Question

How does $\int\limits_0^\infty{{x^{v-1}}{e^{-\frac{\beta}{x}-\gamma x}}dx}=2{({\frac{\beta}{\gamma}})^{\frac{v}{2}}}{K_v}({2\sqrt{\beta\gamma}})$?

score 165 · Answer 1 · 2020-12-02 05:01:48

165

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How does $\int\limits_0^\infty{{x^{v-1}}{e^{-\frac{\beta}{x}-\gamma x}}dx}=2{({\frac{\beta}{\gamma}})^{\frac{v}{2}}}{K_v}({2\sqrt{\beta\gamma}})$?

Published on 02 Dec 2020 - 5:01

score 159 · Answer 2 · 2026-02-23 08:18:51

159

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score 153 · Answer 3 · 2020-12-04 00:30:06

153

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score 614 · Answer 4 · 2020-12-09 12:16:59

614

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score 1.5k · Answer 5 · 2020-12-10 14:02:31

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score 80 · Answer 6 · 2020-12-12 08:02:26

80

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score 65 · Answer 7 · 2026-03-29 20:56:00

65

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score 62 · Answer 8 · 2020-12-14 16:00:45

62

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$J_0(\gamma)=\sum_{m=-\infty}^\infty(J_m(a)J_m(\beta)e^{im\theta})$ and a.β,γ be triangle edges

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score 138 · Answer 9 · 2026-04-01 16:28:55

138

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48

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score 101 · Answer 11 · 2020-12-20 14:41:15

101

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111

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36

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120

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score 223 · Answer 15 · 2020-12-24 14:09:54

223

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List Question

How does $\int\limits_0^\infty{{x^{v-1}}{e^{-\frac{\beta}{x}-\gamma x}}dx}=2{({\frac{\beta}{\gamma}})^{\frac{v}{2}}}{K_v}({2\sqrt{\beta\gamma}})$?

Help Simplifying Function with Bessel Functions

How to derive a Hypergeometric function from the integration of the square of Bessel function

Contour integration of a Bessel function using the Hankel function

Evaluate the Modified Bessel function of the second kind $K_0(z)$ for $z \to 0$

Integrate: $\int\limits_0^{2\pi } {\exp \left( {ar\cos ({\psi _a} - \theta ) + b{r^2}\cos ({\psi _b} - \theta )....} \right)} d\theta $?

Integral of Bessel functions times $x^3$ (diferent eigenvalues)

$J_0(\gamma)=\sum_{m=-\infty}^\infty(J_m(a)J_m(\beta)e^{im\theta})$ and a.β,γ be triangle edges

Solving the wave equation on the unit disk

A Laplace-Bessel transform?

How to integrate $\int_{0}^{\infty} t^{-\frac{3}{2}} e^{-at - \frac{b}{t}}dt$ without gamma function?

On Fourier-Bessel expansion

Finding other linearly indpendent solution other that $J_0(x)$

Is there any analytical approximation to this integral?

Integrals related to the modified Bessel functions of the second kind

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