Math 2.0

Approximations and expansions Definition

Laplace's method

Builds on Asymptotic equivalenceSecond derivativeLocal maximum and minimum

Definition

Let a<ba < b be real numbers and let f ⁣:[a,b]→ℝf \colon [a, b] \to \R be twice continuously differentiable. Assume that ff has a unique global maximum on [a,b][a, b] at a point x0x_0 with a<x0<ba < x_0 < b, and that the second derivative satisfies f′′(x0)<0f''(x_0) < 0. Laplace's method is the approximation of I(M)=∫abeMf(x) dxI(M) = \int_a^b e^{M f(x)}\, dx for large MM by the integral of eM(f(x0)+12f′′(x0)(x−x0)2)e^{M (f(x_0) + \frac12 f''(x_0)(x - x_0)^2)} over ℝ\R. Under these assumptions it gives the asymptotic equivalence ∫abeMf(x) dx∼2πM∣f′′(x0)∣  eMf(x0)(M→∞).\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).

Remarks

Example: Stirling's formula. Substituting t=nst = n s gives n!=∫0∞tne−t dt=nn+1∫0∞en(ln⁡s−s) dsn! = \int_0^\infty t^n e^{-t}\, dt = n^{n+1} \int_0^\infty e^{n(\ln s - s)}\, ds. The function f(s)=ln⁡s−sf(s) = \ln s - s has its maximum at s=1s = 1, with f(1)=−1f(1) = -1 and f′′(1)=−1f''(1) = -1, and after a check that the tails are small the method gives n!∼2πn (n/e)nn! \sim \sqrt{2\pi n}\, (n/e)^n.

Non-example: if the maximum is at an endpoint, for instance f(x)=−xf(x) = -x on [0,1][0, 1], the hypothesis a<x0<ba < x_0 < b fails. Here ∫01e−Mx dx=(1−e−M)/M\int_0^1 e^{-Mx}\, dx = (1 - e^{-M})/M, which decays like 1/M1/M and not like 1/M1/\sqrt{M}.

The same idea applies to ∫abg(x)eMf(x) dx\int_a^b g(x) e^{M f(x)}\, dx with gg continuous and g(x0)≠0g(x_0) \ne 0, where the right side gains the factor g(x0)g(x_0).

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Big O notation, definition, no evidence levelBig O notationdefinitionEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLittle o notation, definition, no evidence levelLittle o notationdefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionAsymptotic equivalence, definition, no evidence levelAsymptotic equivalencedefinitionLocal maximum and minimum, definition, no evidence levelLocal maximum and min…definitionDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionSecond derivative, definition, no evidence levelSecond derivativedefinitionLaplace's method, definition, no evidence levelLaplace's methoddefinition
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  1. Big O notation definition
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  4. Limit point of a set definition
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  7. Limit of a function definition
  8. Asymptotic equivalence definition
  9. Local maximum and minimum definition
  10. Derivative definition
  11. Differentiable function definition
  12. Derivative function definition
  13. Higher derivative definition
  14. Second derivative definition
  15. Laplace's method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace%27s_method
About this page
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MSC41A60, 26A42
DefinesLaplace's method
UsesAsymptotic equivalence, Second derivative, Local maximum and minimum
Changed2 November 2026

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