About Complex Numbers, Real part and Imaginary part (symbolic calculus)

Ok, I need some great help otherwise I’ll go nuts.

I have this function:

 f(s) = \frac{\pi}{2\sqrt{s}} – \frac{\text{arcsec}(\sqrt{s})}{\sqrt{s-1}}

What I have to do is to take $s = +i\omega$ , hence it becomes

 f(s = +i\omega) = \frac{\pi}{2\sqrt{i\omega}} – \frac{\text{arcsec}(\sqrt{i\omega})}{\sqrt{i\omega -1}}

No big deal so far.

Henceforth the pain: I have to separate Real and Imaginary part. Since I ignore how to do it with Mathematica (because it won’t do anything, since I am not able to tell it that $\omega$ is a natural number), I tried to do it by hands.

No big problem in finding that (piece by piece)

 \sqrt{i\omega} = \frac{\sqrt{2\omega}}{2}[1 + i]

and maybe also that

 \sqrt{i\omega -1} = \large e^{\frac{1}{4}\ln(1+\omega^2)[i\sin(\arctan(\frac{\sqrt{1+\omega^2}+1}{\omega^2})) + \cos(\arctan(\frac{\sqrt{1+\omega^2}+1}{\omega^2}))]}

And then again no problem in understanding that

 \sin\arctan(Y)) = \frac{Y}{\sqrt{1 + Y^2}}

and a similar identity for the cosine.

And again: no problem in computing the hellish arc secant term to find

 \text{arcsec}(\sqrt{i\omega}) = \frac{\pi}{2} + \frac{1}{\sqrt{2\omega}}(1 – i) + \frac{1}{2}\sqrt{1 + \frac{1}{\omega^2}} + i\arctan(\omega – \sqrt{\omega^2+1})

Fiiiiine!

But what if I wanted to check?

I don’t claim for the complete commands, I just need some help with the procedure to make Mathematica to do those calculations for me, or at least to simplify a bit the things.

Because now I have to arrange the whole expression, and I will have to find the real and imaginary part of that amusing gizmo.

So to be clear: how could I make Mathematica to find real and imaginary part of a complex numbers with a symbolic calculation? Here $\omega$ is a natural number. Also I’d like to check if what I did is correct or not.

Thank you very much!

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