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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be twice continuously differentiable.\nAssume that $f$ has a unique global [[maximum|04CPG9K1C4]] on $[a, b]$ at a point $x_0$ with $a \u003c x_0 \u003c b$, and that the [[second derivative|0Y9D2FJRQ4]] satisfies $f''(x_0) \u003c 0$.\n*Laplace's method* is the approximation of $I(M) = \\int_a^b e^{M f(x)}\\, dx$ for large $M$ by the integral of $e^{M (f(x_0) + \\frac12 f''(x_0)(x - x_0)^2)}$ over $\\R$.\nUnder these assumptions it gives the [[asymptotic equivalence|00SB5EA3VW]]\n$$\\int_a^b e^{M f(x)}\\, dx \\sim \\sqrt{\\frac{2\\pi}{M \\abs{f''(x_0)}}} \\; e^{M f(x_0)} \\qquad (M \\to \\infty).$$\n\n## Remarks\n\nExample: Stirling's formula.\nSubstituting $t = n s$ gives $n! = \\int_0^\\infty t^n e^{-t}\\, dt = n^{n+1} \\int_0^\\infty e^{n(\\ln s - s)}\\, ds$.\nThe function $f(s) = \\ln s - s$ has its maximum at $s = 1$, with $f(1) = -1$ and $f''(1) = -1$, and after a check that the tails are small the method gives $n! \\sim \\sqrt{2\\pi n}\\, (n/e)^n$.\n\nNon-example: if the maximum is at an endpoint, for instance $f(x) = -x$ on $[0, 1]$, the hypothesis $a \u003c x_0 \u003c b$ fails.\nHere $\\int_0^1 e^{-Mx}\\, dx = (1 - e^{-M})/M$, which decays like $1/M$ and not like $1/\\sqrt{M}$.\n\nThe same idea applies to $\\int_a^b g(x) e^{M f(x)}\\, dx$ with $g$ continuous and $g(x_0) \\ne 0$, where the right side gains the factor $g(x_0)$.\n",
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