Graphical illustration of Stokes phenomenon of integrals with saddles (Q2744106)

The author gives a review on several examples of Stokes phenomenon of the steepest descent method of integrals with saddles from the viewpoint of exact WKB analysis and/or hyperasymptotic analysis. Connections are made between the saddles or equivalently, visibility of singularities on the Borel plane. This gives some insight into the Stokes phenomenon. Computer generated graphics are included in the paper and play an important role in the geometry of the saddles and their steepest descent curves along with their singularities.NEWLINENEWLINENEWLINEWe extract one example among many to illustrate the results contained within the paper. We select the integral equation \(\psi(q, x)={1\over 2} \int^\infty_{-\infty} e^{-xH(u,q)} du\) with \(H(u,q)= u+ q\text{ cosh }u\). There appear two series of infinitely many saddles \(u^{\pm}_j(q)= \log{-\pm\sqrt{1+ q^2}\over q}+ 2n\pi i\), \(n\in\mathbb{Z}\). If \(q\neq \pm i\), \(0\), they are simple saddles. If \(q= \pm i\), they are double saddles. The corresponding singularities are \(\xi^{\pm}_j(q)= \log{-1\pm \sqrt{1+ q^2}\over q}+ 2j\pi i\pm \sqrt{1+ q^2}\). The function \(\psi(q,x)\) is a solution to the second-order equation of WBK type \(\psi''+{1\over q} \psi'- x^2(1+1/q^2) \psi=0\).NEWLINENEWLINENEWLINEComputer graphics for this example proves to be very interesting as well as many computer graphics for the other examples offer the reader an interesting view of the results.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].