Lines Matching +defs:math +defs:sub
46 (setq var (math-read-expr var))
64 (let ((var (math-read-expr var)))
107 (setq var (if (stringp varname) (math-read-expr varname) varname))
117 (setq low (if (stringp lowname) (math-read-expr lowname) lowname))
127 (setq high (if (stringp highname) (math-read-expr highname)
141 (setq step (math-read-expr stepname))
166 (math-read-expr (concat "[" var "]"))
167 (math-read-expr var))))
183 (math-read-expr (concat "[" var "]"))
184 (math-read-expr var))))
194 (let ((var (math-read-expr var)))
203 ;; The following are global variables used by math-derivative and some
205 (defvar math-deriv-var)
206 (defvar math-deriv-total)
207 (defvar math-deriv-symb)
208 (defvar math-decls-cache)
209 (defvar math-decls-all)
211 (defun math-derivative (expr)
212 (cond ((equal expr math-deriv-var)
217 (or (not math-deriv-total)
218 (math-const-var expr)
220 (math-setup-declarations)
222 math-decls-cache)
223 math-decls-all)))))))
226 (math-add (math-derivative (nth 1 expr))
227 (math-derivative (nth 2 expr))))
229 (math-sub (math-derivative (nth 1 expr))
230 (math-derivative (nth 2 expr))))
234 (math-derivative (nth 1 expr))
235 (math-derivative (nth 2 expr))))
237 (math-neg (math-derivative (nth 1 expr))))
239 (math-add (math-mul (nth 2 expr)
240 (math-derivative (nth 1 expr)))
241 (math-mul (nth 1 expr)
242 (math-derivative (nth 2 expr)))))
244 (math-sub (math-div (math-derivative (nth 1 expr))
246 (math-div (math-mul (nth 1 expr)
247 (math-derivative (nth 2 expr)))
248 (math-sqr (nth 2 expr)))))
250 (let ((du (math-derivative (nth 1 expr)))
251 (dv (math-derivative (nth 2 expr))))
253 (setq du (math-mul (nth 2 expr)
254 (math-mul (math-normalize
257 (math-add (nth 2 expr) -1)))
260 (setq dv (math-mul (math-normalize
262 (math-mul expr dv))))
263 (math-add du dv)))
265 (math-derivative (nth 1 expr))) ; a reasonable definition
267 (math-map-vec 'math-derivative expr))
270 (list (car expr) (math-derivative (nth 1 expr))))
273 (let ((d (math-derivative (nth 1 expr))))
274 (if (math-numberp d)
279 (let ((handler (get (car expr) 'math-derivative)))
281 (let ((deriv (math-derivative (nth 1 expr))))
284 (math-mul (funcall handler (nth 1 expr))
286 (let ((handler (get (car expr) 'math-derivative-n)))
289 (and (not (eq math-deriv-symb 'pre-expand))
290 (let ((exp (math-expand-formula expr)))
292 (or (let ((math-deriv-symb 'pre-expand))
293 (catch 'math-deriv (math-derivative expr)))
294 (math-derivative exp)))))
298 (if math-deriv-symb
299 (throw 'math-deriv nil)
300 (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv)
302 math-deriv-var))
308 (or (Math-zerop (setq derv (math-derivative (car arg))))
315 'math-derivative-1)
317 'math-derivative-2)
319 'math-derivative-3)
321 'math-derivative-4)
323 'math-derivative-5))))
325 (math-add
327 (math-mul
332 (if (and math-deriv-symb
335 (throw 'math-deriv nil)
340 (defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb)
341 (let* ((math-deriv-total nil)
342 (res (catch 'math-deriv (math-derivative expr))))
345 (setq res (math-normalize res)))
348 (math-expr-subst res math-deriv-var deriv-value)
351 (defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb)
352 (math-setup-declarations)
353 (let* ((math-deriv-total t)
354 (res (catch 'math-deriv (math-derivative expr))))
357 (setq res (math-normalize res)))
360 (math-expr-subst res math-deriv-var deriv-value)
363 (put 'calcFunc-inv\' 'math-derivative-1
364 (function (lambda (u) (math-neg (math-div 1 (math-sqr u))))))
366 (put 'calcFunc-sqrt\' 'math-derivative-1
367 (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u))))))
369 (put 'calcFunc-deg\' 'math-derivative-1
370 (function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))
372 (put 'calcFunc-rad\' 'math-derivative-1
373 (function (lambda (u) (math-pi-over-180))))
375 (put 'calcFunc-ln\' 'math-derivative-1
376 (function (lambda (u) (math-div 1 u))))
378 (put 'calcFunc-log10\' 'math-derivative-1
380 (math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
383 (put 'calcFunc-lnp1\' 'math-derivative-1
384 (function (lambda (u) (math-div 1 (math-add u 1)))))
386 (put 'calcFunc-log\' 'math-derivative-2
389 (let ((lnv (math-normalize
391 (math-div 1 (math-mul lnv x)))))))
393 (put 'calcFunc-log\'2 'math-derivative-2
396 (math-neg (math-div (list 'calcFunc-log x b)
397 (math-mul lnv b)))))))
399 (put 'calcFunc-exp\' 'math-derivative-1
400 (function (lambda (u) (math-normalize (list 'calcFunc-exp u)))))
402 (put 'calcFunc-expm1\' 'math-derivative-1
403 (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u)))))
405 (put 'calcFunc-sin\' 'math-derivative-1
406 (function (lambda (u) (math-to-radians-2 (math-normalize
409 (put 'calcFunc-cos\' 'math-derivative-1
410 (function (lambda (u) (math-neg (math-to-radians-2
411 (math-normalize
414 (put 'calcFunc-tan\' 'math-derivative-1
415 (function (lambda (u) (math-to-radians-2
416 (math-sqr
417 (math-normalize
420 (put 'calcFunc-sec\' 'math-derivative-1
421 (function (lambda (u) (math-to-radians-2
422 (math-mul
423 (math-normalize
425 (math-normalize
428 (put 'calcFunc-csc\' 'math-derivative-1
429 (function (lambda (u) (math-neg
430 (math-to-radians-2
