Pierre-Simon De Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomer. In 1796 he formulated a nebular hypothesis of cosmic origin and was one of the first scientists to postulate the existence of black holes. He later proposed a solution to irregularities in Newton’s calculations and presented them in a work called Mécanique céleste. According to legend, when asked by Napoleon why God did not appear in his discussion, Laplace replied “I had no need of that hypothesis.” Among many discoveries, Laplace is most notable for the Laplace Transform which aids in solving differential equations by allowing for the transformation of an equation from the time domain to the frequency domain.

For a signal x(t), its Laplace transform X(s) is defined by
[latex]\mathrm{X(s)=}\int\limits_{-\infty}^{\infty}\mathrm{x(t)}e^{-st}dt[/latex]

Example: For a signal x(t) = e^-atu(t) find the Laplace transform X(s).

By definition
[latex]\mathrm{X(s)=}\int\limits_{-\infty}^{\infty}e^{-at}u(t)e^{-st}dt[/latex]

Since u(t) = 0 for t < 0 and u(t) = 1 for t ≥ 0,
[latex]\mathrm{X(s)=}\int\limits_{0}^{\infty}e^{-at}e^{-st}dt = \mathrm{X(s)=}\int\limits_{0}^{\infty}e^{-(s+a)t}dt = -\frac{1}{s+1}e^{-(s+a)t}\big|_{0}^{\infty}[/latex]

Thus we can conclude
[latex]\mathrm{X(s)=}\frac{1}{s + a} \mathrm{for Re(s + a) > 0}[/latex]