= Solution
<Cauchy's integral theorem> states that a <holomorphic function> on a simply connected domain has zero <contour integral> around every closed piecewise smooth curve in that domain. One proof begins with the <Cauchy theorem for a triangle>. Subdivide a triangle into four equal triangles. Their oriented boundary integrals sum to the original integral because internal edges cancel, so choose a nested subtriangle carrying at least one quarter of the preceding integral's modulus. At stage $n$, its perimeter and diameter are $P/2^n,d/2^n$, and the nested triangles converge to a point $z_0$. Complex differentiability gives $f(z)=f(z_0)+f'(z_0)(z-z_0)+\epsilon(z)(z-z_0)$, with $\epsilon(z)\to0$. The affine terms have primitives and integrate to zero. Therefore $|I_n|\leq\sup|\epsilon|\,Pd/4^n$, while $|I_0|\leq4^n|I_n|$. Letting $n\to\infty$ proves $I_0=0$. Triangulation cancels internal edges for polygonal contours; local primitives and deformation through the domain extend the result to the stated piecewise smooth curves. Simple connectedness ensures the necessary contractions do not pass through a missing point.
Apply the theorem to $e^{-az^2}$ on the rectangle between heights zero and $\rho$, with vertical sides at $\operatorname{Re}z=\pm R$. On either vertical side,
$$
|e^{-az^2}|=e^{-aR^2+a(\operatorname{Im}z)^2}\leq e^{-aR^2+a\rho^2}.
$$
The modulus of each side integral is at most $|\rho|e^{-aR^2+a\rho^2}\to0$. Both horizontal integrals converge absolutely. The two horizontal orientations are opposite, so the limit gives $I(\rho,a)=I(0,a)$, also for negative $\rho$. This is <Gaussian contour translation>.
Now complete the square:
$$
-ax^2+i\nu x=-a\left(x-\frac{i\nu}{2a}\right)^2-\frac{\nu^2}{4a}.
$$
The substitution $z=x-i\nu/(2a)$ moves the path to height $-\nu/(2a)$. Its Gaussian integral equals the real-axis integral by the preceding result. Hence
$$
\boxed{\int_{-\infty}^{\infty}e^{-ax^2+i\nu x}dx=\sqrt{\frac\pi a}\,e^{-\nu^2/(4a)},\qquad a>0,\ \nu\in\mathbb R.}
$$
This also gives the <Fourier transform of a Gaussian> in the convention with no prefactor in the forward transform.
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