431 (math-mul
432 (math-normalize
434 (math-normalize
437 (put 'calcFunc-cot\' 'math-derivative-1
438 (function (lambda (u) (math-neg
439 (math-to-radians-2
440 (math-sqr
441 (math-normalize
444 (put 'calcFunc-arcsin\' 'math-derivative-1
446 (math-from-radians-2
447 (math-div 1 (math-normalize
449 (math-sub 1 (math-sqr u)))))))))
451 (put 'calcFunc-arccos\' 'math-derivative-1
453 (math-from-radians-2
454 (math-div -1 (math-normalize
456 (math-sub 1 (math-sqr u)))))))))
458 (put 'calcFunc-arctan\' 'math-derivative-1
459 (function (lambda (u) (math-from-radians-2
460 (math-div 1 (math-add 1 (math-sqr u)))))))
462 (put 'calcFunc-sinh\' 'math-derivative-1
463 (function (lambda (u) (math-normalize (list 'calcFunc-cosh u)))))
465 (put 'calcFunc-cosh\' 'math-derivative-1
466 (function (lambda (u) (math-normalize (list 'calcFunc-sinh u)))))
468 (put 'calcFunc-tanh\' 'math-derivative-1
469 (function (lambda (u) (math-sqr
470 (math-normalize
473 (put 'calcFunc-sech\' 'math-derivative-1
474 (function (lambda (u) (math-neg
475 (math-mul
476 (math-normalize (list 'calcFunc-sech u))
477 (math-normalize (list 'calcFunc-tanh u)))))))
479 (put 'calcFunc-csch\' 'math-derivative-1
480 (function (lambda (u) (math-neg
481 (math-mul
482 (math-normalize (list 'calcFunc-csch u))
483 (math-normalize (list 'calcFunc-coth u)))))))
485 (put 'calcFunc-coth\' 'math-derivative-1
486 (function (lambda (u) (math-neg
487 (math-sqr
488 (math-normalize
491 (put 'calcFunc-arcsinh\' 'math-derivative-1
493 (math-div 1 (math-normalize
495 (math-add (math-sqr u) 1)))))))
497 (put 'calcFunc-arccosh\' 'math-derivative-1
499 (math-div 1 (math-normalize
501 (math-add (math-sqr u) -1)))))))
503 (put 'calcFunc-arctanh\' 'math-derivative-1
504 (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u))))))
506 (put 'calcFunc-bern\'2 'math-derivative-2
508 (math-mul n (list 'calcFunc-bern (math-add n -1) x)))))
510 (put 'calcFunc-euler\'2 'math-derivative-2
512 (math-mul n (list 'calcFunc-euler (math-add n -1) x)))))
514 (put 'calcFunc-gammag\'2 'math-derivative-2
515 (function (lambda (a x) (math-deriv-gamma a x 1))))
517 (put 'calcFunc-gammaG\'2 'math-derivative-2
518 (function (lambda (a x) (math-deriv-gamma a x -1))))
520 (put 'calcFunc-gammaP\'2 'math-derivative-2
521 (function (lambda (a x) (math-deriv-gamma a x
522 (math-div
523 1 (math-normalize
527 (put 'calcFunc-gammaQ\'2 'math-derivative-2
528 (function (lambda (a x) (math-deriv-gamma a x
529 (math-div
530 -1 (math-normalize
534 (defun math-deriv-gamma (a x scale)
535 (math-mul scale
536 (math-mul (math-pow x (math-add a -1))
537 (list 'calcFunc-exp (math-neg x)))))
539 (put 'calcFunc-betaB\' 'math-derivative-3
540 (function (lambda (x a b) (math-deriv-beta x a b 1))))
542 (put 'calcFunc-betaI\' 'math-derivative-3
543 (function (lambda (x a b) (math-deriv-beta x a b
544 (math-div
548 (defun math-deriv-beta (x a b scale)
549 (math-mul (math-mul (math-pow x (math-add a -1))
550 (math-pow (math-sub 1 x) (math-add b -1)))
553 (put 'calcFunc-erf\' 'math-derivative-1
554 (function (lambda (x) (math-div 2
555 (math-mul (list 'calcFunc-exp
556 (math-sqr x))
560 (math-sqrt-pi)))))))
562 (put 'calcFunc-erfc\' 'math-derivative-1
563 (function (lambda (x) (math-div -2
564 (math-mul (list 'calcFunc-exp
565 (math-sqr x))
569 (math-sqrt-pi)))))))
571 (put 'calcFunc-besJ\'2 'math-derivative-2
572 (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ
573 (math-add v -1)
576 (math-add v 1)
580 (put 'calcFunc-besY\'2 'math-derivative-2
581 (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY
582 (math-add v -1)
585 (math-add v 1)
589 (put 'calcFunc-sum 'math-derivative-n
592 (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
593 (throw 'math-deriv nil)
595 (cons (math-derivative (nth 1 expr))
598 (put 'calcFunc-prod 'math-derivative-n
601 (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
602 (throw 'math-deriv nil)
603 (math-mul expr
605 (cons (math-div (math-derivative (nth 1 expr))
609 (put 'calcFunc-integ 'math-derivative-n
613 (if (equal (nth 2 expr) math-deriv-var)
615 (math-normalize
617 (math-derivative (nth 1 expr))
620 (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr)
622 (upper (math-expr-subst (nth 1 expr) (nth 2 expr)
624 (math-add (math-sub (math-mul upper
625 (math-derivative (nth 4 expr)))
626 (math-mul lower
627 (math-derivative (nth 3 expr))))
628 (if (equal (nth 2 expr) math-deriv-var)
630 (math-normalize
632 (math-derivative (nth 1 expr)) (nth 2 expr)
635 (put 'calcFunc-if 'math-derivative-n
640 (math-derivative (nth 2 expr))
641 (math-derivative (nth 3 expr)))))))
643 (put 'calcFunc-subscr 'math-derivative-n
648 (math-derivative (nth 2 expr)))))))
651 (defvar math-integ-var '(var X ---))
652 (defvar math-integ-var-2 '(var Y ---))
653 (defvar math-integ-vars (list 'f math-integ-var math-integ-var-2))
654 (defvar math-integ-var-list (list math-integ-var))
655 (defvar math-integ-var-list-list (list math-integ-var-list))
657 ;; math-integ-depth is a local variable for math-try-integral, but is used
658 ;; by math-integral and math-tracing-integral
659 ;; which are called (directly or indirectly) by math-try-integral.
