Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

268

Let us now analyze the expression on the left side of equation (5.232), and to that end let us observe the boundary conditions (5.230): k 1 y m ( a )+ k 2 y ′ m ( a )= 0 , (5.233) k 1 y n ( a )+ k 2 y ′ n ( a )= 0 . (5.234) Multiplying the first equation by y n and the second by y m , and then subtracting the resulting equations, we obtain k 2 y m ( a ) y ′ n ( a ) − y n ( a ) y ′ m ( a ) = 0 . (5.235) Assuming that k 2̸ = 0we obtain y m ( a ) y ′ n ( a ) − y n ( a ) y ′ m ( a )= 0 . (5.236) Similarly, it can be shown that also y m ( b ) y ′ n ( b ) − y n ( b ) y ′ m ( b )= 0 , (5.237) for l 2̸ = 0. Based on these relations we conclude that b Z a py m y n d y = 0 , for m̸ = n . (5.238) We have thus proved the theorem for k 2̸ = 0 and l 2̸ = 0. Let us observe again conditions (5.233) and (5.234). By multiplying the first condition by y ′ n and the second by y ′ m , and then subtracting the resulting equations, we obtain, for k 1̸ = 0 y m ( a ) y ′ n ( a ) − y n ( a ) y ′ m ( a )= 0 . (5.239) Similarly, for l 1̸ = 0we obtain y m ( b ) y ′ n ( b ) − y n ( b ) y ′ m ( b )= 0 . (5.240) Thus the theorem is proved for this case as well, and given that k 1 and k 2 , that is, l 1 and l 2 can not be both equal to zero, the theorem is proved in its entirety.

Theorem17 If the Sturm–Liouville problem (5.229), (5.230) satisfies the conditions of the previous theorem, and if p̸ = 0 on the entire interval a ≤ x ≤ b , then all principal values of the problem are real.

Proof Let us assume the contrary, namely that λ = α + i β is a principal value of the problem, and that the corresponding principal function has the form

y ( x )= u ( x )+ iv ( x ) .