Laplace integral

A Laplace integral contains a large parameter in an exponential weight, typically in the form $\int a(t)e^{x\varphi (t)}dt$ or ${\int }_0^b{e}^{-xt}f(t)dt$. Its asymptotic behavior is controlled by the points where the real part of the phase is largest.

A linear differential-equation solution may be sought as ${\int }_C{e}^{xt}f(t)dt$. Integration by parts transfers the factor $x$ to derivatives of $f$, giving an equation for the amplitude and boundary terms. Admissible contours must cancel those terms and make the integral converge.