### Mathematical Methods Lecture 28 of 34

December 2, 2011 by K.S. Narain

K.S. Narain , ICTP

This lesson continues with the study of the second order differential equation in complex variable. The solution of this equation around a given point z0 is supposed to be a power series of z-z0, were z0 can be an ordinary point, a regular singular point or a irregular singular point, main attention of this course is given to an ordinary points, and regular singular points. Then a method to find a solution when z0 is an ordinary point was proposed, the main ingredients for this method is to propose a solution of the form U(z)=f(z)Exp(g(p(z))) and then to find a new differential equation for f(z), the solution of this new differential equation is obtained by using a sequence of functions fn(z) that satisfy an auxiliary recursive differential equation then f(z)=? fn(z).Then it is demonstrated, using the Induction theorem that this sum uniformly converge and it is in fact the solution to the problem. So the solution is obtained when f0 is specified, but f0 is selexcted to be linear and has two undetermined constants f0 = C0 + C1(z-z0), C0 is given by the boundary condition for z=z0 C0=f(z0) and C1=df/dz|z=z0 the value of the first derivative at z=z0. As an example the solution to the Hermite's differential equation for complex variable was obtained.

Then in the second part of the lesson attention was given to the solution of the second order differential equation in complex variable in the vicinity of a regular singular point z0. Then a method to solve this problem was explained and as before the solution around the point z0 is supposed to be a power series of z-z0 U(z)=(z-z0)R?Cn(z-z0)n. Then an equation for Cn was derived and it was shown that C0?0. This last result implies that R have to satisfy a quadratic equation thus it can have two solutions R1 and R2. Two interesting cases will be studied in the next lesson when the Re[R1]-Re[R2] is an integer or is not an integer, here Re[*] represents the real part of a complex number.

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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