For integrals of the form:

$I(x) = \int_a^b f(t) e^{x\phi(t)} dt$   as $x\rightarrow \infty$

Contributions near global maxima of $\phi(t)$.

Watson lemma

Special case, for $\phi(t) = t$

Laplace method

1. Restrict integral to a small region (of order $\epsilon$) around maxima of exponential function $\phi$, and confirm we are making an exponentially small error.

2. Expand $f(t)$ and $\phi(t)$ in series valid in this region, so we get a series of integrals.

3. It is then usually easier to evaluate these integrals by extending the limits to infinity (after rescaling), confirming that we are again making an exponentially small error.

4. Confirm assumptions are self-consistent.

Genera Laplace integral

Three cases:

Case 1 The maximum is at $t=a$

$\phi'(a) \leq 0$ (since it is maximum), and we assume it is not $0$, so $\phi'(a) < 0$

$I(x) \sim -\frac{f(a)e^{x\phi(a)}}{x\phi'(a)}$

Case 2 The maximum is at $t=b$

$I(x) \sim \frac{f(b)e^{x\phi(b)}}{x\phi'(b)}$

Case 3 The maximum is at some $t=c$ with $a<c<b$.

$I(x) \sim \frac{\sqrt{2\pi}f(c)e^{x\phi(c)}}{\sqrt{-x\phi''(c)}}$