Lecturer: K.S. Narain , ICTP

This lesson continues with the study of the second order differential equation in **complex variable**. The solution of this equation around a given point z_{0} is supposed to be a power series of z-z_{0}, were z_{0} can be an ordinary point, a regular singular point or a irregular singular point, main attention of this course is given to an ordinary points, and regular singular points. Then a method to find a solution when z_{0} is an ordinary point was proposed, the main ingredients for this method is to propose a solution of the form U(z)=f(z)Exp(g(p(z))) and then to find a new differential equation for f(z), the solution of this new differential equation is obtained by using a sequence of functions f_{n}(z) that satisfy an auxiliary recursive differential equation then f_{}(z)=? f_{n}(z).Then it is demonstrated, using the **Induction** theorem that this sum uniformly converge and it is in fact the solution to the problem. So the solution is obtained when f_{0} is specified, but f_{0} is selexcted to be linear and has two undetermined constants f_{0} = C_{0} + C_{1}(z-z_{0}), C_{0} is given by the boundary condition for z=z_{0} C0=f(z0) and C_{1}=df/dz|_{z=z0} the value of the first derivative at z=z_{0}. As an example the solution to the **Hermite's differential equation** for** complex variable** was obtained.

Then in the second part of the lesson attention was given to the solution of the **second order differential equation** in** complex variable** in the vicinity of a **regular singular point **z_{0}. Then a method to solve this problem was explained and as before the solution around the point z_{0} is supposed to be a power series of z-z_{0} U(z)=(z-z_{0})^{R}?C_{n}(z-z_{0})^{n}. Then an equation for C_{n} was derived and it was shown that C_{0}?0. This last result implies that R have to satisfy a quadratic equation thus it can have two solutions R_{1} and R_{2}. Two interesting cases will be studied in the next lesson when the Re[R_{1}]-Re[R_{2}] is an integer or is not an integer, here Re[*] represents the real part of a complex number.

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4