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Theorem:
![$\displaystyle \zbox {\int_{-\infty}^\infty e^{-p t^2}dt = \sqrt{\frac{\pi}{p}}, \quad \forall p\in \mathbb{C}: \; \mbox{re}\left\{p\right\}>0}$](img2734.png) |
(D.7) |
Proof: Let
denote the integral. Then
where we needed
re
to have
as
. Thus,
![$\displaystyle I(p) = \sqrt{\frac{\pi}{p}}$](img2740.png) |
(D.8) |
as claimed.
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