I am currently working through a derivation of dislocation fields in liquid crystals(physics I know).
In solving the field components of the dislocation fields I've gotten this unwieldy double integral as a means to solve an inverse Fourier transform.
$$ \int_{}^{}\int_{}^{}\frac{iq_{y}}{q^{2}_{y}+\lambda^{2}q^{4}_{x}}\exp[i(q_{y}y +q_{x}x)]dq_{x}dq_{y}$$
Fortunately, I've got what the answer should be, however I am unable to figure out the intermediate steps:
$$\frac{1}{4\sqrt{\pi \lambda |y|}} \exp \left[ - \left(\frac{x^{2}}{4 \lambda |y|} \right) \right] \textrm{sgn}(y)$$
If anyone could help me at least with what integration techniques may be best? I've attempted a few asymptotic complex methods i.e. solving via saddle point method but that doesn't lead anywhere.
Thanks!
This integral is straightforward to evaluate as a 2D Inverse Fourier Transform using Fourier Transform properties and tables of Fourier Transforms, such as those found on this Wikipedia page
$$\begin{align*} I &= \int_{}^{}\int_{}^{}\frac{iq_{y}}{q^{2}_{y}+\lambda^{2}q^{4}_{x}}\exp[i(q_{y}y +q_{x}x)]dq_{x}dq_{y} \ \\ \\ &= 2\pi \int_{}^{}\dfrac{1}{2\lambda q_x^2}\left[\dfrac{1}{2\pi}\int_{}^{}iq_y\frac{2\lambda q_x^2}{q^{2}_{y}+\lambda^{2}q^{4}_{x}}\exp(iq_{y}y )dq_{y}\right]\exp(iq_{x}x)dq_{x} \ \\ \\ &= 2\pi \int_{}^{}\dfrac{1}{2\lambda q_x^2}\dfrac{d}{dy}\left[\exp\left(-\lambda q_x^2|y|\right)\right]\exp(iq_{x}x)dq_{x} \quad \text{(IFT table lookup & derivative property)}\ \\ \\ &= 2\pi \int_{}^{}-\dfrac{1}{2}\mathrm{sgn(y)}\exp\left(-\lambda q_x^2 |y|\right)\exp(iq_{x}x)dq_{x} \ \\ \\ &= -\dfrac{4\pi^2}{2\sqrt{\pi 4\lambda |y|}} \mathrm{sgn(y)}\left[\dfrac{1}{2\pi}\int_{}^{}\sqrt{\pi 4\lambda |y|}\exp\left(-\lambda |y| q_x^2\right)\exp(iq_{x}x)dq_{x}\right] \ \\ \\ &= -\dfrac{4\pi^2}{4\sqrt{\pi\lambda |y|}} \mathrm{sgn(y)} \exp\left(-\dfrac{x^2}{4\lambda |y|}\right) \quad \text{(IFT table lookup)} \end{align*}$$
My answer is off by a factor of $-4\pi^2$, from what you claim should be the answer.
I will assert that provided answer neglected the $\dfrac{1}{2\pi}$ scaling factors required for the Inverse Fourier Transforms, given the convention used, and probably had a sign mistake when taking the derivative.