660 (defvar math-integ-depth)
661 ;; math-integ-level is a local variable for math-try-integral, but is used
662 ;; by math-integral, math-do-integral, math-tracing-integral,
663 ;; math-sub-integration, math-integrate-by-parts and
664 ;; math-integrate-by-substitution, which are called (directly or
665 ;; indirectly) by math-try-integral.
666 (defvar math-integ-level)
667 ;; math-integral-limit is a local variable for calcFunc-integ, but is
668 ;; used by math-tracing-integral, math-sub-integration and
669 ;; math-try-integration.
670 (defvar math-integral-limit)
672 (defmacro math-tracing-integral (&rest parts)
680 '(insert (make-string (- math-integral-limit
681 math-integ-level) 32)
682 (format "%2d " math-integ-depth)
683 (make-string math-integ-level 32)))
696 ;;; ( A [B] ) Same result as for math-cur-record = B.
698 ;; math-cur-record is a local variable for math-try-integral, but is used
699 ;; by math-integral, math-replace-integral-parts and math-integrate-by-parts
700 ;; which are called (directly or indirectly) by math-try-integral, as well as
702 (defvar math-cur-record)
703 ;; math-enable-subst and math-any-substs are local variables for
704 ;; calcFunc-integ, but are used by math-integral and math-try-integral.
705 (defvar math-enable-subst)
706 (defvar math-any-substs)
708 ;; math-integ-msg is a local variable for math-try-integral, but is
709 ;; used (both locally and non-locally) by math-integral.
710 (defvar math-integ-msg)
712 (defvar math-integral-cache nil)
713 (defvar math-integral-cache-state nil)
715 (defun math-integral (expr &optional simplify same-as-above)
716 (let* ((simp math-cur-record)
717 (math-cur-record (assoc expr math-integral-cache))
718 (math-integ-depth (1+ math-integ-depth))
720 (math-tracing-integral "Integrating "
721 (math-format-value expr 1000)
723 (and math-cur-record
725 (math-tracing-integral "Found "
726 (math-format-value (nth 1 math-cur-record) 1000))
727 (and (consp (nth 1 math-cur-record))
728 (math-replace-integral-parts math-cur-record))
729 (math-tracing-integral " => "
730 (math-format-value (nth 1 math-cur-record) 1000)
732 (or (and math-cur-record
733 (not (eq (nth 1 math-cur-record) 'cancelled))
734 (or (not (integerp (nth 1 math-cur-record)))
735 (>= (nth 1 math-cur-record) math-integ-level)))
736 (and (math-integral-contains-parts expr)
742 (let (math-integ-msg)
746 (setq math-integ-msg (format
748 (math-format-flat-expr expr 0)))
749 (message math-integ-msg)))
750 (if math-cur-record
751 (setcar (cdr math-cur-record)
753 (setq math-cur-record
755 math-integral-cache (cons math-cur-record
756 math-integral-cache)))
759 (math-tracing-integral "Simplifying...")
760 (setq simp (math-simplify expr))
763 (math-tracing-integral " no change\n")
764 (math-do-integral expr))
765 (math-tracing-integral " simplified\n")
766 (math-integral simp 'no t))))
767 (or (setq val (math-do-integral expr))
769 (let ((simp (math-simplify expr)))
772 (math-tracing-integral "Trying again after "
774 (setq val (math-integral simp 'no t))))))))
776 (message math-integ-msg)))
777 (setcar (cdr math-cur-record) (or val
778 (if (or math-enable-subst
779 (not math-any-substs))
780 math-integ-level
782 (setq val math-cur-record)
791 (math-tracing-integral "Integral of "
792 (math-format-value expr 1000)
794 (math-format-value val 1000)
798 (defun math-integral-contains-parts (expr)
804 (not (math-integral-contains-parts (car expr)))))
807 (defun math-replace-integral-parts (expr)
817 (math-replace-integral-parts (cons 'foo expr)))
818 (setcar (cdr math-cur-record) 'cancelled)))
819 (math-replace-integral-parts (car expr)))))))
821 (defvar math-linear-subst-tried t
824 ;; The variable math-has-rules is a local variable for math-try-integral,
825 ;; but is used by math-do-integral, which is called (non-directly) by
826 ;; math-try-integral.
