Lecture 8: Integral representations and an introduction to dominant balance скачать в хорошем качестве

Lecture 8: Integral representations and an introduction to dominant balance

In this transitional lecture, we finish the asymptotic analysis of integrals and start the asymptotic analysis of linear ordinary differential equations with variable coefficients. We start by deriving a formula that had been assumed in several previous lectures: an integral representation for the Bessel function J_0(x). The rest of the lecture reviews the concepts of ordinary points, regular singular points, and irregular singular points, and introduces the method of dominant balance, which lets us extract the asymptotic behavior of solutions to an ODE near irregular singular points. The cool thing is we can get the behavior directly from the ODE, without actually solving the ODE or relying on an integral representation for its solution.