= Laplace contour-integral method for a linear differential equation
The Laplace contour-integral method seeks a solution in the form
$$
w(z)=\int_\gamma e^{zt}f(t)\,dt.
$$
Replacing multiplication by $z$ with differentiation of $e^{zt}$ in $t$ and integrating by parts turns the differential equation for $w$ into a first-order equation for $f$. The contour $\gamma$ must make the resulting endpoint term vanish; finite endpoints are often branch points of $f$, while infinite ends must enter sectors where the exponential decays.
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