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qd-bessel.lisp 16.6 KB
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  ;;;; -*- Mode: lisp -*-
  ;;;;
  ;;;; Copyright (c) 2011 Raymond Toy
  ;;;; Permission is hereby granted, free of charge, to any person
  ;;;; obtaining a copy of this software and associated documentation
  ;;;; files (the "Software"), to deal in the Software without
  ;;;; restriction, including without limitation the rights to use,
  ;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell
  ;;;; copies of the Software, and to permit persons to whom the
  ;;;; Software is furnished to do so, subject to the following
  ;;;; conditions:
  ;;;;
  ;;;; The above copyright notice and this permission notice shall be
  ;;;; included in all copies or substantial portions of the Software.
  ;;;;
  ;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
  ;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
  ;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
  ;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
  ;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
  ;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
  ;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
  ;;;; OTHER DEALINGS IN THE SOFTWARE.
  
  (in-package #:oct)
  
  ;;; References:
  ;;;
  ;;; [1] Borwein, Borwein, Crandall, "Effective Laguerre Asymptotics",
  ;;; http://people.reed.edu/~crandall/papers/Laguerre-f.pdf
  ;;;
  ;;; [2] Borwein, Borwein, Chan, "The Evaluation of Bessel Functions
  ;;; via Exp-Arc Integrals", http://web.cs.dal.ca/~jborwein/bessel.pdf
  ;;;
  
  (defvar *debug-exparc* nil)
  
  ;; B[k](p) = 1/2^(k+3/2)*integrate(exp(-p*u)*u^(k-1/2),u,0,1)
  ;;         = 1/2^(k+3/2)/p^(k+1/2)*integrate(t^(k-1/2)*exp(-t),t,0,p)
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  ;;         = 1/2^(k+3/2)/p^(k+1/2) * G(k+1/2, p)
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  ;;
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  ;; where G(a,z) is the lower incomplete gamma function.
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  ;;
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  ;; There is the continued fraction expansion for G(a,z) (see
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  ;; cf-incomplete-gamma in qd-gamma.lisp):
  ;;
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  ;;  G(a,z) = z^a*exp(-z)/ CF
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  ;;
  ;; So
  ;;
  ;;  B[k](p) = 1/2^(k+3/2)/p^(k+1/2)*p^(k+1/2)*exp(-p)/CF
  ;;          = exp(-p)/2^(k+3/2)/CF
  ;;
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  ;;
  ;; Note also that [2] gives a recurrence relationship for B[k](p) in
  ;; eq (2.6), but there is an error there.  The correct relationship is
  ;;
  ;;  B[k](p) = -exp(-p)/(p*sqrt(2)*2^(k+1)) + (k-1/2)*B[k-1](p)/(2*p)
  ;;
  ;; The paper is missing the division by p in the term containing
  ;; B[k-1](p).  This is easily derived from the recurrence relationship
  ;; for the (lower) incomplete gamma function.
  ;;
  ;; Note too that as k increases, the recurrence appears to be unstable
  ;; and B[k](p) begins to increase even though it is strictly bounded.
  ;; (This is also easy to see from the integral.)  Hence, we do not use
  ;; the recursion.  However, it might be stable for use with
  ;; double-float precision; this has not been tested.
  ;;
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  (defun bk (k p)
    (/ (exp (- p))
       (* (sqrt (float 2 (realpart p))) (ash 1 (+ k 1)))
       (let ((a (float (+ k 1/2) (realpart p))))
         (lentz #'(lambda (n)
  		  (+ n a))
  	      #'(lambda (n)
  		  (if (evenp n)
  		      (* (ash n -1) p)
  		      (- (* (+ a (ash n -1)) p))))))))
  
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  ;; Use the recursion
  (defun bk-iter (k p old-bk)
    (with-floating-point-contagion (p old-bk)
      (if (zerop k)
  	(* (sqrt (/ (float-pi p) 8))
  	   (let ((rp (sqrt p)))
  	     (/ (erf rp)
  		rp)))
  	(- (* (- k 1/2)
  	      (/ old-bk (* 2 p)))
  	   (/ (exp (- p))
  	      p
  	      (ash 1 (+ k 1))
  	      (sqrt  (float 2 (realpart p))))))))
  
