Green's Function Homework: Real & Imaginary Parts

[Mancho]

Homework Statement

I'm asked to calculate Green's function's real and imaginary parts.
The expression for the given Green's function is:

g₀₀=[1-(1-4t²(z-E₀)^-2)^1/2]/2t²(z-E₀)-¹ (1)
Where, z is the complex variable: z= E+iO⁺ (2)

Homework Equations

Complex number definition: Z = x + iy, where x is the real part and iy - imaginary.

The Attempt at a Solution

To separate real and imaginary parts I tried to express g₀₀ in the form: g₀₀= x+iy

First I simplified the (1) by multiplying the denominator and numerator by (z-E₀).
The result is g₀₀= 1/2t²[(z-E₀)-(1-4t²)]^1/2.

Then I'm stuck. I don't know how to remove the square root to divide real and imaginary parts. I'm not even sure if it is the pure math problem or if I have to take into consideration anything else.

I would appreciate any help.

 

[tiny-tim]

Welcome to PF!

I'm asked to calculate Green's function's real and imaginary parts.
…
I don't know how to remove the square root to divide real and imaginary parts.

Hi Mancho! Welcome to PF!

Hint: the square-root of re^iθ is (√r)e^iθ/2, and so its real and imaginary parts are (√r)cos(θ/2) and (√r)sin(θ/2)

 

[Mancho]

Thanks a lot tiny-tim!

I guessed from your hint I had to use polar coordinates and it worked great! I got what I was supposed to. Maybe I will upload the solution later when I have time.

Thanks again! :)