Find Fourier Transform of S_ε(x) for Laplace's Equation

[catcherintherye]

I am solving laplaces equation in the half plane and I have the following boundary condition of which I need to find the Fourier transform in the x-direction

$$S_\epsilon(x) = sgn(x)exp(\epsilon|x|), \epsilon >0$$

$$sgn(x)=\left\{\begin{array}{cc}1,&\mbox{ if }<br /> x<0\\-1, & \mbox{ if } x<0\\ 0, &\mbox{ if } x=0\end{array}\right.$$

any hints on how to approach this FT??

 

[mjsd]

there is a typo for the function sgn(x)? note that
$$-S_\epsilon (x) = S_\epsilon (-x)$$ is an odd function... and you should get a Fourier sine series