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  ;; exp-arc I function, as given in the Laguerre paper
  ;;
  ;; I(p, q) = 4*exp(p) * sum(g[k](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf)
  ;;
  ;; where g[k](p) = product(p^2+(2*j-1)^2, j, 1, k) and B[k](p) as above.
  ;;
  ;; For computation, note that g[k](p) = g[k-1](p) * (p^2 + (2*k-1)^2)
  ;; and (2*k)! = (2*k-2)! * (2*k-1) * (2*k).  Then, let
  ;;
  ;;  R[k](p) = g[k](p)/(2*k)!
  ;;
  ;; Then
  ;;
  ;;  R[k](p) = g[k](p)/(2*k)!
  ;;          = g[k-1](p)/(2*k-2)! * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k)
  ;;          = R[k-1](p) * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k)
  ;;
  ;; In the exp-arc paper, the function is defined (equivalently) as
  ;; 
  ;; I(p, q) = 2*%i*exp(p)/q * sum(r[2*k+1](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf)
  ;;
  ;; where r[2*k+1](p) = p*product(p^2 + (2*j-1)^2, j, 1, k)
  ;;
  ;; Let's note some properties of I(p, q).
  ;;
  ;; I(-%i*z, v) = 2*%i*exp(-%i*z)/q * sum(r[2*k+1](-2*%i*v)/(2*k)!*B[k](-%i*z))
  ;;
  ;; Note thate B[k](-%i*z) = 1/2^(k+3/2)*integrate(exp(%i*z*u)*u^(k-1/2),u,0,1)
  ;;                        = conj(B[k](%i*z).
  ;;
  ;; Hence I(-%i*z, v) = conj(I(%i*z, v)) when both z and v are real.
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  ;;
  ;; Also note that when v is an integer of the form (2*m+1)/2, then
  ;;   r[2*k+1](-2*%i*v) = r[2*k+1](-%i*(2*m+1))
  ;;                     = -%i*(2*m+1)*product(-(2*m+1)^2+(2*j-1)^2, j, 1, k)
  ;; so the product is zero when k >= m and the series I(p, q) is
  ;; finite.
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  (defun exp-arc-i (p q)
    (let* ((sqrt2 (sqrt (float 2 (realpart p))))
  	 (exp/p/sqrt2 (/ (exp (- p)) p sqrt2))
  	 (v (* #c(0 -2) q))
  	 (v2 (expt v 2))
  	 (eps (epsilon (realpart p))))
      (when *debug-exparc*
        (format t "sqrt2 = ~S~%" sqrt2)
        (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2))
      (do* ((k 0 (1+ k))
  	  (bk (/ (incomplete-gamma 1/2 p)
  		 2 sqrt2 (sqrt p))
  	      (- (/ (* bk (- k 1/2)) 2 p)
  		 (/ exp/p/sqrt2 (ash 1 (+ k 1)))))
  	  ;; ratio[k] = r[2*k+1](v)/(2*k)!.
  	  ;; r[1] = v and r[2*k+1](v) = r[2*k-1](v)*(v^2 + (2*k-1)^2)
  	  ;; ratio[0] = v
  	  ;; and ratio[k] = r[2*k-1](v)*(v^2+(2*k-1)^2) / ((2*k-2)! * (2*k-1) * 2*k)
  	  ;;              = ratio[k]*(v^2+(2*k-1)^2)/((2*k-1) * 2 * k)
  	  (ratio v
  		 (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2))
  			     (* 2 k (1- (* 2 k))))))
  	  (term (* ratio bk)
  		(* ratio bk))
  	  (sum term (+ sum term)))
  	 ((< (abs term) (* (abs sum) eps))
  	  (* sum #c(0 2) (/ (exp p) q)))
        (when *debug-exparc*
  	(format t "k      = ~D~%" k)
  	(format t " bk    = ~S~%" bk)
  	(format t " ratio = ~S~%" ratio)
  	(format t " term  = ~S~%" term)
  	(format t " sum   - ~S~%" sum)))))
  
