Math 6720 • Applied Complex Variables and Asymptotic Methods

Let \(c=\sqrt {2}-1\). Use the method of steepest descent to show that \[ \int _{-\infty }^{\infty } e^{ikx}(1+x^2)^{-k}\,dx \sim \sqrt {\frac {\pi (1-c)}{k}}\,e^{-ck}(2c)^{-k} \] as \(k\to \infty \). Hint: You do not need to work out the exact steepest descent contour, but give a sketch of what it looks like. You do not need to give full justification for the deformation from the original contour to the steepest descent contour, but at least write down an idea of why it should be true.