Lecturer: K.S. Narain , ICTP

This lesson starts by recalling the **necessary and sufficient condition** for the **Green's function **G(x,y) to exists. Then it was shown that the solution to the **inhomogeneous differential equation** *L*_{x}U(x)=f(x) exist only in the case that f(x) is **orthogonal** to all the solutions V(x) of the **adjoin** **homogeneous differential equation** (zero modes space) *L*_{x}^{+}V(x)=0. Similar condition is valid also for the **adjoin** **inhomogeneous differential equation** *L*_{x}^{+}V(x)=h(x) now h(x) must be **orthogonal** to all the solutions U(x) to the equation *L*_{x}U(x)=0. Then this conditions were tasted in some examples to know if the equation has solution or not and then the **Green's function **G(x,y) were calculated fallowing the method explained in the previous lesson.

To the present moment just **homogeneous** **boundary conditions **have been considered. The second part of the lesson is dedicated to the problem of having **inhomogeneous** **boundary conditions**. To solve this problem the **Green's function **G(x,y) of the **homogeneous** **boundary conditions** can be used. A method to calculate the solution to this problem using G(x,y) is explained and illustrated with one simple example.

The study of **differential equation** including functions of **complex variable** is started that is the **differential** **operator** *L*_{z} = ?^{2}/?z^{2} + p(z)?/?z + q(z). Here it assumed that the coefficients p(z) and q(z) are analytic functions almost everywhere except for some isolated poles. The points where both of them are analytic are called **ordinary points** and the points where one or both have a pole are called **singular points**. The **singular points** can be divided in two types of points, the **regular singular point****, **where p(z) has at most 1st order pole and q(z) has at most 2nd order pole. The second type, that collects all the other possibilities, is the **irregular singular points** and they will not be considered in this course.

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