Tagged Questions

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

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Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
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Limit of complex integral with no primitive

I'm having trouble trying to calculate the following limit. I know the answer is not 0, but after several attempts I am stuck on reducing it. We have $z_0$ as a constant complex number and a fixed ...
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Residue of pole

I am trying to integrate $$\frac{1}{2\pi} \int^\infty_{-\infty} \frac{6e^{-ipt}}{(p+1)^2 +9} dp$$ I am using a D-contour and I am trying to calculate the residue at the pole $p = -1$. I am trying to ...
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I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
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Convolution of complex functions (Laplace Domain)

Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in ...
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Proof that $\int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4$?

Can one prove that $$\int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4?$$ I would prefer using the methods of contour integration.
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Is there anything wrong with the following work on the Argument Principle?

The Argument Principle states that : $$\oint_C {d\over dz}(log (f(z))) \, dz = 2\pi i(N-P)$$ Let $g(z)={d\over dz}\log(f(z))$ If $f: C \to C$ is a continuous function on a directed smooth curve, ...
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how to calculate $\int_{0}^{\infty}\frac{x}{\sqrt{e^x-1}}\mathrm{d}x$

I was trying to solve another integral when then I reached this, I've no idea of how to select the contour for the integration.
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What is the integral of 1/(z-i) over the unit circle?

At present there is a simple pole on the closed contour, so the Residue Theorem appears to be inapplicable. But I want to claim that we can enlarge this circle to make sure that it encloses the ...
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What is the Fourier transform of $\exp(2 \pi i / x)$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
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Integrating secans over the imaginary axis using the residue theorem

I am trying to integrate $\sec(z)$ over the whole imaginary axis using the residue theorem. i.e., I want to calculate the integral $$\int_{\Gamma} \frac{dz}{\cos{z}}$$ where $\Gamma$ is the (open) ...
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Indefinite integral - tricks for expressing solution concisely

Consider the following indefinite integral: $I_n (b) = \int \mathrm{d}x \frac{\sin(nx) \sin(x)}{\cos^2(x)+b^2}$ where $b$ is a constant and $n = 1, 2, 3...$ Is it possible to write the solution ...
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Can any contour integral be calculated by direct methods?

Or is there ever an integral that can only be evaluated using integral theorems? Does substituting the parameterized contour into the equation and evaluating it as a Riemann integral always work?
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Confusion about contour integration of constant function: intuition vs. Residue Theorem

Let's say we have the holomorphic function $$f(z) = 1.$$ Because $f(z)$ has no poles, according the Residue Theorem we have $$\oint_\gamma f(z)\,dz = 0$$ for any closed counterclockwise path $\gamma$. ...
$\int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta$
Let $w$ be a complex number such that $|w| < 1$. Evaluate the integral $$\int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta.$$ I am having a hard time moving forward on this question. I tried ...
Use the residue theorem to compute the real integral: $$I = \int_{0}^{2\pi} \sin^{2n}\theta d\theta$$ I have considered a contour around a unit circle C, and used the substitutions: \$sin\theta = \...