Tagged Questions

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An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
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Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
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integral over product of two bessel functions at discontinuity

The Weber-Schafheitlin integral $$\int_{0}^{\infty}\frac{J_{\mu}(a t)J_{\nu}(bt)}{ t^{\lambda}}$$ where $J_{\mu}(x)$'s are Bessel functions of the first kind, can have delta function singularities ...
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Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
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Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
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How to show that $\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt = \pi^{1/2} \frac{ \Gamma(s-1/2)}{\Gamma(s)}$

I want to compute the integral $$\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt$$ for $s \in \mathbb{C}$ such that the integral converges ($\mathrm{Re}(s) > 1/2$ I think) in terms of the Gamma ...
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Solution of definite integrals involving incomplete Gamma function

The solution of the integral $$\int_0^{\infty}e^{-\beta x}\gamma(\nu,\alpha \sqrt x)dx$$ is given as ...
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Solution of some Bessel integrals

The solution of the integration $\int_0^\infty e^{-\alpha x}J_v(\beta x)x^{\mu-1}dx$ is given in a standard form. Can I use the same result when the upper limit of the integration is finite? The ...
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Gaussian-Like integral

What is the integral of this? $$\int_0^\infty xe^{-(ax^2+bx)}\,\mathrm{d}x$$ $a$ and $b$ are positive integers.
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Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1$

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
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How to evaluate the integral involving two Bessel functions and following elementary function?

Let's have the integral $$\int \limits_{0}^{\infty} J_{0}(kr) J_{0}(kr')\frac{kdk}{k^2 + a^2}.$$ How to evaluate it? I failed when was trying using integral representations for the Bessel ...
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How to evaluate following integral

I tried to evaluate integral $$I = \int \limits_{0}^{\infty}\frac{J_{\nu}(bx)x^{\nu + 1}dx}{(x^{2} + a^{2})^{\mu + 1}},$$ but I failed. I can't represent Bessel function as a series, because ...
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Closed-form Solution of an Integral Equation

Recently, I get stuck in an equation, and I would like to obtain an closed-form solution of an equation, which is given as $$\int_1^\infty\frac{x}{t^\alpha-x}dt=c,$$ where $\alpha>1$ and $c>0$ ...
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Limit of the integral, integral of the limit (no dominated convergence)

I'm working on the integral: $\lim_{x \rightarrow 0} \int_0^T \frac{1}{\sqrt{\pi t}} \exp \left(-\frac{x^2}{t}-t\right) \left(1-\frac{t}{T} \right) \mbox{d}t$ I'ld like to inverse the integral and ...
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Integral of function with a pretty long name

I am looking for a compact result of this integral: $$\int_R^\infty k_n(x) \, dx,$$ where $k_n$ is the modified spherical Bessel function of the second kind (explanation to this function). ...
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Oscillating integral

I want to calculate $$\int _0^\infty e^{-iyx}\sqrt{x(x+2)}\, dx$$ in the sense of distributions, at least for $y\ne 0$. Now, I happen to know the following integral representation for the modified ...
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A method for calculating this integral hermite polynomials

I need proof this, $\int_{-\infty}^{\infty}e^{-x^2}H_n^{2}(x)x^2dx=2^nn!\sqrt{\pi}(n+\frac{1}{2})$ This is the idea: Multiply ...
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Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
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Is there a formula for this integral

Is there a formula for the following integral? $$I(a,b)=\int_0^1 t^{-3/2}(1-t)^{-1/2}\exp\left(-\frac{a^2}{t}-\frac{b^2}{1-t} \right)dt$$ where $a,b$ are non-zero real numbers.
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Improper integral and special functions

I'd like to have an expression of the following integral: $$\int_0^{+\infty} \left(\sqrt{1+x^4} - x^2\right) dx$$ in terms of some special functions (but not in the form given by Wolfram Alpha).
Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$
I need help with solving this integral: $$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$ where $\text{Li}_{s}(z)$ is the polylogarithm.