Perturbative Methods for Modelling (M2)

Example 6(g), p. 268

The modified Bessel function $\fs2 K_\nu(x)$  has the following integral representation for $\fs2 x>0$ : $\fs2 K_\nu(x) =\,\int_0^\infty e^{-x\, \cosh t} \,\cosh(\nu \,t) \,dt$.

Using Laplace's method, show that $\fs2 K_\nu(x)\sin \sqrt{{\pi \over 2\,x}} \, e^{-x}$  for $\fs2 x\to +\infty$.

Example 10,  p.275

The Gamma function $\fs2 \Gamma(x)$ has the following integral representation for $\fs2 x>0$: $\fs2 \Gamma(x) = \int_0^\infty e^{-t} \, t^{x-1}\, dt$. Using Laplace's method, show the Stirling's formula $\fs2 \Gamma(x) \sim x^x \, e^{-x}\, \sqrt{2\, \pi \over x}$ for $\fs2 x\to +\infty$.