両側Laplace変換

$ \mathcal L_B(f):s\mapsto\int_\R e^{-st}f(t)\mathrm dt

$ \mathcal L_B(f)(s)=\int_0^\infty e^{-st}f(t)\mathrm dt-\int_0^{-\infty} e^{-st}f(t)\mathrm dt

$ =\int_0^\infty e^{-st}f(t)\mathrm dt+\int_0^\infty e^{st}f(-t)\mathrm dt

$ =\int_0^\infty e^{-st}f(t)\mathrm dt+\int_0^\infty e^{-(-s)t}f(-t)\mathrm dt

$ =\mathcal L(f)(s)+\mathcal L(t\mapsto f(-t))(s)