the method of steepest descent
When you invert a Laplace transform through the Bromwich integral, or evaluate an inverse Fourier integral whose exponent grows and decays rather than merely oscillating, you face an integral of the form integral of A(z) e^(t phi(z)) dz along a contour in the complex plane, with t large. The method of steepest descent is the technique for estimating such integrals. Its slogan: deform the contour so that almost all the action happens at a single point, then read off the answer from the behaviour there.
The strategy uses the freedom of complex analysis. By Cauchy's theorem you may bend the contour (without crossing singularities) to any equivalent path. The clever choice is to route it through a saddle point of the exponent — a place where phi'(z) = 0 — and, more precisely, along the path of steepest descent, the direction in which the real part of phi falls off fastest on each side. On that path e^(t phi(z)) is largest at the saddle and drops sharply away, so for large t the whole integral is concentrated in a tiny neighbourhood of the saddle. Expanding phi to second order there turns the integral into a Gaussian you can evaluate exactly, yielding a leading asymptotic term of the form (constant) times e^(t phi(saddle)) / sqrt(t). It is the complex-plane sibling of Laplace's method for real integrals, and the close cousin of stationary phase (which is the special case where the relevant exponent is purely imaginary).
In the PDE toolkit this is how you extract the large-time behaviour of a Laplace- or Fourier-inverted solution — the long-time decay of a diffusion or telegrapher signal, the asymptotics of special functions like Bessel and Airy functions that appear as fundamental solutions, and the rates at which transients die away. The saddle point's location often has a direct meaning, picking out the dominant frequency or decay rate. The honest caveat: finding the right contour deformation and the dominant saddle (there can be several, and which one counts can switch as a parameter varies — a Stokes phenomenon) is genuine craft, and the method gives an asymptotic approximation valid for large t, not an exact closed form.
The Bessel function J_0(x) for large x has an integral representation with a complex phase; routing the contour through its saddle points and expanding gives J_0(x) approximately equal to sqrt(2/(pi x)) cos(x - pi/4). That tidy 1/sqrt(x) decay with a phase shift — central to wave problems on a disk — drops straight out of the saddle-point analysis.
Bend the contour through the saddle, where the integrand peaks; the saddle alone gives the large-t answer.
It is an asymptotic method: it captures the leading behaviour as t grows, not an exact value, and the deformed contour must avoid the integrand's singularities. When two saddles exchange dominance as a parameter changes (a Stokes phenomenon), the simple one-saddle formula can fail.