Perturbative Methods for Modelling (M2)

Example 1, p. 281
Example 10 p. 296

Using the steepest descent method, show that:

(a) $\fs2 I(x) = \int_0^1 \ln t \,e^{i\,x\,t}\,dt \sim - {i\,\ln x\over x}-{i\,\gamma+\pi/2 \over x} + i\,e^{i\,x} \sum_{n=1}^{\infty}{(-i)^n\,(n-1)! \over x^{n+1}}$  for $\fs2 x\to +\infty$ , where Euler's constant is  $\fs2 \gamma =-\int_0^\infty  e^{-u} \, \ln u\,du =0.5772...$

(b) $\fs2 I(x) = \int_0^1 e^{-4\,x\,t^2}\,\cos(5\,x\,t-x\,t^3)\,dt \sim {1\over 2} \,e^{-x}\,\sqrt{\pi/x}$  for $\fs2 x\to +\infty$