Basis Eigenfunctions

subject to the boundary conditions

It immediately follows from Equations (12.234) and (12.235) that

(12.232) |

Hence, we deduce that the are real. It follows that we can choose the to be real functions. Equations (12.234) and (12.235) also yield

(12.233) |

which implies that the are positive. Integration of Equation (12.235), subject to the boundary condition (12.236), gives

(12.234) |

Because is positive, this implies that

Finally, Equations (12.234) and (12.235) yield

(12.236) |

It follows that

(12.237) |

if . As is well known (Riley 1974), if and are linearly independent solutions of (12.235) corresponding to the same eigenvalue, , then it is always possible to choose linear combinations of them that satisfy

(12.238) |

This argument can be extended to multiple linearly independent solutions corresponding to the same eigenvalue. Hence, we conclude that it is possible to choose the such that they satisfy the orthonormality condition

Let be a well-behaved function. Suppose that

We can automatically satisfy the previous boundary condition by writing

(12.241) |

(Note that is undetermined to an arbitrary additive constant which is chosen so as to ensure that ) Here, is the smallest eigenvalue of Equation (12.235), the next smallest eigenvalue, and so on. It follows from Equation (12.244) that

(12.242) |

Suppose that the are well-behaved solutions of the eigenvalue equation

subject to the boundary conditions

Using analogous arguments to those employed previously, we can show that the are real and positive, and that the can be chosen so as to satisfy the orthonormality constraint

Let be a well-behaved function. Suppose that

We can automatically satisfy the previous boundary condition by writing

(12.247) |

It follows from Equation (12.250) that

(12.248) |