### Mathematical Methods Lecture 27 of 34

November 30, 2011 by K.S. Narain

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 LxU(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) Lx+V(x)=0. Similar condition is valid also for the adjoin inhomogeneous differential equation Lx+V(x)=h(x) now h(x) must be orthogonal to all the solutions U(x) to the equation LxU(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 Lz = ?2/?z2 + 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.

Mathematics for Physicists (Text Book from Google e-books preview)Written by<span class="addmd"> Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4</span>

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