- #1
fluidistic
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Homework Statement
Hi guys! I'm basically stuck at "starting" (ouch!) on the following problem:
Using the integral representation of the Bessel function [itex]J_0 (x)=\frac{1}{\pi} \int _0 ^\pi \cos ( x\sin \theta ) d \theta[/itex], find its Laplace transform.
Homework Equations
[itex]\mathbb{L} [f(x)]=\int _0 ^{\infty} e^{-sx} f(x)dx[/itex].
The Attempt at a Solution
So I simply applied the formula above and could not solve the integral in theta.
Namely [itex]\mathbb{L} [J_0 (x)]=\frac{1}{\pi} \int _0^{\infty} e^{-sx} \int _0 ^ \pi \cos (x \sin \theta ))d\theta dx[/itex].
My only idea is to evaluate the theta integral first, treating x as a constant. So I've something of the form [itex]\int \cos (k \sin \theta ))d\theta[/itex] to calculate. I am not sure this is the way to go. And if it is, I don't have any idea on how to evaluate the integral.
Any idea is welcome.