Help with Fourier transform of T'(x)/x

[geetar_king]

Homework Statement

T(x,t)

What is the Fourier transform of
<br /> \frac{1}{x}\frac{\partial T}{\partial x}<br />

<br /> F(\frac{1}{x}\frac{\partial T}{\partial x}) = \int^{\infty}_{-\infty} \frac{1}{x}\frac{\partial T}{\partial x} e^{i \theta x}dx = ??<br />

Homework Equations

The Attempt at a Solution

Can this be split up using convolution into...

F(\frac{1}{x}\frac{\partial T}{\partial x}) = F(\frac{1}{x})F(\frac{\partial T}{\partial x}) =\int^{\infty}_{-\infty} \frac{1}{x} e^{i \theta x}dx \int^{\infty}_{-\infty} \frac{\partial T}{\partial x} e^{i \theta x}dx<br />

 

[e(ho0n3]

You will need to prove that the convolution of 1/x and dT/dx is (1/x)(dT/dx).

 

[geetar_king]

Okay thanks. I was trying to solve
<br /> \frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}<br /> <br />

for <br /> 0 &lt; x &lt; \infty <br /> and initial condition like <br /> T(x,0) = g(x) <br />
with boundary conditions <br /> T(\infty,t) = C_{1} <br /> and <br /> T(0,t) = f(t)<br />

I got stuck with separation of variables and method of characteristics so I was going to try Fourier method