- #1
Combinatus
- 42
- 1
Change of variables for integral
Determine whether the following equality holds:
[itex]\displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(x^2+4z^2)}{4z}}}{2z} \, \mathrm{d}z = \displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(1+z^2)x}{2z}}}{2z} \, \mathrm{d}z, \forall x,z \in \mathbb R_+[/itex]
I obtained the first integral after fiddling around with the integral
[itex]K_0(x) := \displaystyle\int_0^{\infty} e^{-x\cdot\cosh{z}} \, \mathrm{d}z[/itex]
using the substitution [itex]\tau := \dfrac{xe^z}{2}[/itex]. Note that [itex]K_0(x)[/itex] is a modified Bessel function of the second kind of order 0. The second integral was something I came across after playing around with Rohatgi's "product convolution" integral for independent exponential random variables for a while. Mathematica claims that the second integral is also [itex]K_0(x)[/itex], and numerical integration seems to show that the original equality holds. The integrals look similar enough that some simple transformation seems relevant, but I haven't managed to find one yet.
Obviously, determining how Mathematica concluded that the second integral is [itex]K_0(x)[/itex] would be helpful, too.
Hints?
Homework Statement
Determine whether the following equality holds:
[itex]\displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(x^2+4z^2)}{4z}}}{2z} \, \mathrm{d}z = \displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(1+z^2)x}{2z}}}{2z} \, \mathrm{d}z, \forall x,z \in \mathbb R_+[/itex]
Homework Equations
The Attempt at a Solution
I obtained the first integral after fiddling around with the integral
[itex]K_0(x) := \displaystyle\int_0^{\infty} e^{-x\cdot\cosh{z}} \, \mathrm{d}z[/itex]
using the substitution [itex]\tau := \dfrac{xe^z}{2}[/itex]. Note that [itex]K_0(x)[/itex] is a modified Bessel function of the second kind of order 0. The second integral was something I came across after playing around with Rohatgi's "product convolution" integral for independent exponential random variables for a while. Mathematica claims that the second integral is also [itex]K_0(x)[/itex], and numerical integration seems to show that the original equality holds. The integrals look similar enough that some simple transformation seems relevant, but I haven't managed to find one yet.
Obviously, determining how Mathematica concluded that the second integral is [itex]K_0(x)[/itex] would be helpful, too.
Hints?
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