# Fourier integral help

1. Apr 4, 2008

### mhill

1. The problem statement, all variables and given/known data

calculate the fourier transform $$\int_{-\infty}^{\infty}dx e^{iux} (a+x)^{-1/2}$$ and $$\int_{-\infty}^{\infty}dx e^{iux} (a-x)^{-1/2}$$

2. Relevant equations

see statement above

3. The attempt at a solution

i believe that making a change of variable (a+x)=t or (a-x)=t depending of the problem i can convert the initial problem into this one

$$\int_{-\infty}^{\infty}dx e^{iut} (t)^{-1/2}$$

and using the property of the square root $$(-x)^{1/2}=ix^{1/2}$$ with "i" the root of -1 then this integral is equal to the real valued integral

$$\int_{0}^{\infty}dt sin(ut) (t)^{-1/2}$$ which is for me easier to calculate, my question is if the reasoning made is correct or if i forgot a constant in the results.

also using distribution theory and tables i reachet to the result

$$\int_{-\infty}^{\infty}dx e^{iux} (x)^{-1/2}=C(-iu)^{-1/2}sgn(u)$$ with 'C' a constant relating $$\Gamma (-1/2)$$ and $$-i\pi$$ sgn(x) is the sign function that values -1 for negative values and 1 for positive ones.

Last edited: Apr 4, 2008