     Next: Consistency with Biorthonormality Up: Perturbation of the Previous: Perturbation of the

# Recursion relations in Component Form

We will now present formulae for calculating , and . We do this by expanding ( ) and ( ) in the and basis (zeroth order basis) with the following definition: We now define an operator so that where is the projection onto the orthogonal   complement of . In other words, if the domain of is restricted to make one to one, then is the inverse of .

By applying on equation ( ), the multipole functions can be generated according to  Or expressed in the zeroth order basis for ,  where are the matrix elements of the operator , and are the eigenvalues of .

These relations will generate the multipole functions from the seeds . If then and may be chosen arbitrarily subject to To complete the prescription, we need a formula for calculating the , and we also need to check that the biorthonormality condition ( )   is satisfied in the case . For convenience of notation, we will define for every n less than zero. This means that all the power series expansions hold for every integer n. Expanding this in powers of , one gets The second term can be expressed as The second two terms vanish because of the biorthonormality condition ( ) and the eigenvalue equation ( ). The final step is to evaluate this equation by the recursion equation ( ): A similar calculation using the adjoint equation gives an   alternative formula for . Thus the problem has been reduced to evaluating the matrix element .

For example, if the spectrum of is non-degenerate,   the first few coefficients of the recursion series are    In this case, the arbitrary additive terms are proportional to . The rank n tensors are the coefficients of these terms, and may be chosen to be zero.     Next: Consistency with Biorthonormality Up: Perturbation of the Previous: Perturbation of the

Russell Standish
Thu May 18 11:43:52 EST 1995