How to Handle the Distribution 1/(x-i0)^2?

[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?

 

[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).

 

[George Jones]

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

I think this means "tempered distribution" and "regular distribution that is also a tempered distribution."

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

I think so. This is usually denoted $x- i \epsilon$.

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).

But I think the idea behind the question is to gain familiarity with distribution theory.

Any idea?

$$\frac{1}{\left( x - i \epsilon \right)^2} = - \frac{d}{dx} \left[ \frac{1}{ x - i \epsilon} \right]$$

 

[ziojoe]

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