Mathematical Methods Lecture 29 of 34

December 5, 2011 by K.S. Narain

K.S. Narain , ICTP

This lesson continues with the study of the solution for the second order differential equation in complex variable in the vicinity of a regular singular point z0. It was started form the fact that the solution is of the form U(z)=(z-z0)R?Cn(z-z0)n and that R have two solutions R1 and R2. Then the equation for determining Cn was analyzed and it was shown that defining R1 the solution such that Re[R1]>Re[R2] and then taken the solution R1 we can always find Cn in terms of C0. If we choose R2 then for the case when Re[R1]-Re[R2] is an integer number k then it is not possible to determine Cn for n?k. Fallowing it was demonstrated that for R=R1 The sum U1(z)=(z-z0)R1?Cn(R1)(z-z0)n uniformly converge.

Then the convergence for the second case R=R2 is studied, in this case the previous demonstration holds for all cases but for the particular case when the Re[R1]-Re[R2] is an integer. For solve this problem the method of variation of constants is used. This method has been already used in previous lessons and consist in find a solution U2(z)=U1(z)h(z) and then find the h(z) that satisfy the obtained differential equation. Then an expression for h(z) and therefore for U2(z) was found and finally two examples of this type of solutions were studied.

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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