;;;; -*- 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) (named-readtables:in-readtable oct-readtable)) ;; Theta functions ;; ;; theta[1](z,q) = 2*sum((-1)^n*q^((n+1/2)^2)*sin((2*n+1)*z), n, 0, inf) ;; ;; theta[2](z,q) = 2*sum(q^((n+1/2)^2)*cos((2*n+1)*z), n, 0, inf) ;; ;; theta[3](z,q) = 1+2*sum(q^(n*n)*cos(2*n*z), n, 1, inf) ;; ;; theta[4](z,q) = 1+2*sum((-1)^n*q^(n*n)*cos(2*n*z), n, 1, inf) ;; ;; where q is the nome, related to parameter tau by q = ;; exp(%i*%pi*tau), or %pi*tau = log(q)/%i. ;; ;; In all cases |q| < 1. ;; The algorithms for computing the theta functions were given to me ;; by Richard Gosper (yes, that Richard Gosper). These came from ;; package for maxima for the theta functions. ;; e1 M[1,3] + e2 M[2,3] + e3, where M = prod(mat(a11 ... a23 0 0 1)) ;; where fun(k,matfn) supplies the upper six a[ij](k) to matfn. ;; ;; This is clearer if you look at the formulas below for the theta functions. (defun 3by3rec (e1 e2 e3 fun) (do ((k 0 (+ k 1))) ((= e3 (funcall fun k #'(lambda (a11 a12 a13 a21 a22 a23) ;&opt (a31 0) (a32 0) (a33 1) (psetf e1 (+ (* a11 e1) (* a21 e2)) e2 (+ (* a12 e1) (* a22 e2)) e3 (+ (* a13 e1) (* a23 e2) e3)) (+ e3 (abs e1) (abs e2))))) e3))) ;; inf [ 2 n 1/4 ] ;; /===\ [ - 2 q cos(2 z) 1 2 q ] ;; | | [ ] ;;[sin(z), sin(z), 0] | | [ 4 n - 2 ] = [0, 0, theta (z, q)] ;; | | [ - q 0 0 ] 1 ;; n = 1 [ ] ;; [ 0 0 1 ] (defun elliptic-theta-1 (z q) "Elliptic theta function 1 theta1(z, q) = 2*q^(1/4)*sum((-1)^n*q^(n*(n+1))*sin((2*n+1)*z), n, 0, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision)) (s (sin z)) (q^2 (* q q)) (q^4 (* q^2 q^2)) (-q^4n-2 (/ -1 q^2)) (-2q^2ncos (* -2 (cos (* 2 z)))) (2q^1/4 (* 2 (sqrt (sqrt q))))) (3by3rec s s 0 #'(lambda (k matfun) (declare (ignore k)) (funcall matfun (setf -2q^2ncos (* q^2 -2q^2ncos)) 1 2q^1/4 (setf -q^4n-2 (* q^4 -q^4n-2)) 0 0))))) ;; inf [ 2 k + 1 ] ;; /===\ [ 2 q cos(2 z) 1 2 ] ;; | | [ ] ;;[q cos(2 z), 1, 1] | | [ 4 k ] = [0, 0, theta (z)] ;; | | [ - q 0 0 ] 3 ;; k = 1 [ ] ;; [ 0 0 1 ] (defun elliptic-theta-3 (z q) "Elliptic theta function 3 theta3(z, q) = 1 + 2 * sum(q^(n^2)*cos(2*n*z), n, 1, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision)) (q^2 (* q q)) (q^2k 1.0) (cos (cos (* 2 z)))) (3by3rec (* q cos) 1 1 #'(lambda (k matfun) (declare (ignore k)) (funcall matfun (* 2 (* (setf q^2k (* q^2 q^2k)) q cos)) 1 2 (- (* q^2k q^2k)) 0 0))))) ;; theta[2](z,q) = theta[1](z+%pi/2, q) (defun elliptic-theta-2 (z q) "Elliptic theta function 2 theta2(z, q) = 2*q^(1/4)*sum(q^(n*(n+1))*cos((2*n+1)*z), n, 0, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision))) (elliptic-theta-1 (+ z (/ (float-pi z) 2)) q))) ;; theta[4](z,q) = theta[3](z+%pi/2,q) (defun elliptic-theta-4 (z q) "Elliptic theta function 4 theta4(z, q) = 1 + 2*sum((-1)^n*q^(n^2)*cos(2*n*z), n, 1, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision))) (elliptic-theta-3 (+ z (/ (float-pi z) 2)) q))) (defun elliptic-theta (n z q) "Elliptic Theta function n where n = 1, 2, 3, or 4." (ecase n (1 (elliptic-theta-1 z q)) (2 (elliptic-theta-2 z q)) (3 (elliptic-theta-3 z q)) (4 (elliptic-theta-4 z q)))) ;; The nome, q, is given by q = exp(-%pi*K'/K) where K and %i*K' are ;; the quarter periods. (defun elliptic-nome (m) "Compute the elliptic nome, q, from the parameter m" (exp (- (/ (* (float-pi m) (elliptic-k (- 1 m))) (elliptic-k m)))))