deletions | additions       diff --git a/We_can_define_the_Fourier__.tex b/We_can_define_the_Fourier__.tex  index 67fa06b..28babe6 100644  --- a/We_can_define_the_Fourier__.tex  +++ b/We_can_define_the_Fourier__.tex 
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\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]}.  \end{align} The second term in this equation involving $(f+f_k)$ is negligible for our purposes, so only the first term is required.  This integral is not analytic, but integral of cubic trigonometric functions (through e.g.\ use of \href{http://docs.sympy.org/latest/index.html}{sympy}) can be written as  \begin{equation}  \int_x^y \cos{(t^3)} {\mathrm{d}}t = \left( y\, {}_1F_2(1/6; 1/2, 7/6; -y^6/4) - x\, {}_1F_2(1/6; 1/2, 7/6; -x^6/4)\right)  \end{equation}  and  \begin{equation}  \int_x^y \sin{(t^3)} {\mathrm{d}}t = \frac{1}{4}\left( y^4\, {}_1F_2(2/3; 3/2, 5/3; -y^6/4) - x^4\, {}_1F_2(2/3; 3/2, 5/3; -x^6/4) \right),  \end{equation}  where ${}_1F_2$ is a \href{http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F2/06/ShowAll.html}{generalised hypergeometric function}.