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

  • #1

Homework Statement



[tex]T(x,t)[/tex]

What is the fourier transform of
[tex]
\frac{1}{x}\frac{\partial T}{\partial x}
[/tex]

[tex]
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 = ??
[/tex]

Homework Equations





The Attempt at a Solution



Can this be split up using convolution into...

[tex] 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
[/tex]
 

Answers and Replies

  • #2
1,357
0
You will need to prove that the convolution of 1/x and dT/dx is (1/x)(dT/dx).
 
  • #3
Okay thanks. I was trying to solve
[tex]
\frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}

[/tex]

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

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

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