Position to momentum space in three dimensions

[Cleo]

Homework Statement

Mathematical complication solving Q.M. problem.

Relevant Equations

Im [(ip/ħ-1/a)^(-2)]

Hi! I am trying to change the hydrogen ground state wave funcion from position to momentum space, so i solved the integral
Ψ(p)=(2πħ)^(-3/2)

(πa^3)^(-1/2)​

∫∫∫e^(prcosθ/ħ) e^(-r/a) senθ r^2 dΦdθdr
and got 4πħ(2πħ)^(-3/2) p^(-1) (πa^3)^(-1/2) I am [(ip/ħ-1/a)^(-2)], which according to the professor's solution is ok, but then i don't know which is the imaginary part that i need to pick. p is the momentum, i is the imaginary unit, a is Bohr radius, r is the radial coordinate, and I am means imaginary part of what is in the brackets.
So basically mi question is: How do i separate the imaginary part from the real part in this expression?:

(ip/ħ-1/a)^(-2)​

The professor said that the final result is Ψ(p)=(8πħ^4)/[(2πħ)^(-3/2) a^2 (4πa)^(1/2) (p^2+ħ^2/a^2)^2], but i have no clue of how he did that final step. I guess it shouldn't be very difficoult but i have been long trying to figure out how to do it and for some reason i don't get it. It would be very nice if some of you could help me. Thanks!

 

[vela]

To get $i$ out of the denominator, multiply the top and bottom by the complex conjugate:
$$\frac{1}{a+bi} = \frac{a-bi}{(a+bi)(a-bi)} = \frac{a}{a^2+b^2} - i \frac{b}{a^2+b^2}$$

 

[Cleo]

Ahhh thanks a lot!