827 (defvar math-has-rules)
829 ;; math-old-integ is a local variable for math-do-integral, but is
830 ;; used by math-sub-integration.
831 (defvar math-old-integ)
833 ;; The variables math-t1, math-t2 and math-t3 are local to
834 ;; math-do-integral, math-try-solve-for and math-decompose-poly, but
836 ;; math-do-integral calls math-do-integral-methods;
837 ;; math-try-solve-for calls math-try-solve-prod,
838 ;; math-solve-find-root-term and math-solve-find-root-in-prod;
839 ;; math-decompose-poly calls math-solve-poly-funny-powers and
840 ;; math-solve-crunch-poly.
841 (defvar math-t1)
842 (defvar math-t2)
843 (defvar math-t3)
845 (defun math-do-integral (expr)
846 (let ((math-linear-subst-tried nil)
847 math-t1 math-t2)
848 (or (cond ((not (math-expr-contains expr math-integ-var))
849 (math-mul expr math-integ-var))
850 ((equal expr math-integ-var)
851 (math-div (math-sqr expr) 2))
853 (and (setq math-t1 (math-integral (nth 1 expr)))
854 (setq math-t2 (math-integral (nth 2 expr)))
855 (math-add math-t1 math-t2)))
857 (and (setq math-t1 (math-integral (nth 1 expr)))
858 (setq math-t2 (math-integral (nth 2 expr)))
859 (math-sub math-t1 math-t2)))
861 (and (setq math-t1 (math-integral (nth 1 expr)))
862 (math-neg math-t1)))
864 (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
865 (and (setq math-t1 (math-integral (nth 2 expr)))
866 (math-mul (nth 1 expr) math-t1)))
867 ((not (math-expr-contains (nth 2 expr) math-integ-var))
868 (and (setq math-t1 (math-integral (nth 1 expr)))
869 (math-mul math-t1 (nth 2 expr))))
871 (math-integral (list (car (nth 1 expr))
872 (math-mul (nth 1 (nth 1 expr))
874 (math-mul (nth 2 (nth 1 expr))
878 (math-integral (list (car (nth 2 expr))
879 (math-mul (nth 1 (nth 2 expr))
881 (math-mul (nth 2 (nth 2 expr))
885 (cond ((and (not (math-expr-contains (nth 1 expr)
886 math-integ-var))
887 (not (math-equal-int (nth 1 expr) 1)))
888 (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr))))
889 (math-mul (nth 1 expr) math-t1)))
890 ((not (math-expr-contains (nth 2 expr) math-integ-var))
891 (and (setq math-t1 (math-integral (nth 1 expr)))
892 (math-div math-t1 (nth 2 expr))))
894 (not (math-expr-contains (nth 1 (nth 1 expr))
895 math-integ-var)))
896 (and (setq math-t1 (math-integral
897 (math-div (nth 2 (nth 1 expr))
899 (math-mul math-t1 (nth 1 (nth 1 expr)))))
901 (not (math-expr-contains (nth 2 (nth 1 expr))
902 math-integ-var)))
903 (and (setq math-t1 (math-integral
904 (math-div (nth 1 (nth 1 expr))
906 (math-mul math-t1 (nth 2 (nth 1 expr)))))
908 (not (math-expr-contains (nth 1 (nth 2 expr))
909 math-integ-var)))
910 (and (setq math-t1 (math-integral
911 (math-div (nth 1 expr)
913 (math-div math-t1 (nth 1 (nth 2 expr)))))
915 (not (math-expr-contains (nth 2 (nth 2 expr))
916 math-integ-var)))
917 (and (setq math-t1 (math-integral
918 (math-div (nth 1 expr)
920 (math-div math-t1 (nth 2 (nth 2 expr)))))
922 (math-integral
923 (math-mul (nth 1 expr)
925 (math-neg (nth 1 (nth 2 expr)))))))))
927 (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
928 (or (and (setq math-t1 (math-is-polynomial (nth 2 expr)
929 math-integ-var 1))
930 (math-div expr
931 (math-mul (nth 1 math-t1)
932 (math-normalize
935 (math-integral
937 (math-mul (nth 2 expr)
938 (math-normalize
942 ((not (math-expr-contains (nth 2 expr) math-integ-var))
944 (math-integral
945 (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr))))
947 (or (and (setq math-t1 (math-is-polynomial (nth 1 expr)
948 math-integ-var
950 (setq math-t2 (math-add (nth 2 expr) 1))
951 (math-div (math-pow (nth 1 expr) math-t2)
952 (math-mul math-t2 (nth 1 math-t1))))
954 (math-integral
955 (math-div 1
956 (math-pow (nth 1 expr)
957 (math-neg
963 (and (setq math-t1 (math-is-polynomial expr math-integ-var 20))
966 (while math-t1
967 (if (setq accum (math-add accum
968 (math-div (math-mul (car math-t1)
969 (math-pow
970 math-integ-var
973 math-t1 (cdr math-t1))
980 (setq math-t1 (get (car expr) 'math-integral))
982 (while (and math-t1
983 (not (setq math-t2 (funcall (car math-t1)
985 (setq math-t1 (cdr math-t1)))
986 (and math-t2 (math-normalize math-t2)))))
989 (setq math-t1 (get (car expr) 'math-integral-2))
991 (while (and math-t1
992 (not (setq math-t2 (funcall (car math-t1)
995 (setq math-t1 (cdr math-t1)))
996 (and math-t2 (math-normalize math-t2))))))
999 (and (math-ratpoly-p expr math-integ-var)
