I´m having a problem with the value of the expression(adsbygoogle = window.adsbygoogle || []).push({});

##F(it)-F(-it)##, found on the Abel-Plana formula, where $$F(z)=\sqrt{z^2 + A^2}$$, with ##A## being a positive real number (F(z) is analytic in the right half-plane).

Well, I know the result is ##F(it)-F(-it)=2i\sqrt{t^2 -A^2}##, for ##t>A##

Starting from the fact that the function has branch points ##z=\pm iA## I´d have to go around around these points to obtain the above result. However, to obtain it, I should have

$$F(-it)=-F(it)$$, which means ##F(it)=i\sqrt{t^2 -A^2}## and ##F(-it)=-i\sqrt{t^2 -A^2}##.

I honestly can't see why that happens and I can't formulate a proof for it.

Any help would be appreciated.

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# I Help with expression ##F(it)-F(-it)## in the Abel-Plana form

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