deletions | additions       diff --git a/Equation_ref_eq_fourier3_is__.tex b/Equation_ref_eq_fourier3_is__.tex  index dde289e..3fc8f1b 100644  --- a/Equation_ref_eq_fourier3_is__.tex  +++ b/Equation_ref_eq_fourier3_is__.tex 
...
Equation~\ref{eq:fourier3} is potentially \href{http://math.stackexchange.com/a/1470809/95147}{soluble} analytically using incomplete Bessel functions and hypergeometric equations. However, it may be easier to create a lookup table of numerical solutions in $z$ and $\alpha$ space. First, it may also be better to split the integral into the real and imaginary parts to solve each individually. So, we have  \begin{align}  H_k(f) =& y_k\left(\frac{\pi}{3}\ddot{f_k}\right)^{-1/3}e^{i\left(\phi_k-\pi f \Delta t + \beta \right)}\Bigg[\int_{0}^{z_2} \cos{\left(z^3 + \alpha z\right)} {\rm d}z +\nonumber \\  &  i \int_{0}^{z_2} \sin{\left(z^3 + \alpha z\right)} {\rm d}z - \nonumber \\ & \int_{0}^{z_1} &\int_{0}^{z_1}  \cos{\left(z^3 + \alpha z\right)} {\rm d}z -\nonumber \\  &  i\int_{0}^{z_1} \sin{\left(z^3 + \alpha z\right)} {\rm d}z \Bigg]. \end{align}