1000 (setq math-t1 (calcFunc-apart expr math-integ-var))
1001 (not (equal math-t1 expr))
1002 (math-integral math-t1))
1005 (and math-has-rules
1006 (let ((math-old-integ (symbol-function 'calcFunc-integ))
1007 (input (list 'calcFunc-integtry expr math-integ-var))
1011 (fset 'calcFunc-integ 'math-sub-integration)
1012 (setq res (math-rewrite input
1015 (fset 'calcFunc-integ math-old-integ)
1017 (if (setq part (math-expr-calls
1023 (math-expr-subst
1025 math-integ-var)))
1027 (math-integrate-by-substitution
1031 math-integ-var))
1033 (if (not (math-expr-calls res
1037 (fset 'calcFunc-integ math-old-integ))))
1044 (if (equal (car p) math-integ-var)
1046 (if (math-expr-contains (car p) math-integ-var)
1060 (and (> math-integ-level 0)
1061 (math-do-integral-methods expr))
1064 (let ((res (math-expand-formula expr)))
1066 (math-integral res))))))
1068 (defun math-sub-integration (expr &rest rest)
1070 (and (< math-integ-level math-integral-limit)
1071 (eq (car rest) math-integ-var)))
1072 (math-integral expr)
1073 (let ((res (apply math-old-integ expr rest)))
1074 (and (or (= math-integ-level math-integral-limit)
1075 (not (math-expr-calls res 'calcFunc-integ)))
1079 ;; math-so-far is a local variable for math-do-integral-methods, but
1080 ;; is used by math-integ-try-linear-substitutions and
1081 ;; math-integ-try-substitutions.
1082 (defvar math-so-far)
1084 ;; math-integ-expr is a local variable for math-do-integral-methods,
1085 ;; but is used by math-integ-try-linear-substitutions and
1086 ;; math-integ-try-substitutions.
1087 (defvar math-integ-expr)
1089 (defun math-do-integral-methods (math-integ-expr)
1090 (let ((math-so-far math-integ-var-list-list)
1093 ;; Integration by substitution, for various likely sub-expressions.
1094 ;; (In first pass, we look only for sub-exprs that are linear in X.)
1095 (or (math-integ-try-linear-substitutions math-integ-expr)
1096 (math-integ-try-substitutions math-integ-expr)
1099 (and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr))))
1107 (equal (nth 1 (car p)) math-integ-var))
1110 (or (and (math-integ-parts-easy math-integ-expr)
1111 (math-integ-try-parts math-integ-expr t))
1112 (math-integrate-by-good-substitution
1113 math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2)))))
1125 (equal (nth 1 (car p)) math-integ-var))
1128 (or (and (math-integ-parts-easy math-integ-expr)
1129 (math-integ-try-parts math-integ-expr t))
1130 (math-integrate-by-good-substitution
1131 math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2)))))
1135 (setq math-t1 nil)
1137 (or (equal (car p) math-integ-var)
1139 (setq math-t1 (math-is-polynomial
1140 (nth 1 (setq math-t2 (car p)))
1141 math-integ-var 2)))))
1143 (and (null p) math-t1))
1144 (if (cdr (cdr math-t1))
1145 (if (math-guess-if-neg (nth 2 math-t1))
1146 (let* ((c (math-sqrt (math-neg (nth 2 math-t1))))
1147 (d (math-div (nth 1 math-t1) (math-mul -2 c)))
1148 (a (math-sqrt (math-add (car math-t1) (math-sqr d)))))
1149 (math-integrate-by-good-substitution
1150 math-integ-expr (list 'calcFunc-arcsin
1151 (math-div-thru
1152 (math-add (math-mul c math-integ-var) d)
1154 (let* ((c (math-sqrt (nth 2 math-t1)))
1155 (d (math-div (nth 1 math-t1) (math-mul 2 c)))
1156 (aa (math-sub (car math-t1) (math-sqr d))))
1158 (math-integrate-by-good-substitution
1159 math-integ-expr (math-add (math-mul c math-integ-var) d))
1160 (if (math-guess-if-neg aa)
1161 (math-integrate-by-good-substitution
1162 math-integ-expr (list 'calcFunc-arccosh
1163 (math-div-thru
1164 (math-add (math-mul c math-integ-var)
1166 (math-sqrt (math-neg aa)))))
1167 (math-integrate-by-good-substitution
1168 math-integ-expr (list 'calcFunc-arcsinh
1169 (math-div-thru
1170 (math-add (math-mul c math-integ-var)
1172 (math-sqrt aa))))))))
1173 (math-integrate-by-good-substitution math-integ-expr math-t2)) )
1176 (math-integ-try-parts math-integ-expr)
1181 (defun math-integ-parts-easy (expr)
1184 (and (math-integ-parts-easy (nth 1 expr))
1185 (math-integ-parts-easy (nth 2 expr))))
1187 (and (math-integ-parts-easy (nth 1 expr))
1188 (math-atomic-factorp (nth 2 expr))))
1191 (math-integ-parts-easy (nth 1 expr))))
1193 (math-integ-parts-easy (nth 1 expr)))
1196 ;; math-prev-parts-v is local to calcFunc-integ (as well as
1197 ;; math-integrate-by-parts), but is used by math-integ-try-parts.
