There is a Laplace transform where the integration is performed over [math](-\infty,\infty).[/math]It is the bilateral Laplace transform (or two sided Laplace transform.)

[math]\displaystyle F(s)=\int_{-\infty}^{\infty}{e^{-st}f(t)\text{d}t.}\tag*{(A)}[/math]

This transform possesses precisely the same complex inversion formula as the ordinary Laplace transform.

Theorem

Let [math]f(t) [/math]be locally integrable and let the integral (A) converge absolutely for [math]s=x, x [/math]real. Then

[math]\displaystyle \frac{f(t^+)+f(t^-)}{2}=\lim_{Y\to\infty}\frac{1}{2\pi \text{i}}\int_{x-\text{i} Y}^{x+\text{i} Y}{e^{ts}F(s)\text{d}t,}\tag*{}[/math]

holds at every point where [math]f[/math] is of bounded variation in some neighborhood of [math]t[/math].

The bilateral transform reduces to the ordinary transform when the support of [math]f\subset [0,\infty)[/math]

Obviously, the bilateral Laplace transform is a disguised form of the exponential Fourier . transform.

The bilateral transform has been neglected. Almost all of the math literature deals with the ordinary Laplace transform. Doetsch*has a few pages about it, and there is even an interesting and obscure book devoted to the subject**. But, tragically, the bilateral transform has never really caught on. I suspect the reason is the Laplace transform is often applied to dynamic problems and in those problems time begins at zero. The ordinary Laplace transform makes it easier to incorporate initial conditions.

[math]REFERENCES\tag*{}[/math]

*Doetsch, Gustav, “Introduction to the theory and application of the Laplace transformation,” Springer-Verlag (1974), p. 155 ff.

**Pol, Balthsar van der, “ Operational calculus based on the two-sided Laplace integral,” Cambridge University Press (1955).