Mathematical methods Lecture 30 of 34

November 8, 2012 by Multimedia Publications and Printing Services

Lecturer: K.S. Narain , ICTP

This lesson starts by recalling the solution for the second order differential equation in complex variable in the vicinity of a regular singular point z0 studied in the previous lesson. Then a definition was introduced, for the case that R1 is an integer then the solution U1(z) is called meromorphic if also R1>0 then U1(z) is analytic. Then if R2 is an integer and R1 is not an integer the solution U2(z) is meromorphic and if R2>0 then U2(z) is analytic. Thus if one of the R is a non negative integer then there exist an analytic solution. Then the Fuchsian equations were studied, this equations are second order differential equation where p(z) and q(z) have only isolated regular singular points including infinity.

This lesson focused in the case of only three isolated regular singular points but is regular at infinity, this give a large class of equations called hypergeometric equations and the solutions hypergeometric solutions. To solve this kind of differential equations the fallowing change of variables is proposed z = 1/z’ using this trick it is shown that the requirement that the infinity is a regular point impose very strong conditions on p(z) and q(z).  Thus the most general hypergeometric equation was written with 9 numbers of complex parameters to be determined and subject to one complex condition so just 8 of them are free this is called the Riemann's  differential equation the solution to this equation is founded with the help of the Riemann's P symbol. It was shown that the Riemann's P symbol possesses a symmetry under the action of fractional linear transformations.

Then the solution of the form U(z)=(z-z1)R(z-z2)S(z-z3)TV(z) is proposed where z1,z2 and z3 are the three isolated regular singular points. By doing this the same type of equation is founded for V(z) where all the initial parameters are shifted by the values of R,S and T but due to the condition this parameters satisfy we have that R+S+T=0 this implies that the behavior of  U(z) and V(z) as z goes to infinity is the same. Finally a solution for a equation with  z1=0, z2=infinity, z3=1 was derived as a hypergeometric function.

Mathematics for Physicists (Text Book from Google e-books preview)

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

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