Laplace's Method - Example 1: Stirling's Approximation

Example 1: Stirling's Approximation

Laplace's method can be used to derive Stirling's approximation

for a large integer N.

From the definition of the Gamma function, we have

Now we change variables, letting

so that

Plug these values back in to obtain

\begin{align} N! & = \int_0^\infty e^{-N z} \left(N z \right)^N N \, dz \\ & = N^{N+1}\int_0^\infty e^{-N z} z^N \, dz \\ & = N^{N+1}\int_0^\infty e^{-N z} e^{N\ln z} \, dz \\ & = N^{N+1}\int_0^\infty e^{N(\ln z-z)} \, dz. \end{align}

This integral has the form necessary for Laplace's method with

which is twice-differentiable:

The maximum of ƒ(z) lies at z0 = 1, and the second derivative of ƒ(z) has the value −1 at this point. Therefore, we obtain

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