Solution for $int_0^infty e^{-(ct)^alpha} cos(x t) dt$
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I'm trying to evaluate the integral:
$int_0^infty e^{-(ct)^alpha} cos(x t) dt$, where $alpha$ and $c$ are parameters.
This integral arises from trying to solve for the probability density for a symmetric $alpha$-stable probability distribution.
Can this integral be expressed in terms of some special functions, or at least does it have a ready-made numerical solution method?
probability-distributions definite-integrals
asked Nov 15 at 23:28
Carlos Danger
1327
add a comment |
up vote
1
down vote
favorite
I'm trying to evaluate the integral:
$int_0^infty e^{-(ct)^alpha} cos(x t) dt$, where $alpha$ and $c$ are parameters.
This integral arises from trying to solve for the probability density for a symmetric $alpha$-stable probability distribution.
Can this integral be expressed in terms of some special functions, or at least does it have a ready-made numerical solution method?
probability-distributions definite-integrals
asked Nov 15 at 23:28
Carlos Danger
1327
I do not think these can be calculated explicitly.
– Will M.
Nov 15 at 23:42
1
For $alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $alpha$, I have no idea.
– Andy Walls
Nov 16 at 0:00
Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals?
– Moti
Nov 16 at 2:08
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm trying to evaluate the integral:
$int_0^infty e^{-(ct)^alpha} cos(x t) dt$, where $alpha$ and $c$ are parameters.
This integral arises from trying to solve for the probability density for a symmetric $alpha$-stable probability distribution.
Can this integral be expressed in terms of some special functions, or at least does it have a ready-made numerical solution method?
probability-distributions definite-integrals
asked Nov 15 at 23:28
Carlos Danger
1327
I'm trying to evaluate the integral:
$int_0^infty e^{-(ct)^alpha} cos(x t) dt$, where $alpha$ and $c$ are parameters.
This integral arises from trying to solve for the probability density for a symmetric $alpha$-stable probability distribution.
Can this integral be expressed in terms of some special functions, or at least does it have a ready-made numerical solution method?
probability-distributions definite-integrals
probability-distributions definite-integrals
asked Nov 15 at 23:28
Carlos Danger
1327
asked Nov 15 at 23:28
Carlos Danger
1327
edited Nov 15 at 23:38
asked Nov 15 at 23:28
Carlos Danger
1327
asked Nov 15 at 23:28
Carlos Danger
1327
asked Nov 15 at 23:28
Carlos Danger
1327
1327
I do not think these can be calculated explicitly.
– Will M.
Nov 15 at 23:42
1
For $alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $alpha$, I have no idea.
– Andy Walls
Nov 16 at 0:00
Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals?
– Moti
Nov 16 at 2:08
add a comment |
I do not think these can be calculated explicitly.
– Will M.
Nov 15 at 23:42
1
For $alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $alpha$, I have no idea.
– Andy Walls
Nov 16 at 0:00
Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals?
– Moti
Nov 16 at 2:08
I do not think these can be calculated explicitly.
– Will M.
Nov 15 at 23:42
I do not think these can be calculated explicitly.
– Will M.
Nov 15 at 23:42
1
1
For $alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $alpha$, I have no idea.
– Andy Walls
Nov 16 at 0:00
For $alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $alpha$, I have no idea.
– Andy Walls
Nov 16 at 0:00
Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals?
– Moti
Nov 16 at 2:08
Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals?
– Moti
Nov 16 at 2:08
add a comment |
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I do not think these can be calculated explicitly.
– Will M.
Nov 15 at 23:42
1
For $alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $alpha$, I have no idea.
– Andy Walls
Nov 16 at 0:00
Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals?
– Moti
Nov 16 at 2:08