  (defun exp-arc-i-2 (p q)
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    (let* ((v (* #c(0 -2) q))
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  	 (v2 (expt v 2))
  	 (eps (epsilon (realpart p))))
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      (do* ((k 0 (1+ k))
  	  (bk (bk 0 p)
  	      (bk k p))
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  	  ;; Compute g[k](p)/(2*k)!, not r[2*k+1](p)/(2*k)!
  	  (ratio 1
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  		 (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2))
  			     (* 2 k (1- (* 2 k))))))
  	  (term (* ratio bk)
  		(* ratio bk))
  	  (sum term (+ sum term)))
  	 ((< (abs term) (* (abs sum) eps))
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  	  (when *debug-exparc*
  	    (format t "Final k= ~D~%" k)
  	    (format t " bk    = ~S~%" bk)
  	    (format t " ratio = ~S~%" ratio)
  	    (format t " term  = ~S~%" term)
  	    (format t " sum   - ~S~%" sum))
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  	  (* sum 4 (exp p)))
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        (when *debug-exparc*
  	(format t "k      = ~D~%" k)
  	(format t " bk    = ~S~%" bk)
  	(format t " ratio = ~S~%" ratio)
  	(format t " term  = ~S~%" term)
  	(format t " sum   - ~S~%" sum)))))
  
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  (defun exp-arc-i-3 (p q)
    (let* ((v (* #c(0 -2) q))
  	 (v2 (expt v 2))
  	 (eps (epsilon (realpart p))))
      (do* ((k 0 (1+ k))
  	  (bk (bk 0 p)
  	      (bk-iter k p bk))
  	  ;; Compute g[k](p)/(2*k)!, not r[2*k+1](p)/(2*k)!
  	  (ratio 1
  		 (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2))
  			     (* 2 k (1- (* 2 k))))))
  	  (term (* ratio bk)
  		(* ratio bk))
  	  (sum term (+ sum term)))
  	 ((< (abs term) (* (abs sum) eps))
  	  (when *debug-exparc*
  	    (format t "Final k= ~D~%" k)
  	    (format t " bk    = ~S~%" bk)
  	    (format t " ratio = ~S~%" ratio)
  	    (format t " term  = ~S~%" term)
  	    (format t " sum   - ~S~%" sum))
  	  (* sum 4 (exp p)))
        (when *debug-exparc*
  	(format t "k      = ~D~%" k)
  	(format t " bk    = ~S~%" bk)
  	(format t " ratio = ~S~%" ratio)
  	(format t " term  = ~S~%" term)
  	(format t " sum   - ~S~%" sum)))))
  
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  ;; Not really just for Bessel J for integer orders, but in that case,
  ;; this is all that's needed to compute Bessel J.  For other values,
  ;; this is just part of the computation needed.
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  ;;
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  ;; Compute
  ;;
  ;;  1/(2*%pi) * (exp(-%i*v*%pi/2) * I(%i*z, v) + exp(%i*v*%pi/2) * I(-%i*z, v))
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  (defun integer-bessel-j-exp-arc (v z)
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    (let* ((iz (* #c(0 1) z))
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  	 (i+ (exp-arc-i-2 iz v)))
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      (cond ((and (= v (ftruncate v)) (realp z))
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  	   ;; We can simplify the result
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  	   (let ((c (exp (* v (float-pi i+) #c(0 -1/2)))))
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  	     (/ (+ (* c i+)
  		   (* (conjugate c) (conjugate i+)))
  		(float-pi i+)
  		2)))
  	  (t
  	   (let ((i- (exp-arc-i-2 (- iz ) v)))
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  	     (/ (+ (* (exp (* v (float-pi i+) #c(0 -1/2)))
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  		      i+)
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  		   (* (exp (* v (float-pi i+) #c(0 1/2)))
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  		      i-))
  		(float-pi i+)
  		2))))))
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  ;; alpha[n](z) = integrate(exp(-z*s)*s^n, s, 0, 1/2)
  ;; beta[n](z)  = integrate(exp(-z*s)*s^n, s, -1/2, 1/2)
  ;;
  ;; The recurrence in [2] is
  ;;
  ;; alpha[n](z) = - exp(-z/2)/2^n/z + n/z*alpha[n-1](z)
  ;; beta[n]z)   = ((-1)^n*exp(z/2)-exp(-z/2))/2^n/z + n/z*beta[n-1](z)
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  ;;             = (-1)^n/(2^n)*2*sinh(z/2)/z + n/z*beta[n-1](z)
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  ;;
  ;; We also note that
  ;;
  ;; alpha[n](z) = G(n+1,z/2)/z^(n+1)
  ;; beta[n](z)  = G(n+1,z/2)/z^(n+1) - G(n+1,-z/2)/z^(n+1)
  
