)$ be a smooth function on $(X,2X)$ and satisfy there $f^{(j)}(u)\ll _jH^j/X^j$ for all $j$. Let $$\mathfrak g(s)=\frac {\Gamma (s/2)^3}{\Gamma ((1-s)/2)^3}\approx t^{3(...
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Strategy for integrating product of powers and trigonometric functions
I want to compute the simple-looking definite integral $$\int_0^\infty \frac{\cos\omega t}{t^a}\,dt,\quad 0<a<1\,,$$
but I have no idea where to begin. Mathematica and integration tables give ...
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Solving $\frac{d^2u}{dt^2} + u = e^{-t}\cos{t}$ using the Laplace Transform
I'm trying to solve the following ODE using the Laplace Transform:
$$\frac{d^2u}{dt^2} + u = e^{-t}\cos{t}$$
With $u(0)$ and $\frac{du}{dt}(0)$ given like that.
I have found that the transform of $e^{-...
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Hints to solve an equation containing $\int\frac{dx}{x\sqrt{(1+x^{-4})^n-1}}$ for $n<0$
I have an integral equation I need to solve. One of the sides has an integral of the form:
$$\int\frac{dx}{x\sqrt{(1+x^{-4})^n-1}}$$
where, for my case, $n < 0$. Mathematica yields no solution for ...
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Growth rate of Mellin transform
For $x>0$ and $0<\sigma <1/6$ consider $$\int _{\sigma \pm i\infty }\underbrace {\frac {\Gamma ^3(s/2)}{\Gamma ^3((1-s)/2)}}_{=:G}\frac {ds}{x^s}$$
which is absolutely convergent since $G\...
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An attempt at generalizing a family of integrals $\int_0^{\alpha}{\frac{\sin \left( \pi x \right)}{x^x\left( \alpha -x \right) ^{\alpha -x}}}\, dx$
Main question
$$
\int_0^1{\frac{\sin \left( \pi x \right)}{x^x\left( 1-x \right) ^{1-x}}}\mathrm{d}x
$$
This is a famous integral. It can be evaluated by contour integration, and it equals to $\pi/e$.
...
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A dodgy contour integration method giving the correct result
Consider the following integral
$$I = \int_{-\infty}^\infty \frac{x}{\sinh x}dx$$
Using contour integration with some rectangular contour, it is not too hard to show that the integral evaluates to $\...
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Evaluating $\int_{-\infty}^{\infty}\frac{\ln\left(1+x^{8}\right)}{x^{2}\left(1+x^{2}\right)^{2}}dx$
(Motivation) Here is an integral I made up for fun:
$$\int_{-\infty}^{\infty}\frac{\ln\left(1+x^{8}\right)}{x^{2}\left(1+x^{2}\right)^{2}}dx.$$
WolframAlpha doesn't seem to come up with a closed form, ...
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How to set up the contour for $ \int_{-\infty}^{+\infty} \frac{e^{iax}}{1+x^4}dx $?
I'm trying to solve the following integral:
$$ \int_{-\infty}^{+\infty} \frac{e^{iax}}{1+x^4}dx $$
From what I understand, since we have the complex exponential we consider the poles not only in the ...
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Analytic continuation of $\int_{-\infty + i \cdot p}^{\infty + i \cdot p} \exp\left[ -w^{2} +\left|w-y- i\cdot p\right|-y^{2}\right]\operatorname{d}w$
I am dealing with the following integral defined in Eq. $(40)$. I am kind of sure that steps from Eqs. $(40)$ to $(41)$, $(41)$ to $(42)$ are both fine.
I tested on Mathematica step from Eq. $(42)$ to ...
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Evaluation of the infinite series $\sum_{n=1}^\infty (-1)^{n-1} \frac{\overline{H}_n}{n \binom{2n}{n}}$
I am trying to evaluate the infinite series
$$\sum_{n=1}^\infty (-1)^{n-1} \frac{\overline{H}_n}{n \binom{2n}{n}}$$
where $\overline{H}_n = 1 - \frac{1}{2}+ \frac{1}{3} - \ldots+ \frac{(-1)^{n-1}}{n}$ ...
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Gauss' formula on positively oriented closed contour
Show for a positively oriented simple closed contour $C$ that the area of the region $G \subset{\mathbb{C}}$ enclosed by C is given by $\frac{1}{2i}\int_C \bar{z}\text{ d}z$. Use the Gauss' integral ...
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Hyperbolic sums $S(n,k)=\sum_{m=1}^{\infty} \frac{1}{\cosh(\pi m)^{n} \sinh(\pi m)^{k}}$
It can be verified that
$$
\sum_{n=1}^{\infty}\frac{1}{\cosh(\pi n)^3\sinh(\pi n)^2}
=\frac{11}{12}-\frac{3K}{2\pi}+\frac{K^2}{2\pi^2}-\frac{K^3}{\pi^3}$$
where $K=\frac{\Gamma\left ( \frac14 \right )^...
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Calculate $\oint\frac{z^{2}}{z-4}dz$ over a contour C which is a circle with $\left|z\right|=1$ in anticlock-wise direction
Question:
Calculate $\oint\frac{z^{2}}{z-4}dz$ over a contour C which is a circle with $\left|z\right|=1$ in anticlock-wise direction.
My Approach:
Using the Cauchy-Integral Formula $f\left(z_{0}\...
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what is wrong with this understanding of contour intergration?
problem statement:
integrate $f(z)=2$ over a curve $C$ in an anticlockwise sense, where $C$ is a semi-circular arc centered at the origin with radius $2$.
math representation:
$$
\int_C f(z) \, dz ;...
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Prove complex integral equality
Suppose $\triangle$ is the open unit disk and $\overline{\triangle}$ be it’s closure (closed unit disk). Let $f$ be holomorphic in an open set containing the set $D = \mathbb{C} - \overline{\triangle}$...
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Inverse fourier transform of $1/(\sqrt x+ia)$
I am trying to analytically calculate $$\int_{-\infty}^\infty d\omega \frac{e^{-i\omega t}}{\sqrt{\omega}+ia},$$
where $a>0$. The integrand has a branch cut with the branch point at $\omega=0$. ...
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Hypergeometric Fox H-function representation with Mellin-Barnes-type contour integrals problem.
I need help with the double complex integral (Mellin-Barnes type). I am solving the following integral:
\begin{equation} \int_0^{\infty}x^{b-1}\mbox{}_1{\rm{F}}_1(a,b, -c x)e^{-s x^\delta}e^{-\...
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