Fourier Transform (Another question):

MathematicalPhysicist

I need to find the fourier transform of f(x) which is given by the equation:
[tex]-\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|)[/tex]

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?

thanks in advance.

jhicks

Without doing the fourier transform, it looks to me like you'll need to use the property that [itex]\delta(x'-x)f(x')=f(x)[/itex]. 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.

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?

jhicks

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

MathematicalPhysicist

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