# Problem in distribution theory

1. Jul 8, 2009

### ziojoe

I have a little problem with the following exercise:
"Consider the temperate distribution

$$f\left(x\right)=\frac{1}{\left(x-i0\right)^2}$$

Write f(x) like function of elementary temperate distributions and calculate its Fourier-transform."
I am almost sure I have to use the identity

$$\frac{1}{x-i0}=PP\frac{1}{x}+i\pi\delta\left(x\right)$$

But the square makes appear terms like $$\delta^2\left(x\right)$$, that is not a distribution.

Any idea?

2. Jul 9, 2009

### turin

What is "the temperate distribution", and what is an "elementary temperate distribution"?

What is "i0"? E.g. is 0 simply an arbitrarily small positive number?

The original equation defines a function with one double pole (assuming my example interpretation of 0 above). I don't see any need to use distribution theory. You can evaluate the Fourier transform using the Cauchy Integral Theorem (assuming my example interpretation of 0 above).

3. Jul 9, 2009

### George Jones

Staff Emeritus
I think this means "tempered distribution" and "regular distribution that is also a tempered distribution."
I think so. This is usually denoted $x- i \epsilon$.
But I think the idea behind the question is to gain familiarity with distribution theory.
$$\frac{1}{\left( x - i \epsilon \right)^2} = - \frac{d}{dx} \left[ \frac{1}{ x - i \epsilon} \right]$$

4. Jul 10, 2009

### ziojoe

Thanks, that was exactly the answer I got myself after a while. Thanks again.