Fourier transform (FT) of a Hyperbolic secant

[vytrvalost]

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

 

[mighty2000]

per

Hello vyper,

Thank you for your question. The Fourier transform of the hyperbolic secant function can be derived using the definition of the Fourier transform and some properties of hyperbolic functions. Here is the step-by-step process:

1. First, recall the definition of the Fourier transform:

F(u) = ∫f(x)e^(-iux)dx

where f(x) is the function to be transformed and F(u) is the transformed function.

2. Next, we need to express the hyperbolic secant function in terms of exponential functions. Using the identity sech(x) = 2/(e^x + e^-x), we can write the hyperbolic secant as:

sech(x) = 1/2 * (e^x + e^-x)

3. Now, substituting this into the Fourier transform definition, we get:

F(u) = ∫(1/2 * (e^x + e^-x))e^(-iux)dx

4. Simplifying the expression, we get:

F(u) = (1/2) * ∫e^(-ix(u+1))dx + (1/2) * ∫e^(-ix(u-1))dx

5. Using the property ∫e^(ax)dx = 1/ia * e^(ax), we can evaluate the integrals:

F(u) = (1/2i(u+1)) * e^(-ix(u+1)) + (1/2i(u-1)) * e^(-ix(u-1))

6. Now, we can use the fact that e^(-ix(u+1)) = e^(-iux) * e^-ix and e^(-ix(u-1)) = e^(-iux) * e^ix to rewrite the expression as:

F(u) = (1/2i(u+1)) * e^(-iux) * e^-ix + (1/2i(u-1)) * e^(-iux) * e^ix

7. Factoring out e^(-iux), we get:

F(u) = e^(-iux) * (1/2i(u+1)) * e^-ix + e^(-iux) * (1/2i(u-1)) * e^ix

8. Using the property e^(-