  (defun alpha (n z)
    (let ((n (float n (realpart z))))
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      (/ (incomplete-gamma (1+ n) (/ z 2))
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         (expt z (1+ n)))))
  
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  (defun alpha-iter (n z alpha-old)
    (if (zerop n)
        ;; (1- exp(-z/2))/z.
        (/ (- 1 (exp (* z -1/2)))
  	 z)
        (- (* (/ n z) alpha-old)
  	 (/ (exp (- (* z 1/2)))
  	    z
  	    (ash 1 n)))))
  
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  (defun beta (n z)
    (let ((n (float n (realpart z))))
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      (/ (- (incomplete-gamma (1+ n) (/ z 2))
  	  (incomplete-gamma (1+ n) (/ z -2)))
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         (expt z (1+ n)))))
  
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  (defun beta-iter (n z old-beta)
    (if (zerop n)
        ;; integrate(exp(-z*s),s,-1/2,1/2)
        ;;   = (exp(z/2)-exp(-z/2)/z
        ;;   = 2*sinh(z/2)/z
        ;;   = sinh(z/2)/(z/2)
        (* 2 (/ (sinh (* 1/2 z)) z))
        (+ (* n (/ old-beta z))
  	 (* (/ (sinh (* 1/2 z)) (* 1/2 z))
  	    (scale-float (float (if (evenp n) 1 -1) (realpart z)) (- n))))))
  
  
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  ;; a[0](k,v) := (k+sqrt(k^2+1))^(-v);
  ;; a[1](k,v) := -v*a[0](k,v)/sqrt(k^2+1);
  ;; a[n](k,v) := 1/(k^2+1)/(n-1)/n*((v^2-(n-2)^2)*a[n-2](k,v)-k*(n-1)*(2*n-3)*a[n-1](k,v));
  
  ;; Convert this to iteration instead of using this quick-and-dirty
  ;; memoization?
  (let ((hash (make-hash-table :test 'equal)))
    (defun an-clrhash ()
      (clrhash hash))
    (defun an-dump-hash ()
      (maphash #'(lambda (k v)
  		 (format t "~S -> ~S~%" k v))
  	     hash))
    (defun an (n k v)
      (or (gethash (list n k v) hash)
  	(let ((result
  		(cond ((= n 0)
  		       (expt (+ k (sqrt (float (1+ (* k k)) (realpart v)))) (- v)))
  		      ((= n 1)
  		       (- (/ (* v (an 0 k v))
  			     (sqrt (float (1+ (* k k)) (realpart v))))))
  		      (t
  		       (/ (- (* (- (* v v) (expt (- n 2) 2)) (an (- n 2) k v))
  			     (* k (- n 1) (+ n n -3) (an (- n 1) k v)))
  			  (+ 1 (* k k))
  			  (- n 1)
  			  n)))))
  	  (setf (gethash (list n k v) hash) result)
  	  result))))
  
  ;; SUM-AN computes the series
  ;;
  ;; sum(exp(-k*z)*a[n](k,v), k, 1, N)
  ;;
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  #+nil
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  (defun sum-an (big-n n v z)
    (let ((sum 0))
      (loop for k from 1 upto big-n
  	  do
  	     (incf sum (* (exp (- (* k z)))
  			  (an n k v))))
      sum))
  
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  ;; Like above, but we just stop when the terms no longer contribute to
  ;; the sum.
  (defun sum-an (big-n n v z)
    (let ((eps (epsilon (realpart z))))
      (do* ((k 1 (+ 1 k))
  	  (term (* (exp (- (* k z)))
  		   (an n k v))
  		(* (exp (- (* k z)))
  		   (an n k v)))
  	  (sum term (+ sum term)))
  	 ((or (<= (abs term) (* eps (abs sum)))
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  	      (>= k big-n))
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  	  sum))))
  
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  ;; SUM-AB computes the series
  ;;
  ;; sum(alpha[n](z)*a[n](0,v) + beta[n](z)*sum_an(N, n, v, z), n, 0, inf)
  (defun sum-ab (big-n v z)
    (let ((eps (epsilon (realpart z))))
      (an-clrhash)
      (do* ((n 0 (+ 1 n))
  	  (term (+ (* (alpha n z) (an n 0 v))
  		   (* (beta n z) (sum-an big-n n v z)))
  		(+ (* (alpha n z) (an n 0 v))
  		   (* (beta n z) (sum-an big-n n v z))))
  	  (sum term (+ sum term)))
  	 ((<= (abs term) (* eps (abs sum)))
  	  sum)
        (when nil
  	(format t "n = ~D~%" n)
  	(format t " term = ~S~%" term)
  	(format t " sum  = ~S~%" sum)))))
  
