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Manuscript Title: A code to evaluate prolate and oblate spheroidal harmonics.
Authors: A. Gil, J. Segura
Program title: DPROH, DOBLH
Catalogue identifier: ADHD_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 108(1998)267
Programming language: Fortran.
Computer: HP Model 715/100.
Operating system: UNIX.
Keywords: General purpose, Function, Prolate and oblate, Spheroidal harmonics, Legendre functions, Continued fraction.
Classification: 4.7.

Nature of problem:
We include two codes in order to evaluate:
  1. Prolate spheroidal harmonics (subroutine DPROH)
  2. Oblate spheroidal harmonics (subroutine DOBLH)
The two codes evaluate spheroidal harmonics of the first and second kinds for a given order m, from the lowest (positive) degree (n = m) to a maximum degree n = m + NMAX in the same run.
The algorithms find their application in problems with prolate and oblate spheroidal geometries respectively. We show as an example the application of the subroutine DOBLH to the evaluation of the surface charge density on a conducting disk of radius a due to a point charge q at distance l from the disk along its axis of symmetry.

Solution method:
The codes evaluate first kind spheroidal harmonics {Pmn} through forward recurrence over n starting from the calculation of the lowest degree (n = m, n = m + 1) P's and then, after using a continued fraction for the second kind spheroidal harmonics {Qmn} and the Wronskian relation, applies backward recurrence for the Q's. This algorithm does not require any trial values to start the recurrences nor any renormalization.

The maximum degree (order) that can be reached with our method, for a given order (degree) m (n) and for a fixed real positive value of x, is provided by the maximum real number defined in our machine. The code for prolate spheroidal harmonics is to be used for real x > 1 while the code for oblate spheroidal harmonics can be used for real x > 0.

Running time:
Around 0.1 ms to evaluate {Pmm+N,Qmm+N} N = 0,1,...,10 for a precision ~ 10**-15; see text (LONG WRITE-UP: section 6).