Editing Two-sided Laplace transform (section)

==Properties==

The following properties can be found in {{harvtxt|Bracewell|2000}} and {{harvtxt|Oppenheim|Willsky|1997}}

{| class="wikitable" 
|+ Properties of the bilateral Laplace transform
|-
! Property !!  Time domain !! {{math|''s''}} domain  !! Strip of convergence  !!  Comment
|-
| Definition
| <math> f(t) </math>
| <math> F(s) = \mathcal{B}\{f\}(s) = \int_{-\infty}^{\infty} f(t) \, e^{-st} \, dt </math>
| <math> \alpha < \Re s < \beta </math>
|
|-
| Time scaling
| <math>f(at)</math>
| <math> \frac{1}{|a|} F \left ({s \over a} \right)</math>
| <math> \alpha < a^{-1} \, \Re s < \beta </math>
| <math> a \in\mathbb{R} </math>
|-
| Reversal
| <math> f(-t) </math>
| <math> F(-s)</math>
| <math> -\beta < \Re s < -\alpha </math>
| 
|-
| Frequency-domain derivative
| <math> t f(t) </math>
| <math> -F'(s) </math>
| <math> \alpha < \Re s < \beta </math>
|
|-
| Frequency-domain general derivative
| <math> t^{n} f(t) </math>
| <math> (-1)^{n} \, F^{(n)}(s) </math>
| <math> \alpha < \Re s < \beta </math>
| 
|-
| Derivative
| <math> f'(t) </math>
| <math> s F(s) </math>
| <math> \alpha < \Re s < \beta </math>
|
|-
| General derivative
| <math> f^{(n)}(t) </math>
| <math> s^n \, F(s) </math>
| <math> \alpha < \Re s < \beta </math>
| 
|-
| Frequency-domain integration 
| <math> \frac{1}{t}\,f(t) </math>
| <math> \int_s^\infty F(\sigma)\, d\sigma </math>
| 
| only valid if the integral exists
|-
| Time-domain integral
| <math> \int_{-\infty}^t f(\tau)\, d\tau </math>
| <math> {1 \over s} F(s) </math>
| <math> \max(\alpha,0) < \real s < \beta </math>
|
|-
| Time-domain integral
| <math> \int_{t}^{\infty} f(\tau)\, d\tau </math>
| <math> {1 \over s} F(s) </math>
| <math> \alpha < \real s < \min(\beta,0) </math>
|
|-
| Frequency shifting
| <math> e^{at} \, f(t) </math>
| <math> F(s - a) </math>
| <math> \alpha + \Re a < \Re s < \beta + \Re a </math>
| 
|-
| Time shifting
| <math> f(t - a) </math>
| <math> e^{-as} \, F(s) </math>
| <math> \alpha < \Re s < \beta </math>
| <math> a\in\mathbb{R} </math>
|-
| Modulation
| <math> \cos(at)\,f(t) </math>
| <math> \tfrac{1}{2} F(s-ia) + \tfrac{1}{2} F(s+ia) </math>
| <math> \alpha < \Re s < \beta </math>
| <math> a\in\mathbb{R} </math>
|-
| Finite difference
| <math> f(t+\tfrac{1}{2}a)-f(t-\tfrac{1}{2}a) </math>
| <math> 2 \sinh(\tfrac{1}{2} a s) \, F(s) </math>
| <math> \alpha < \Re s < \beta </math>
| <math> a\in\mathbb{R} </math>
|-
| Multiplication
| <math>f(t)\,g(t)</math>
| <math> \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty}F(\sigma)G(s - \sigma)\,d\sigma \ </math>
| <math> \alpha_f+\alpha_g < \Re s < \beta_f+\beta_g </math>
| <math> \alpha_f < c < \beta_f </math>. The integration is done along the vertical line {{nowrap|1=Re(''σ'') = ''c''}} inside the region of convergence.
|-
| Complex conjugation
| <math> \overline{f(t)} </math>
| <math> \overline{F(\overline{s})} </math>
| <math> \alpha < \Re s < \beta </math>
|
|-
| [[Convolution]]
| <math> (f * g)(t) = \int_{-\infty}^{\infty} f(\tau)\,g(t - \tau)\,d\tau </math>
| <math> F(s) \cdot G(s) \ </math>
| <math> \max(\alpha_f,\alpha_g) < \Re s < \min(\beta_f,\beta_g) </math>
| 
|-
| [[Cross-correlation]]
| <math> (f\star g)(t) = \int_{-\infty}^{\infty} \overline{f(\tau)}\,g(t + \tau)\,d\tau </math>
| <math> \overline{F(-\overline{s})} \cdot G(s) </math>
| <math> \max(-\beta_f,\alpha_g) < \Re s < \min(-\alpha_f,\beta_g) </math>
| 
|}

Most properties of the bilateral Laplace transform are very similar to properties of the unilateral Laplace transform,
but there are some important differences:
{| class="wikitable"
 |+ '''Properties of the unilateral transform vs. properties of the bilateral transform'''
 !
 	! unilateral time domain
 	! bilateral time domain
	! unilateral-'s' domain
 	! bilateral-'s' domain

 |-
 ! [[Derivative|Differentiation]]
	| <math> f'(t) \ </math>
	| <math> f'(t) \ </math>
	| <math>  s F(s) - f(0) \ </math>
	| <math>  s F(s) \ </math>
 |-
 ! Second-order [[Derivative|differentiation]]
	| <math> f''(t) \ </math>
	| <math> f''(t) \ </math>
	| <math>  s^2 F(s) - s f(0) - f'(0) \ </math>
	| <math>  s^2 F(s) \ </math>
 |-
 ! [[Convolution]]
	| <math> \int_0^{t} f(\tau) \, g(t-\tau) \, d\tau \ </math>
	| <math> \int_{-\infty}^{\infty} f(\tau) \, g(t-\tau) \,d\tau \ </math>
	| <math>  F(s) \cdot G(s) \ </math>
	| <math>  F(s) \cdot G(s) \ </math>
 |-
|}

===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>

===Uniqueness===

For any two functions <math display="inline"> f,g </math> for which the two-sided Laplace transforms <math display="inline"> \mathcal{T} \{f\}, \mathcal{T} \{g\} </math> exist, if <math display="inline"> \mathcal{T}\{f\} = \mathcal{T} \{g\}, </math> i.e. <math display="inline"> \mathcal{T}\{f\}(s) = \mathcal{T}\{g\}(s) </math> for every value of <math display="inline"> s\in\mathbb R, </math> then <math display="inline"> f=g </math> [[almost everywhere]].