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  (defun sum-ab-2 (big-n v z)
    (let ((eps (epsilon (realpart z))))
      (an-clrhash)
      (do* ((n 0 (+ 1 n))
  	  (alphan (alpha-iter 0 z 0)
  		  (alpha-iter n z alphan))
  	  (betan (beta-iter 0 z 0)
  		 (beta-iter n z betan))
  	  (term (+ (* alphan (an n 0 v))
  		   (* betan (sum-an big-n n v z)))
  		(+ (* alphan (an n 0 v))
  		   (* betan (sum-an big-n n v z))))
  	  (sum term (+ sum term)))
  	 ((<= (abs term) (* eps (abs sum)))
  	  sum)
        (when nil
  	(format t "n = ~D~%" n)
  	(format t " term = ~S~%" term)
  	(format t " sum  = ~S~%" sum)))))
  
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  ;; Convert to iteration instead of this quick-and-dirty memoization?
  (let ((hash (make-hash-table :test 'equal)))
    (defun %big-a-clrhash ()
      (clrhash hash))
    (defun %big-a-dump-hash ()
      (maphash #'(lambda (k v)
  		 (format t "~S -> ~S~%" k v))
  	     hash))
    (defun %big-a (n v)
      (or (gethash (list n v) hash)
  	(let ((result
  		(cond ((zerop n)
  		       (expt 2 (- v)))
  		      (t
  		       (* (%big-a (- n 1) v)
  			  (/ (* (+ v n n -2) (+ v n n -1))
  			     (* 4 n (+ n v))))))))
  	  (setf (gethash (list n v) hash) result)
  	  result))))
  
  ;; Computes A[n](v) =
  ;; (-1)^n*v*2^(-v)*pochhammer(v+n+1,n-1)/(2^(2*n)*n!)  If v is a
  ;; negative integer -m, use A[n](-m) = (-1)^(m+1)*A[n-m](m) for n >=
  ;; m.
  (defun big-a (n v)
    (let ((m (ftruncate v)))
      (cond ((and (= m v) (minusp m))
  	   (if (< n m)
  	       (%big-a n v)
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  	       (let ((result (%big-a (+ n m) (- v))))
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  		 (if (oddp (truncate m))
  		     result
  		     (- result)))))
  	  (t
  	   (%big-a n v)))))
  
