Fourier transform (FT) of a Hyperbolic secant

  1. Hi,

    Did anyone know how to do the Fourier transform of the hyperbolic
    secant? I know the answer; it's given in the text (I'm reading
    Ablowitz, Fokas, Complex Variables), it's another hyperbolic secant,
    but I want to know how to do it. My dilemma is:

    a) what contour to use? I'm having difficulty showing that the
    contour integration goes to zero on the semicircle as the radius R
    goes to infinity, particularly in the theta, angle, dependence in the
    bottom, denominator, for e^z + e^{-z}

    b) There are an infinite number of singularities along the imaginary
    axis. Do I sum them up?

    Thank you for your time my friend, I wanted to throw this out there. -vy
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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