This course is concerned with asymptotic expansions. In an asymptotic expansion a function of interest is written as the sum of an explicit expression and a remainder term, for example the Taylor expansion f (x) = f (0) + f ′(0)x + O(x^2), for short f (x) ∼ f (0) + f ′(0)x. We can also consider asymptotic series where the right hand side is replaced by an infinite series. This statement claims nothing about the convergence of the series on the right hand side. It only means that the difference between a partial sum of the series and the left hand side is smaller than all terms in the partial sum. Surprisingly it can happen that asymptotic expansions are more useful in applications than convergent series. We will discuss applications of these ideas to ordinary differential equations whose solutions have complicated behaviour near a point. A famous example is the harmless-lookingequation d^2 y/ dx^2 = xy, the Airy equation. The prerequisites for this course are the basic courses (Grundvorlesungen). A knowledge of complex analysis (Funktionentheorie) could be helpful but is not necessary in order to follow the course. The course is aimed at MSc students but could also be followed by BSc or MEd students. The main source for the course is the book of Wasow quoted below.
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover (1987)