- #1
JasonPhysicist
- 13
- 0
I´m having a problem with the value of the expression
##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.
##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.