Editing Two-sided Laplace transform (section)

===Parseval's theorem and Plancherel's theorem===

Let <math>f_1(t)</math> and <math>f_2(t)</math> be functions with bilateral Laplace transforms
<math>F_1(s)</math> and <math>F_2(s)</math> in the strips of convergence 
<math>\alpha_{1,2}<\real s<\beta_{1,2}</math>. 
Let <math>c\in\mathbb{R}</math> with <math>\max(-\beta_1,\alpha_2)<c<\min(-\alpha_1,\beta_2)</math>.
Then [[Parseval's theorem]] holds:
<ref>{{harvnb|LePage|1980|loc=Chapter 11-3, p.340}}</ref>
:<math>
\int_{-\infty}^{\infty} \overline{f_1(t)}\,f_2(t)\,dt = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \overline{F_1(-\overline{s})}\,F_2(s)\,ds
</math>
This theorem is proved by applying the inverse Laplace transform on the convolution theorem in form of the cross-correlation.

Let <math>f(t)</math> be a function with bilateral Laplace transform <math>F(s)</math>
in the strip of convergence <math>\alpha<\Re s<\beta</math>.
Let <math>c\in\mathbb{R}</math> with <math> \alpha<c<\beta </math>.
Then the [[Plancherel theorem]] holds:
<ref>{{harvnb|Widder|1941|loc=Chapter VI, §8, p.246}}</ref>
:<math>
\int_{-\infty}^{\infty} e^{-2c\,t} \, |f(t)|^2  \,dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(c+ir)|^2 \, dr
</math>