# Fourier transform of function decaying at $ae^{-bx^2+cy^2}$

I'm a bit stumped on a problem. The problem is as follows:

Suppose $f(z)$ is entire and $|f(x+iy)|\leq ae^{-bx^2+cy^2}$ for $a,b,c>0$. If $\hat{f}(z)=\int_{-\infty}^\infty f(x)e^{-2\pi ixz}dx$, then $|\hat{f}(\zeta+i\eta)|\leq a'e^{-b'\zeta^2+c'\eta^2}$.

There's a hint that tells me to assume $\zeta>0$ and change the contour of integration to $x-iy$ for $y>0$ fixed, and $x$ between $-\infty$ and $\infty$ to show that $\hat{f}=O(e^{-b'\zeta^2})$. Then it follows that $\hat{f}(\zeta)=O(e^{-2\pi y\zeta}e^{cy^2})$, and the result follows if we choose $y=d\zeta$ with $d$ a small constant.

I've tried using residues to evaluate $\hat{f}$, but with no luck. I also don't know how to use the hint-- if there's anybody who could help elucidate or tell me what to do here, that would be greatly appreciated, as I'm quite confused. Thanks in advance :)

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In particular following your hint you should move the integration contour away from the real axis along $x-i y$ with $y>0$ fixed, i.e., $$\hat{f}(z)=\int_{-\infty}^\infty f(x)e^{-2\pi ixz}dx = \int_{-\infty}^\infty f(x-i y)e^{-2\pi i(x-i y)z}dx = \cdots$$