Lecturer: K.S. Narain , ICTP

This lesson continues with the study of the **Green's function** and at the first step it was demonstrated that the **necessary and sufficient condition** for G(x,y) to exist is that there is not a trivial solution to the equation *L*_{x}U(x)=0. To calculate the **Green's function** for a second order **differential operator** it was used the fact that the first derivative of the **Step function** H(x-y) is the **Dirac delta function** ?(x-y), then G(x,y) must be continuous at x=y, **continuity condition**, but its first derivative must behave like H(x-y) **discontinuity condition** and then its second derivative will be ?(x-y). On the other hand for x?y G(x,y) as a function of x must be the solution to the homogeneous equation. Finally by using a linear combination of the two possible solutionsĀ U_{1}(x) and U_{2}(x) to the **homogeneous differential equation** with coefficients that are function of y and using the previous conditions the green function G(x,y) is obtained and it was demonstrated that the solution is unique. Finally G(x,y) was obtained for some simples examples.

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