Inverse Fourier Tranform of Transmission Lines Wave Equation

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Homework Statement


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From the derivation of v(x,t) and i(x,t) I am stuck on how the inverse Fourier transform of e^(-jwx/u) was calculated. I am trying to understand how the PDE was fully solved here: http://fourier.eng.hmc.edu/e84/lectures/transmission_line/node1.html

Homework Equations


Not sure what rule to implement here, working using laplace transforms (s = jw) to perform Fourier transformations to avoid integration.
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The Attempt at a Solution


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Your notation and writing are a mess. You have f as a function of x in some places, of t in other places, and x and t in yet other places. Why don't you start by clearly defining what the problem is and writing it out carefully.
 
And type it instead of posting an image.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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