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[Maxima-commits] CVS: maxima/src ellipt.lisp,1.56,1.57
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From: Raymond T. <rt...@us...> - 2009-02-28 17:18:55
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Update of /cvsroot/maxima/maxima/src In directory 23jxhf1.ch3.sourceforge.com:/tmp/cvs-serv18659 Modified Files: ellipt.lisp Log Message: Teach maxima how to compute jacobi_<foo>(inverse_jacobi_<bar>). Index: ellipt.lisp =================================================================== RCS file: /cvsroot/maxima/maxima/src/ellipt.lisp,v retrieving revision 1.56 retrieving revision 1.57 diff -u -d -r1.56 -r1.57 --- ellipt.lisp 27 Feb 2009 17:55:04 -0000 1.56 +++ ellipt.lisp 28 Feb 2009 17:18:46 -0000 1.57 @@ -692,10 +692,63 @@ (neg (cons-exp '%jacobi_sn (neg u) m))) ((and $triginverses (listp u) - (eq (caar u) '%inverse_jacobi_sn) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) (alike1 (third u) m)) - ;; jacobi_sn(inverse_jacobi_sn(u,m), m) = u - (second u)) + (let ((inv-arg (second u))) + (ecase (caar u) + (%inverse_jacobi_sn + ;; jacobi_sn(inverse_jacobi_sn(u,m), m) = u + inv-arg) + (%inverse_jacobi_ns + ;; inverse_jacobi_ns(u,m) = inverse_jacobi_sn(1/u,m) + (div 1 inv-arg)) + (%inverse_jacobi_cn + ;; sn(x)^2 + cn(x)^2 = 1 so sn(x) = sqrt(1-cn(x)^2) + (power (sub 1 (mul inv-arg inv-arg)) 1//2)) + (%inverse_jacobi_nc + ;; inverse_jacobi_nc(u) = inverse_jacobi_cn(1/u) + ($jacobi_sn ($inverse_jacobi_cn (div 1 inv-arg) m) + m)) + (%inverse_jacobi_dn + ;; dn(x)^2 + m*sn(x)^2 = 1 so + ;; sn(x) = 1/sqrt(m)*sqrt(1-dn(x)^2) + (mul (div 1 (power m 1//2)) + (power (sub 1 (mul inv-arg inv-arg)) 1//2))) + (%inverse_jacobi_nd + ;; inverse_jacobi_nd(u) = inverse_jacobi_dn(1/u) + ($jacobi_sn ($inverse_jacobi_dn (div 1 inv-arg) m) + m)) + (%inverse_jacobi_sc + ;; See below for inverse_jacobi_sc. + (div inv-arg (power (add 1 (mul inv-arg inv-arg)) 1//2))) + (%inverse_jacobi_cs + ;; inverse_jacobi_cs(u) = inverse_jacobi_sc(1/u) + ($jacobi_sn ($inverse_jacobi_sc (div 1 inv-arg) m) + m)) + (%inverse_jacobi_sd + ;; See below for inverse_jacobi_sd + (div inv-arg (power (add 1 (mul m (mul inv-arg inv-arg))) 1//2))) + (%inverse_jacobi_ds + ;; inverse_jacobi_ds(u) = inverse_jacobi_sd(1/u) + ($jacobi_sn ($inverse_jacobi_sd (div 1 inv-arg) m) + m)) + (%inverse_jacobi_cd + ;; See below + (div (power (sub 1 (mul inv-arg inv-arg)) 1//2) + (power (sub 1 (mul m (mul inv-arg inv-arg))) 1//2))) + (%inverse_jacobi_dc + ($jacobi_dn ($inverse_jacobi_cd (div 1 inv-arg) m) m))))) ;; A&S 16.20.1 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) (mul '$%i @@ -785,10 +838,26 @@ (cons-exp '%jacobi_cn (neg u) m)) ((and $triginverses (listp u) - (eq (caar u) '%inverse_jacobi_cn) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) (alike1 (third u) m)) - ;; jacobi_cn(inverse_jacobi_cn(u,m), m) = u - (second u)) + (cond ((eq (caar u) '%inverse_jacobi_cn) + (second u)) + (t + ;; I'm lazy. Use cn(x) = sqrt(1-sn(x)^2). Hope + ;; this is right. + (power (sub 1 (power ($jacobi_sn u (third u)) 2)) + 1//2)))) ;; A&S 16.20.2 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) (cons-exp '%jacobi_nc (coeff u '$%i 1) (add 1 (neg m)))) @@ -882,10 +951,28 @@ (cons-exp '%jacobi_dn (neg u) m)) ((and $triginverses (listp u) - (eq (caar u) '%inverse_jacobi_dn) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) (alike1 (third u) m)) - ;; jacobi_dn(inverse_jacobi_dn(u,m), m) = u - (second u)) + (cond ((eq (caar u) '%inverse_jacobi_dn) + ;; jacobi_dn(inverse_jacobi_dn(u,m), m) = u + (second u)) + (t + ;; Express in terms of sn: + ;; dn(x) = sqrt(1-m*sn(x)^2) + (power (sub 1 (mul m + (power ($jacobi_sn u m) 2))) + 1//2)))) ((zerop1 ($ratsimp (sub u (power (sub 1 m) 1//2)))) ;; A&S 16.5.3 ;; dn(sqrt(1-m),m) = K(m) @@ -2686,6 +2773,27 @@ ((and $trigsign (mminusp* u)) ;; ns is odd (neg (cons-exp '%jacobi_ns (neg u) m))) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_ns) + (second u)) + (t + ;; Express in terms of sn: + ;; ns(x) = 1/sn(x) + (div 1 ($jacobi_sn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; ns(i*u) = 1/sn(i*u) = -i/sc(u,m1) = -i*cs(u,m1) @@ -2784,7 +2892,28 @@ ((and $trigsign (mminusp* u)) ;; nc is even (cons-exp '%jacobi_nc (neg u) m)) - ;; A&S 16.20 (Jacobi's Imaginary transformation) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_nc) + (second u)) + (t + ;; Express in terms of cn: + ;; nc(x) = 1/cn(x) + (div 1 ($jacobi_cn u m))))) + ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; nc(i*u) = 1/cn(i*u) = 1/nc(u,1-m) = cn(u,1-m) (cons-exp '%jacobi_cn (coeff u '$%i 1) (add 1 (neg m)))) @@ -2893,7 +3022,28 @@ ((and $trigsign (mminusp* u)) ;; nd is even (cons-exp '%jacobi_nd (neg u) m)) - ;; A&S 16.20 (Jacobi's Imaginary transformation) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_nd) + (second u)) + (t + ;; Express in terms of dn: + ;; nd(x) = 1/dn(x) + (div 1 ($jacobi_dn u m))))) + ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; nd(i*u) = 1/dn(i*u) = 1/dc(u,1-m) = cd(u,1-m) (cons-exp '%jacobi_cd (coeff u '$%i 1) (add 1 (neg m)))) @@ -2999,6 +3149,28 @@ ((and $trigsign (mminusp* u)) ;; sc is odd (neg (cons-exp '%jacobi_sc (neg u) m))) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_sc) + (second u)) + (t + ;; Express in terms of sn and cn + ;; sc(x) = sn(x)/cn(x) + (div ($jacobi_sn u m) + ($jacobi_cn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; sc(i*u) = sn(i*u)/cn(i*u) = i*sc(u,m1)/nc(u,m1) = i*sn(u,m1) @@ -3105,6 +3277,27 @@ ((and $trigsign (mminusp* u)) ;; sd is odd (neg (cons-exp '%jacobi_sd (neg u) m))) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_sd) + (second u)) + (t + ;; Express in terms of sn and dn + (div ($jacobi_sn u m) + ($jacobi_dn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; sd(i*u) = sn(i*u)/dn(i*u) = i*sc(u,m1)/dc(u,m1) = i*sd(u,m1) @@ -3241,6 +3434,27 @@ ((and $trigsign (mminusp* u)) ;; cs is odd (neg (cons-exp '%jacobi_cs (neg u) m))) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_cs) + (second u)) + (t + ;; Express in terms of cn an sn + (div ($jacobi_cn u m) + ($jacobi_sn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; cs(i*u) = cn(i*u)/sn(i*u) = -i*nc(u,m1)/sc(u,m1) = -i*ns(u,m1) @@ -3353,6 +3567,27 @@ ((and $trigsign (mminusp* u)) ;; cd is even (cons-exp '%jacobi_cd (neg u) m)) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_cd) + (second u)) + (t + ;; Express in terms of cn and dn + (div ($jacobi_cn u m) + ($jacobi_dn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; cd(i*u) = cn(i*u)/dn(i*u) = nc(u,m1)/dc(u,m1) = nd(u,m1) @@ -3477,6 +3712,27 @@ (dbz-err1 'jacobi_ds)) ((and $trigsign (mminusp* u)) (neg (cons-exp '%jacobi_ds (neg u) m))) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_ds) + (second u)) + (t + ;; Express in terms of dn and sn + (div ($jacobi_dn u m) + ($jacobi_sn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; ds(i*u) = dn(i*u)/sn(i*u) = -i*dc(u,m1)/sc(u,m1) = -i*ds(u,m1) @@ -3614,6 +3870,27 @@ 1) ((and $trigsign (mminusp* u)) (cons-exp '%jacobi_dc (neg u) m)) + ((and $triginverses + (listp u) + (member (caar u) '(%inverse_jacobi_sn + %inverse_jacobi_ns + %inverse_jacobi_cn + %inverse_jacobi_nc + %inverse_jacobi_dn + %inverse_jacobi_nd + %inverse_jacobi_sc + %inverse_jacobi_cs + %inverse_jacobi_sd + %inverse_jacobi_ds + %inverse_jacobi_cd + %inverse_jacobi_dc)) + (alike1 (third u) m)) + (cond ((eq (caar u) '%inverse_jacobi_dc) + (second u)) + (t + ;; Express in terms of dn and cn + (div ($jacobi_dn u m) + ($jacobi_cn u m))))) ;; A&S 16.20 (Jacobi's Imaginary transformation) ((and $%iargs (multiplep u '$%i)) ;; dc(i*u) = dn(i*u)/cn(i*u) = dc(u,m1)/nc(u,m1) = dn(u,m1) |