In several problems in mathematical physics it is necessary to study the differential

Chapter 5, Problem 19

(choose chapter or problem)

In several problems in mathematical physics it is necessary to study the differential equation x(1 x)y + [ (1 + + )x]y y = 0, (i) where , , and are constants. This equation is known as the hypergeometric equation. (a) Show that x = 0 is a regular singular point and that the roots of the indicial equation are 0 and 1 . (b) Show that x = 1 is a regular singular point and that the roots of the indicial equation are 0 and . (c) Assuming that 1 is not a positive integer, show that, in the neighborhood of x = 0, one solution of Eq. (i) is y1(x) = 1 + 1! x + ( + 1)( + 1) ( + 1)2! x2 + . What would you expect the radius of convergence of this series to be? (d) Assuming that 1 is not an integer or zero, show that a second solution for 0 < x < 1 is y2(x) = x1 1 + ( + 1)( + 1) (2 )1! x + ( + 1)( + 2)( + 1)( + 2) (2 )(3 )2! x2 + . (e) Show that the point at infinity is a regular singular point and that the roots of the indicial equation are and . See of Section 5.4.