1198 (defvar math-prev-parts-v)
1200 ;; math-good-parts is local to calcFunc-integ (as well as
1201 ;; math-integ-try-parts), but is used by math-integrate-by-parts.
1202 (defvar math-good-parts)
1205 (defun math-integ-try-parts (expr &optional math-good-parts)
1211 (math-integral exp)))
1213 (let ((first-bad (or (math-polynomial-p (nth 1 expr)
1214 math-integ-var)
1215 (equal (nth 2 expr) math-prev-parts-v))))
1217 (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))
1218 (math-integrate-by-parts (nth 2 expr) (nth 1 expr))
1220 (math-integrate-by-parts (nth 1 expr) (nth 2 expr))))))
1222 (math-expr-contains (nth 1 expr) math-integ-var)
1223 (let ((recip (math-div 1 (nth 2 expr))))
1224 (or (math-integrate-by-parts (nth 1 expr) recip)
1225 (math-integrate-by-parts recip (nth 1 expr)))))
1227 (math-integrate-by-parts (math-pow (nth 1 expr)
1228 (math-sub (nth 2 expr) 1))
1231 (defun math-integrate-by-parts (u vprime)
1232 (let ((math-integ-level (if (or math-good-parts
1233 (math-polynomial-p u math-integ-var))
1234 math-integ-level
1235 (1- math-integ-level)))
1236 (math-doing-parts t)
1238 (and (>= math-integ-level 0)
1241 (setcar (cdr math-cur-record) 'parts)
1242 (math-tracing-integral "Integrating by parts, u = "
1243 (math-format-value u 1000)
1245 (math-format-value vprime 1000)
1247 (and (setq v (math-integral vprime))
1248 (setq temp (calcFunc-deriv u math-integ-var nil t))
1249 (setq temp (let ((math-prev-parts-v v))
1250 (math-integral (math-mul v temp) 'yes)))
1251 (setq temp (math-sub (math-mul u v) temp))
1252 (if (eq (nth 1 math-cur-record) 'parts)
1254 (setq v (list 'var 'PARTS math-cur-record)
1256 (math-simplify-extended
1257 (math-solve-for (math-sub v temp) 0 v nil)))
1259 (math-zerop (nth 2 temp)))
1261 (setcar (cdr math-cur-record) 'busy)))))
1265 (defun math-integrate-by-substitution (expr u &optional user uinv uinvprime)
1266 (and (> math-integ-level 0)
1267 (let ((math-integ-level (max (- math-integ-level 2) 0)))
1268 (math-integrate-by-good-substitution expr u user uinv uinvprime))))
1270 (defun math-integrate-by-good-substitution (expr u &optional user
1272 (let ((math-living-dangerously t)
1275 (math-expr-subst uinv math-integ-var
1276 math-integ-var-2)
1278 (math-solve-for u
1279 math-integ-var-2
1280 math-integ-var nil))))
1282 (math-tracing-integral "Integrating by substitution, u = "
1283 (math-format-value u 1000)
1286 math-integ-var nil
1288 (setq temp (math-integral (math-expr-subst
1289 (math-expr-subst
1290 (math-expr-subst
1291 (math-div expr deriv)
1293 math-integ-var-2)
1294 math-integ-var
1296 math-integ-var-2
1297 math-integ-var)
1301 math-integ-var-2
1302 math-integ-var
1304 (setq temp (math-integral (math-mul
1305 (math-expr-subst
1306 (math-expr-subst
1307 (math-expr-subst
1310 math-integ-var-2)
1311 math-integ-var
1313 math-integ-var-2
1314 math-integ-var)
1317 (math-simplify-extended
1318 (math-expr-subst temp math-integ-var u)))))
1321 (defun math-integ-try-linear-substitutions (sub-expr)
1322 (setq math-linear-subst-tried t)
1323 (and (not (Math-primp sub-expr))
1324 (or (and (not (memq (car sub-expr) '(+ - * / neg)))
1325 (not (and (eq (car sub-expr) '^)
1326 (integerp (nth 2 sub-expr))))
1327 (math-expr-contains sub-expr math-integ-var)
1329 (while (and (setq sub-expr (cdr sub-expr))
1330 (or (not (math-linear-in (car sub-expr)
1331 math-integ-var))
1332 (assoc (car sub-expr) math-so-far)
1334 (setq math-so-far (cons (list (car sub-expr))
1335 math-so-far))
1337 (math-integrate-by-substitution
1338 math-integ-expr (car sub-expr))))))))
1341 (while (and (setq sub-expr (cdr sub-expr))
1342 (not (setq res (math-integ-try-linear-substitutions
1343 (car sub-expr))))))
1346 ;;; Recursively try different substitutions based on various sub-expressions.
