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

  1. Aug 7, 2017 #1
    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.
     
  2. jcsd
  3. Aug 11, 2017 #2
    I think ##F(it)-F(-it)=0##, no? This is unsurprising given that the only occurrence of z in the formula is squared, and ##z^2## is an even function. In any case:

    ##F(it)=\sqrt{(it)^2+A^2}=\sqrt{-t^2+A^2}##
    and
    ##F(-it)=\sqrt{(-it)^2+A^2}=\sqrt{-t^2+A^2}##
     
    Last edited: Aug 11, 2017
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