The Gaussian $e^{-\pi x^2}$ is its own Fourier transform for complex frequencies

Prove by justifying all steps that for all \(\xi \in \mathbb { C }\) we have \(e ^ { - \pi \xi ^ { 2 } } = \int _ { - \infty } ^ { \infty } e ^ { - \pi x ^ { 2 } } e ^ { 2 \pi i x \xi } d x\)

Hint: You may use that fact in Example 1 on p. 42 of the textbook without proof, i.e., you may assume the above is true for real values of \(\xi .\)