1347 (defun math-integ-try-substitutions (sub-expr &optional allow-rat)
1348 (and (not (Math-primp sub-expr))
1349 (not (assoc sub-expr math-so-far))
1350 (math-expr-contains sub-expr math-integ-var)
1351 (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg)))
1352 (not (and (eq (car sub-expr) '^)
1353 (integerp (nth 2 sub-expr)))))
1356 (not (eq sub-expr math-integ-expr))
1357 (or (math-integrate-by-substitution math-integ-expr sub-expr)
1358 (and (eq (car sub-expr) '^)
1359 (integerp (nth 2 sub-expr))
1360 (< (nth 2 sub-expr) 0)
1361 (math-integ-try-substitutions
1362 (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr)))
1365 (setq math-so-far (cons (list sub-expr) math-so-far))
1366 (while (and (setq sub-expr (cdr sub-expr))
1367 (not (setq res (math-integ-try-substitutions
1368 (car sub-expr) allow-rat)))))
1371 ;; The variable math-expr-parts is local to math-expr-rational-in,
1372 ;; but is used by math-expr-rational-in-rec
1373 (defvar math-expr-parts)
1375 (defun math-expr-rational-in (expr)
1376 (let ((math-expr-parts nil))
1377 (math-expr-rational-in-rec expr)
1378 (mapcar 'car math-expr-parts)))
1380 (defun math-expr-rational-in-rec (expr)
1382 (and (equal expr math-integ-var)
1383 (not (assoc expr math-expr-parts))
1384 (setq math-expr-parts (cons (list expr) math-expr-parts))))
1387 (math-expr-rational-in-rec (nth 1 expr))
1388 (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr))))
1390 (eq (math-quarter-integer (nth 2 expr)) 2))
1391 (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr))))
1393 (and (not (assoc expr math-expr-parts))
1394 (math-expr-contains expr math-integ-var)
1395 (setq math-expr-parts (cons (list expr) math-expr-parts))))))
1397 (defun math-expr-calls (expr funcs &optional arg-contains)
1401 (eq (math-quarter-integer (nth 2 expr)) 2)))
1403 (math-expr-contains expr arg-contains))
1408 (not (setq res (math-expr-calls
1412 (defun math-fix-const-terms (expr except-vars)
1413 (cond ((not (math-expr-depends expr except-vars)) 0)
1416 (math-add (math-fix-const-terms (nth 1 expr) except-vars)
1417 (math-fix-const-terms (nth 2 expr) except-vars)))
1419 (math-sub (math-fix-const-terms (nth 1 expr) except-vars)
1420 (math-fix-const-terms (nth 2 expr) except-vars)))
1428 (let ((p math-integral-cache)
1429 math-cur-record)
1434 (setq math-cur-record (car p))
1435 (or arg (math-replace-integral-parts math-cur-record))
1436 (insert (math-format-flat-expr (car math-cur-record) 0)
1438 (if (symbolp (nth 1 math-cur-record))
1439 (concat "(" (symbol-name (nth 1 math-cur-record)) ")")
1440 (math-format-flat-expr (nth 1 math-cur-record) 0))
1446 ;; The variable math-max-integral-limit is local to calcFunc-integ,
1447 ;; but is used by math-try-integral.
1448 (defvar math-max-integral-limit)
1450 (defun math-try-integral (expr)
1451 (let ((math-integ-level math-integral-limit)
1452 (math-integ-depth 0)
1453 (math-integ-msg "Working...done")
1454 (math-cur-record nil) ; a technicality
1455 (math-integrating t)
1458 (math-has-rules (calc-has-rules 'var-IntegRules)))
1459 (or (math-integral expr 'yes)
1460 (and math-any-substs
1461 (setq math-enable-subst t)
1462 (math-integral expr 'yes))
1463 (and (> math-max-integral-limit math-integral-limit)
1464 (setq math-integral-limit math-max-integral-limit
1465 math-integ-level math-integral-limit)
1466 (math-integral expr 'yes)))))
1474 (math-add (calcFunc-integ (nth 1 expr) var low high)
1477 (math-sub (calcFunc-integ (nth 1 expr) var low high)
1480 (math-neg (calcFunc-integ (nth 1 expr) var low high)))
1482 (not (math-expr-contains (nth 1 expr) var)))
1483 (math-mul (nth 1 expr) (calcFunc-integ (nth 2 expr) var low high)))
1485 (not (math-expr-contains (nth 2 expr) var)))
1486 (math-mul (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr)))
1488 (not (math-expr-contains (nth 1 expr) var))
1489 (not (math-equal-int (nth 1 expr) 1)))
1490 (math-mul (nth 1 expr)
1491 (calcFunc-integ (math-div 1 (nth 2 expr)) var low high)))
1493 (not (math-expr-contains (nth 2 expr) var)))
1494 (math-div (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr)))
1497 (not (math-expr-contains (nth 1 (nth 1 expr)) var)))
1498 (math-mul (nth 1 (nth 1 expr))
1499 (calcFunc-integ (math-div (nth 2 (nth 1 expr)) (nth 2 expr))
1503 (not (math-expr-contains (nth 2 (nth 1 expr)) var)))
1504 (math-mul (nth 2 (nth 1 expr))
1505 (calcFunc-integ (math-div (nth 1 (nth 1 expr)) (nth 2 expr))
1509 (not (math-expr-contains (nth 1 (nth 2 expr)) var)))
1510 (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 2 (nth 2 expr)))
1515 (not (math-expr-contains (nth 2 (nth 2 expr)) var)))
1516 (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 1 (nth 2 expr)))
1529 (or (equal state math-integral-cache-state)
