Laplace 變換
Laplace transform$ {\cal L}\lbrack f\rbrack
$ {\cal L}\lbrack f\rbrack(s):=\int^\infty_0 f(t)e^{-st}dt.
兩側 Laplace 變換$ {\cal B}\lbrack f\rbrack
$ {\cal B}\lbrack f\rbrack(t):=\int_{-\infty}^\infty f(t)e^{-st}dt.
$ {\cal L}\lbrack f(t)\rbrack={\cal B}\lbrack f(t)H(t)\rbrack.
$ {\cal B}\lbrack f\rbrack(s)={\cal L}\lbrack f(t)\rbrack(s)+{\cal L}\lbrack f(-t)\rbrack(-s).
$ {\cal F}\lbrack f\rbrack=\frac 1{\sqrt{2\pi}}{\cal B}\lbrack f\rbrack(s).
$ M_X(s)={\cal B}\lbrack f\rbrack(-s).
$ {\cal M}\lbrack f\rbrack(s):=\int_0^\infty f(x)x^{s-1}dx.
$ {\cal M}\lbrack f\rbrack(s)={\cal B}\lbrack f(e^{-x})\rbrack(s).
$ {\cal B}\lbrack f\rbrack(s)={\cal M}\lbrack f(-\log x)\rbrack(s).