Tagged Questions

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

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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 ...
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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 ...
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Integration of a analytic function

here is the problem I currently try to solve: $$\int\limits_{-\infty}^{+\infty}\left((1+ixa^2)^{-\frac{n_1}{2}}\cdot(1+ixb^2)^{-\frac{n_2}{2}}\right)e^{icx} \mathrm{d}x$$ with $a,b,c\geq0$ (real ...
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Generalization to this integral

$$\int_0^\infty \frac{\ln(1 + x^a)x^s}{1+x^2} \ dx$$ Actually the problem was $\displaystyle \int_0^\infty \frac{\ln(1 + x^a)}{(1+x^2)\ln(x)} \ dx$. But I guess the form of a Mellin Transform ...
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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 ...
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Evaluate $\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx$

Prove that $$\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx=\frac{\pi\sqrt{2}}{2}\log\left(1+\frac{\sqrt{2}}{2}\right).$$ I managed to prove this result with some ...
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Evaluating the integral $\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt$

It is stated that (for $\lambda>0$) $$\frac{1}{\pi}\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt = 1-\frac{|\mu|}{\lambda}$$ for $0\leq|\mu|\leq\lambda$, and zero ...
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Problem with setting limits on a Line Integral

Problem: Evaluate the line integral of the vector field $$f(x,y)=(x^2-2xy)i+(y^2-2xy)j$$ from $(-1,1)$ to $(1,1)$ along the parabola $y=x^2$. I've never tried to compute Line Integrals before ...
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Question about Meijer-G definition and identity

I'm trying to wrap my mind around computation involving the Meijer $G$ function, as defined here. (Edit: I'm actually using a somewhat mixed notation using the definitions from MathWorld and the ...
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Branch cuts for the contour integral $\int_{-1}^{1} \frac{\ln(x+a)}{(x+b) \, \sqrt{1-x^{2}}} \, dx$

How can the branch cut be handled in the contour integral, for $|b| \leq 1, \, a > 1$, $$\int_{-1}^{1} \frac{\ln(x+a)}{(x+b) \, \sqrt{1-x^{2}}} \, dx \quad ?$$ If $a=1$ can the value of the ...
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What is a “Contour Integral” and how do I evaluate one?

A very general question, I apologize, but as you read this, hopefully you get what I'm asking. Recently, Bernoulli Numbers have caught my eye, for I am studying infinite series' and it is a part of ...
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Integrate $\int_{-\infty}^{\infty} \frac{\cosh(\beta x)}{1+\cosh( \beta x )} e^{-x^2} x^2 \rm{d}x$

Integrate $$\int_{-\infty}^{\infty} \frac{\cosh(\beta x)}{1+\cosh( \beta x )} e^{-x^2} x^2 \rm{d}x,$$ with $\beta \in \mathbb{R}$ and $\beta > 0$. Numerical integration shows that this ...
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Integral using Cauchy's integral formula and residue theorem

So, I'm having trouble getting the correct value for the integral $\int_0^{2\pi} \frac{\cos^2(3\theta)}{5-4\cos(2\theta)}\mathrm{d}\theta$. I substitute the exponential form of cosine into the ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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Translated complex gaussian-type integral: $\int_0^{\infty} \exp(i(t-\alpha)^2) dt$

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 ...
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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 ...
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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) = ...
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??