deletions | additions       diff --git a/We_can_define_the_Fourier__.tex b/We_can_define_the_Fourier__.tex  index 278309e..24f3a10 100644  --- a/We_can_define_the_Fourier__.tex  +++ b/We_can_define_the_Fourier__.tex 
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We can define the Fourier transform of our signal, which has complex amplitude at time $t_k$ of $y_k$ as  \begin{align}  H_k(f) \approx & y_k e^{i\phi_k-i\pi f \Delta t} \int_{-\Delta t/2}^{\Delta t/2} \exp{\left[2\pi i \left((f_k-f)t + \frac{t^2}{2}\dot{f}_k + \frac{t^3}{6}\ddot{f}_k \right) \right]} \nonumber \\  & + y_k^{*}e^{-i\phi_k-i\pi f \Delta t} \int_{-\Delta t/2}^{\Delta t/2} \exp{\left[-2\pi i \left((f_k+f)t - \frac{t^2}{2}\dot{f}_k - \frac{t^3}{6}\ddot{f}_k \right) \right]}. \label{eq:fourier}  \end{align}  The second term in this equation involving $(f+f_k)$ is negligible for our purposes, so only the first term is required. 
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\end{equation}  where ${}_1F_2$ is a \href{http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F2/06/ShowAll.html}{generalised hypergeometric function}. So, we need to convert our equation into an equivalent form.  To factor equation~\ref{eq:fourier} into a form that we can use with the hypergeometric functions we want to have  \begin{equation}  z^3 = 2\pi\left((f_k-f)t + \frac{t^2}{2}\dot{f}_k + \frac{t^3}{6}\ddot{f}_k \right).  \end{equation}  Firstly, we can factor it into sine and cosine parts to give  \begin{align}  H_k(f) \approx & y_k e^{i\phi_k-i\pi f \Delta t} \Bigg( \int_{-\Delta t/2}^{\Delta t/2} \cos{\left[2\pi \left((f_k-f)t + \frac{t^2}{2}\dot{f}_k + \frac{t^3}{6}\ddot{f}_k \right) \right]} {\rm d}t \nonumber \\