Lecturer: K.S. Narain , ICTP

This lesson continues with the study of the solution for the **second order differential equation** in** complex variable** in the vicinity of a **regular singular point ** z_{0}. It was started form the fact that the solution is of the form U(z)=(z-z_{0})^{R}?C_{n}(z-z_{0})^{n} and that R have two solutions R_{1} and R_{2}. Then the equation for determining C_{n} was analyzed and it was shown that defining R_{1} the solution such that Re[R_{1}]>Re[R_{2}] and then taken the solution R_{1} we can always find C_{n} in terms of C_{0.} If we choose R_{2 }then for the case when Re[R_{1}]-Re[R_{2}] is an integer number k then it is not possible to determine C_{n} for n?k. Fallowing it was demonstrated that for R=R_{1} The sum U_{1}(z)=(z-z_{0})^{R1}?C_{n}(R_{1})(z-z_{0})^{n} uniformly converge.

Then the convergence for the second case R=R_{2 }is studied, in this case the previous demonstration holds for all cases but for the particular case when the Re[R_{1}]-Re[R_{2}] is an integer. For solve this problem the method of **variation of constants** is used. This method has been already used in previous lessons and consist in find a solution U_{2}(z)=U_{1}(z)h(z) and then find the h(z) that satisfy the obtained **differential equation. **Then an expression for h(z) and therefore for U_{2}(z) was found and finally two examples of this type of solutions were studied.

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