Lecturer: K.S. Narain , ICTP

This lesson starts by recalling the solution for the **second order differential equation** in** **** complex variable** in the vicinity of a

**regular singular point**z

_{0}studied in the previous lesson. Then a definition was introduced, for the case that R

_{1}is an integer then the solution U

_{1}(z) is called

**meromorphic**if also R

_{1}>0 then U

_{1}(z) is analytic. Then if R

_{2}is an integer and R

_{1}is not an integer the solution U

_{2}(z) is

**meromorphic**and if R

_{2}>0 then U

_{2}(z) is analytic. Thus if one of the R is a non negative integer then there exist an analytic solution. Then the

**Fuchsian equations**were

**studied, this equations are**

**second order differential equation**where p(z) and q(z) have only isolated

**regular singular points**including infinity.

This lesson focused in the case of only three isolated **regular singular points **but is regular at infinity, this give** **a large class of equations** **called **hypergeometric equations **and the solutions **hypergeometric solutions. **To solve this kind of differential equations the fallowing change of variables is proposed z = 1/z’ using this trick it is shown that the requirement that the infinity is a **regular point** impose very strong conditions on p(z) and q(z). Thus the most general **hypergeometric equation **was written with 9 numbers of complex parameters to be determined and subject to one complex condition so just 8 of them are free this is called the **Riemann's differential equation **the solution to this equation is founded with the help of the **Riemann's P symbol. **It was shown that the** Riemann's P symbol **possesses a symmetry under the action of

**fractional linear transformations.**Then the solution of the form U(z)=(z-z_{1})^{R}(z-z_{2})^{S}(z-z_{3})^{T}V(z) is proposed** **where z_{1},z_{2 and }z_{3} are the three isolated **regular singular points**.** **By doing this the same type of equation is founded for V(z) where all the initial parameters are shifted by the values of R,S and T but due to the condition this parameters satisfy we have that R+S+T=0 this implies that the behavior of U(z) and V(z) as z goes to infinity is the same. Finally a solution for a equation with z_{1}=0, z_{2}=infinity, z_{3}=1 was derived as a **hypergeometric function.**

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