the saddle point method
The saddle point method is the same idea as the method of steepest descent, named after the geometric feature that does all the work. Picture the surface that the real part of g(z) traces out over the complex plane: it is a landscape of hills and valleys. A holomorphic function can never have an ordinary peak or pit — the maximum modulus principle forbids it — so the only critical points of that landscape are saddles, places shaped like a mountain pass that go up in two directions and down in the other two. The integral e^(N g(z)) is dominated by what happens at such a pass.
Concretely, a saddle point is a point z_0 where g'(z_0) = 0. Through it run two special directions: a ridge along which the real part of g increases, and a valley along which it decreases — the steepest-descent direction. The method deforms the integration contour to cross the saddle along the valley, perpendicular to the ridge, so the integrand has a clean single maximum there. Expanding g(z) approximately as g(z_0) + (1/2) g''(z_0)(z - z_0)^2 turns the local integral into a Gaussian, and the leading term is e^(N g(z_0)) times the square root of (2 pi / (N |g''(z_0)|)), with the orientation of the path fixing the phase. When there are several saddles you take the one with the largest real part of g(z_0), or, if more than one contributes, you sum them.
The names 'saddle point method' and 'method of steepest descent' are used interchangeably; some authors reserve 'stationary phase' for the purely oscillatory cousin where the saddle sits on a contour along which g is imaginary. The honest point is that the whole technique rests on two pillars of complex analysis: Cauchy's theorem lets you move the contour at will, and the holomorphy of g guarantees the landscape has only saddles, never peaks, so a single dominant pass really does control the answer.
For the Bessel function J_n(x) at large order one writes an integral representation whose exponent has a saddle where the derivative vanishes; sliding the contour over that saddle gives the function's leading exponential growth or decay. The same machinery, applied to (integral of e^(N(log s - s)) ds), reproduces Stirling's formula from the single saddle at s = 1.
A mountain-pass saddle, crossed along its valley, controls the integral.
A degenerate saddle, where g''(z_0) = 0 as well, breaks the simple square-root formula; you then expand to the next order (a coalescence of saddles is exactly what produces the Airy function).