Question

Closed-form integral $\int_{-1}^1 e^{a x^2 + bx}/\sqrt{1-x^2}dx$

score 235 · Answer 1 · 2016-08-03 13:42:23

235

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Closed-form integral $\int_{-1}^1 e^{a x^2 + bx}/\sqrt{1-x^2}dx$

Published on 03 Aug 2016 - 13:42

score 3.5k · Answer 2 · 2016-08-03 23:46:04

3.5k

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Prove equation using Bessel functions: $\frac{d}{dx}[xJ_n(x)J_{n+1}(x)]=x[J_n^{2}(x)-J_{n+1}^{2}(x)]$

Published on 03 Aug 2016 - 23:46

score 413 · Answer 3 · 2016-08-04 06:47:52

413

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Proof $\int_0^\infty \frac{\exp(-\sqrt{x^2+y^2})}{x^2+y^2}dx = \frac{\pi}{2}\left(\frac{1}{y} - K_0(y)L_{-1}(y) - K_1(y)L_{0}(y)\right)$

Published on 04 Aug 2016 - 6:47

score 147 · Answer 4 · 2016-08-10 15:15:21

147

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Help with calculating Fourier transform of $\left(\frac{x-i}{x+i}\right)^{n}\cdot \frac{1}{(x^2+1)^{s}}$

Published on 10 Aug 2016 - 15:15

score 310 · Answer 5 · 2016-08-11 08:15:51

310

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Solving the Integral $\int_0^{\infty} \, \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right) \, \mathrm{cos}(b \, x) \,\mathrm{d}x$

Published on 11 Aug 2016 - 8:15

score 169 · Answer 6 · 2026-03-25 03:18:16

169

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Definite Integral resulting in Bessel and Struve Function

Published on 25 Mar 2026 - 3:18

score 476 · Answer 7 · 2016-08-15 14:22:00

476

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Integral of product of Bessel functions

Published on 15 Aug 2016 - 14:22

score 386 · Answer 8 · 2016-08-15 15:26:00

386

Views

Hankel's expansion for the first Bessel function

Published on 15 Aug 2016 - 15:26

score 72 · Answer 9 · 2016-08-15 22:26:02

72

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Determine $a_1$ by substituting the bessel function into the ODE

Published on 15 Aug 2016 - 22:26

score 201 · Answer 10 · 2016-08-16 16:53:34

201

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Evaluate integral $\int_{-\pi/2}^{\pi/2} \cos \left[\frac{\pi n}{2} +\left(a \sin t+b \cos t \right) \right] dt$, Bessel?

Published on 16 Aug 2016 - 16:53

score 113 · Answer 11 · 2016-08-26 17:07:30

113

Views

Help with integral involving the Bessel Functions of the First kind

Published on 26 Aug 2016 - 17:07

score 1.2k · Answer 12 · 2016-08-28 09:59:57

1.2k

Views

Fourier Cosine Transform (Parseval Identity) for definite integral

Published on 28 Aug 2016 - 9:59

score 202 · Answer 13 · 2016-08-29 15:48:29

202

Views

Evaluate the integral $\int_0^\pi \sin{(x \cos{t}})\cos{t}\; dt$

Published on 29 Aug 2016 - 15:48

score 762 · Answer 14 · 2016-08-30 06:48:40

762

Views

How to integrate $\int_1^\infty e^{-\alpha x}J_0(\beta\sqrt{x^2-1})\mathrm{d}x \,$?

Published on 30 Aug 2016 - 6:48

score 564 · Answer 15 · 2026-03-28 17:42:05

564

Views

Asymptotics of Gaussian integral over the unit sphere

Published on 28 Mar 2026 - 17:42

List Question

Closed-form integral $\int_{-1}^1 e^{a x^2 + bx}/\sqrt{1-x^2}dx$

Prove equation using Bessel functions: $\frac{d}{dx}[xJ_n(x)J_{n+1}(x)]=x[J_n^{2}(x)-J_{n+1}^{2}(x)]$

Proof $\int_0^\infty \frac{\exp(-\sqrt{x^2+y^2})}{x^2+y^2}dx = \frac{\pi}{2}\left(\frac{1}{y} - K_0(y)L_{-1}(y) - K_1(y)L_{0}(y)\right)$

Help with calculating Fourier transform of $\left(\frac{x-i}{x+i}\right)^{n}\cdot \frac{1}{(x^2+1)^{s}}$

Solving the Integral $\int_0^{\infty} \, \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right) \, \mathrm{cos}(b \, x) \,\mathrm{d}x$

Definite Integral resulting in Bessel and Struve Function

Integral of product of Bessel functions

Hankel's expansion for the first Bessel function

Determine $a_1$ by substituting the bessel function into the ODE

Evaluate integral $\int_{-\pi/2}^{\pi/2} \cos \left[\frac{\pi n}{2} +\left(a \sin t+b \cos t \right) \right] dt$, Bessel?

Help with integral involving the Bessel Functions of the First kind

Fourier Cosine Transform (Parseval Identity) for definite integral

Evaluate the integral $\int_0^\pi \sin{(x \cos{t}})\cos{t}\; dt$

How to integrate $\int_1^\infty e^{-\alpha x}J_0(\beta\sqrt{x^2-1})\mathrm{d}x \,$?

Asymptotics of Gaussian integral over the unit sphere

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