# Fourier Transform (Another question):

1. Mar 8, 2008

### MathematicalPhysicist

I need to find the fourier transform of f(x) which is given by the equation:
$$-\frac{d^2f(x)}{dx^2}+\frac{1}{a^3}\int_{-\infty}^{\infty}dx'exp(-\lambda|x-x'|)f(x')=\frac{b}{a^2}exp(-\lambda|x|)$$

ofcourse Iv'e taken the fourier tarnsform of both sides, but I don't see how to calcualte the fourier tranform of the integral in the above equation, I feel I need to use the definition of dirac's delta function, but don't see how to do this, any ideas, hints?

Last edited: Mar 8, 2008
2. Mar 8, 2008

### jhicks

Without doing the fourier transform, it looks to me like you'll need to use the property that $\delta(x'-x)f(x')=f(x)$. Just note which of x and x' is actually a variable for integration and which is the "constant" inside the integration. Use Parseval's Theorem.

Last edited: Mar 8, 2008
3. Mar 8, 2008

### MathematicalPhysicist

I'm not sure it's correct I got that the F.T of the integral without the constant 1/a^3 is:
2pi*f(0)*e^(a-\lambda|x|), is this correct?

4. Mar 8, 2008

### jhicks

Well actually disregard my previous advice, sorry! Note that the integral is equivalent to $e^{-\lambda |x|} \ast f(x)$ where the operator is convolution. In the frequency domain, this becomes multiplication. Now you simply need to know $e^{-\lambda |x|} \Leftrightarrow \frac{2\lambda}{\lambda^2 + \omega^2}$ which is actually the same answer as you'd arrive by from what I previously said, but in much less time.

Last edited: Mar 8, 2008
5. Mar 8, 2008

### MathematicalPhysicist

so using convolution you say, ok I'll try it, thanks.