hypergeometric_appell.lisp

;;;; Copyright (c) 2015 Russell Andrew Edson
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;;;; Code for approximations of the Appell hypergeometric function.
;;;; TODO
;;;; Date: 27/09/2015

(in-package :cl-mathspecialfunctions)

;;; The Appell Hypergeometric function, F1(a,b,bb,c,x,y). We can compute
;;; this as a truncation of an infinite sum of Gauss hypergeometric
;;; functions -- the given parameter num-terms specifies how many terms
;;; of the summation to use (default is 20).
;;;
;;; Code Usage Examples:
;;;
;;;
(defun hypergeometric-appell (a b bb c x y &optional (num-terms 20))
  "Returns Appell's Hypergeometric function, F1(a,b,bb,c,x,y)."
  ;; As each term in the series contains a ratio of factorials in a, b
  ;; and c, we can keep track of the previous term each time and update
  ;; it, instead of computing factorials each time.
  (let ((partial-sum (hypergeometric-gauss a bb c y))
        (abcx-term 1))
    (loop for m from 1 to num-terms do
         (setf abcx-term (* abcx-term
                            (/ (* (+ a m -1) (+ b m -1)) (+ c m -1))
                            x
                            (/ 1 m)))
         (setf partial-sum
               (+ partial-sum
                  (* abcx-term
                     (hypergeometric-gauss (+ a m) bb (+ c m) y)))))
    partial-sum))