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

[geetar_king]

Homework Statement

$$T(x,t)$$

What is the Fourier transform of
$$\frac{1}{x}\frac{\partial T}{\partial x}$$

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

Homework Equations

The Attempt at a Solution

Can this be split up using convolution into...

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

 

[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
$$\frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}<br />$$

for $$0 < x < \infty$$ and initial condition like $$T(x,0) = g(x)$$
with boundary conditions $$T(\infty,t) = C_{1}$$ and $$T(0,t) = f(t)$$

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