  ;; I[n](t, z, v) = exp(-t*z)/t^(2*n+v-1) *
  ;;                  integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf)
  ;;
  ;; Use the substitution u=1+s to get a new integral
  ;;
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  ;;   integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf)
  ;;     = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf)
  ;;     = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z)
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  ;;
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  ;; Thus,
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  ;;
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  ;;   I[n](t, z, v) = z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z)
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  ;;
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  (defun big-i (n theta z v)
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    (let* ((a (- 1 v n n)))
      (* (expt z (- a))
         (incomplete-gamma-tail a (* theta z)))))
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  (defun sum-big-ia (big-n v z)
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    (let ((big-n-1/2 (+ big-n 1/2))
  	(eps (epsilon z)))
      (do* ((n 0 (1+ n))
  	  (term (* (big-a 0 v)
  		   (big-i 0 big-n-1/2 z v))
  		(* (big-a n v)
  		   (big-i n big-n-1/2 z v)))
  	  (sum term (+ sum term)))
  	 ((<= (abs term) (* eps (abs sum)))
  	  sum)
        #+nil
        (progn
  	(format t "n = ~D~%" n)
  	(format t " term = ~S~%" term)
  	(format t " sum  = ~S~%" sum)))))
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  ;; Series for bessel J:
  ;;
  ;; (z/2)^v*sum((-1)^k/Gamma(k+v+1)/k!*(z^2//4)^k, k, 0, inf)
  (defun s-bessel-j (v z)
    (with-floating-point-contagion (v z)
      (let ((z2/4 (* z z 1/4))
  	  (eps (epsilon z)))
        (do* ((k 0 (+ 1 k))
  	    (f (gamma (+ v 1))
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  	       (* k (+ v k)))
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  	    (term (/ f)
  		  (/ (* (- term) z2/4) f))
  	    (sum term (+ sum term)))
  	   ((<= (abs term) (* eps (abs sum)))
  	    (* sum (expt (* z 1/2) v)))
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  	#+nil
  	(progn
  	  (format t "k = ~D~%" k)
  	  (format t " f    = ~S~%" f)
  	  (format t " term = ~S~%" term)
  	  (format t " sum  = ~S~%" sum))))))
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  ;; 
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  ;; TODO:
  ;;  o For |z| <= 1 use the series.
  ;;  o Currently accuracy is not good for large z and half-integer
  ;;    order.
  ;;  o For real v and z, return a real number instead of complex.
  ;;  o Handle the case of Re(z) < 0. (The formulas are for Re(z) > 0:
  ;;    bessel_j(v,z*exp(m*%pi*%i)) = exp(m*v*%pi*%i)*bessel_j(v, z)
  ;;  o The paper suggests using
  ;;      bessel_i(v,z) = exp(-v*%pi*%i/2)*bessel_j(v, %i*z)
  ;;    when Im(z) >> Re(z)
  ;; 
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  (defvar *big-n* 100)
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  (defun bessel-j (v z)
    (let ((vv (ftruncate v)))
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      ;; Clear the caches for now.
      (an-clrhash)
      (%big-a-clrhash)
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      (cond ((and (= vv v) (realp z))
  	   ;; v is an integer and z is real
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  	   (integer-bessel-j-exp-arc v z))
  	  (t
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  	   ;; Need to fine-tune the value of big-n.
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  	   (let ((big-n *big-n*)
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  		 (vpi (* v (float-pi (realpart z)))))
  	     (+ (integer-bessel-j-exp-arc v z)
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  		(if (= vv v)
  		    0
  		    (* z
  		       (/ (sin vpi) vpi)
  		       (+ (/ -1 z)
  			  (sum-ab big-n v z)
  			  (sum-big-ia big-n v z))))))))))
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  ;; Bessel Y
  ;;
  ;; bessel_y(v, z) = 1/(2*%pi*%i)*(exp(-%i*v*%pi/2)*I(%i*v,z) - exp(%i*v*%pi/2)*I(-%i*z, v))
  ;;                   + z/v/%pi*((1-cos(v*%pi)/z) + S(N,z,v)*cos(v*%pi)-S(N,z,-v))
  ;;
  ;; where
  ;;
  ;;   S(N,z,v) = sum(alpha[n](z)*a[n](0,v) + beta[n](z)*sum(exp(-k*z)*a[n](k,v),k,1,N),n,0,inf)
  ;;               + sum(A[n](v)*I[n](N+1/2,z,v),n,0,inf)
  ;;
  (defun bessel-y (v z)
    (flet ((ipart (v z)
  	   (let* ((iz (* #c(0 1) z))
  		  (c+ (exp (* v (float-pi z) 1/2)))
  		  (c- (exp (* v (float-pi z) -1/2)))
  		  (i+ (exp-arc-i-2 iz v))
  		  (i- (exp-arc-i-2 (- iz) v)))
  	     (/ (- (* c- i+) (* c+ i-))
  		(* #c(0 2) (float-pi z)))))
  	 (s (big-n z v)
  	   (+ (sum-ab big-n v z)
  	      (sum-big-ia big-n v z))))
      (let* ((big-n 100)
  	   (vpi (* v (float-pi z)))
  	   (c (cos vpi)))
        (+ (ipart v z)
  	 (* (/ z vpi)
  	    (+ (/ (- 1 c)
  		  z)
  	       (* c
  		  (s big-n z v))
  	       (- (s big-n z (- v)))))))))
  	   
    
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  (defun paris-series (v z n)
    (labels ((pochhammer (a k)
  	     (/ (gamma (+ a k))
  		(gamma a)))
  	   (a (v k)
  	     (* (/ (pochhammer (+ 1/2 v) k)
  		   (gamma (float (1+ k) z)))
  		(pochhammer (- 1/2 v) k))))
      (* (loop for k from 0 below n
  	     sum (* (/ (a v k)
  		       (expt (* 2 z) k))
  		    (/ (cf-incomplete-gamma (+ k v 1/2) (* 2 z))
  		       (gamma (+ k v 1/2)))))
         (/ (exp z)
  	  (sqrt (* 2 (float-pi z) z))))))