1530 (setq math-integral-cache-state state
1531 math-integral-cache nil)))
1532 (let* ((math-max-integral-limit (or (and (natnump var-IntegLimit)
1535 (math-integral-limit 1)
1536 (sexpr (math-expr-subst expr var math-integ-var))
1539 (math-any-substs t)
1540 (math-enable-subst nil)
1541 (math-prev-parts-v nil)
1542 (math-doing-parts nil)
1543 (math-good-parts nil)
1560 (math-try-integral sexpr))
1563 (math-try-integral sexpr))))
1567 (setq res (math-rewrite res '(var IntegAfterRules
1569 (math-simplify
1571 (math-sub (math-expr-subst res math-integ-var high)
1572 (math-expr-subst res math-integ-var low))
1573 (setq res (math-fix-const-terms res math-integ-vars))
1575 (math-expr-subst res math-integ-var low)
1576 (math-expr-subst res math-integ-var var)))))
1582 (math-defintegral calcFunc-inv
1583 (math-integral (math-div 1 u)))
1585 (math-defintegral calcFunc-conj
1586 (let ((int (math-integral u)))
1590 (math-defintegral calcFunc-deg
1591 (let ((int (math-integral u)))
1595 (math-defintegral calcFunc-rad
1596 (let ((int (math-integral u)))
1600 (math-defintegral calcFunc-re
1601 (let ((int (math-integral u)))
1605 (math-defintegral calcFunc-im
1606 (let ((int (math-integral u)))
1610 (math-defintegral calcFunc-sqrt
1611 (and (equal u math-integ-var)
1612 (math-mul '(frac 2 3)
1613 (list 'calcFunc-sqrt (math-pow u 3)))))
1615 (math-defintegral calcFunc-exp
1616 (or (and (equal u math-integ-var)
1618 (let ((p (math-is-polynomial u math-integ-var 2)))
1620 (let ((sqa (math-sqrt (math-neg (nth 2 p)))))
1621 (math-div
1622 (math-mul
1623 (math-mul (math-div (list 'calcFunc-sqrt '(var pi var-pi))
1625 (math-normalize
1627 (math-div (math-sub (math-mul (car p)
1629 (math-div
1630 (math-sqr (nth 1 p))
1634 (math-sub (math-mul sqa math-integ-var)
1635 (math-div (nth 1 p) (math-mul 2 sqa)))))
1638 (math-defintegral calcFunc-ln
1639 (or (and (equal u math-integ-var)
1640 (math-sub (math-mul u (list 'calcFunc-ln u)) u))
1642 (math-integral (math-add (list 'calcFunc-ln (nth 1 u))
1645 (math-integral (math-sub (list 'calcFunc-ln (nth 1 u))
1648 (math-integral (math-mul (nth 2 u)
1651 (math-defintegral calcFunc-log10
1652 (and (equal u math-integ-var)
1653 (math-sub (math-mul u (list 'calcFunc-ln u))
1654 (math-div u (list 'calcFunc-ln 10)))))
1656 (math-defintegral-2 calcFunc-log
1657 (math-integral (math-div (list 'calcFunc-ln u)
1660 (math-defintegral calcFunc-sin
1661 (or (and (equal u math-integ-var)
1662 (math-neg (math-from-radians-2 (list 'calcFunc-cos u))))
1663 (and (nth 2 (math-is-polynomial u math-integ-var 2))
1664 (math-integral (math-to-exponentials (list 'calcFunc-sin u))))))
1666 (math-defintegral calcFunc-cos
1667 (or (and (equal u math-integ-var)
1668 (math-from-radians-2 (list 'calcFunc-sin u)))
1669 (and (nth 2 (math-is-polynomial u math-integ-var 2))
1670 (math-integral (math-to-exponentials (list 'calcFunc-cos u))))))
1672 (math-defintegral calcFunc-tan
1673 (and (equal u math-integ-var)
1674 (math-from-radians-2
1677 (math-defintegral calcFunc-sec
1678 (and (equal u math-integ-var)
1679 (math-from-radians-2
1681 (math-add
1685 (math-defintegral calcFunc-csc
1686 (and (equal u math-integ-var)
1687 (math-from-radians-2
1689 (math-sub
1693 (math-defintegral calcFunc-cot
1694 (and (equal u math-integ-var)
1695 (math-from-radians-2
1698 (math-defintegral calcFunc-arcsin
1699 (and (equal u math-integ-var)
1700 (math-add (math-mul u (list 'calcFunc-arcsin u))
1701 (math-from-radians-2
1702 (list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))))
1704 (math-defintegral calcFunc-arccos
1705 (and (equal u math-integ-var)
1706 (math-sub (math-mul u (list 'calcFunc-arccos u))
1707 (math-from-radians-2
1708 (list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))))
1710 (math-defintegral calcFunc-arctan
1711 (and (equal u math-integ-var)
1712 (math-sub (math-mul u (list 'calcFunc-arctan u))
1713 (math-from-radians-2
1714 (math-div (list 'calcFunc-ln (math-add 1 (math-sqr u)))
1717 (math-defintegral calcFunc-sinh
1718 (and (equal u math-integ-var)
1721 (math-defintegral calcFunc-cosh
1722 (and (equal u math-integ-var)
1725 (math-defintegral calcFunc-tanh
1726 (and (equal u math-integ-var)
1729 (math-defintegral calcFunc-sech
1730 (and (equal u math-integ-var)
1733 (math-defintegral calcFunc-csch
1734 (and (equal u math-integ-var)
1735 (list 'calcFunc-ln (list 'calcFunc-tanh (math-div u 2)))))
1737 (math-defintegral calcFunc-coth
1738 (and (equal u math-integ-var)
1741 (math-defintegral calcFunc-arcsinh
1742 (and (equal u math-integ-var)
1743 (math-sub (math-mul u (list 'calcFunc-arcsinh u))
1744 (list 'calcFunc-sqrt (math-add (math-sqr u) 1)))))
1746 (math-defintegral calcFunc-arccosh
1747 (and (equal u math-integ-var)
1748 (math-sub (math-mul u (list 'calcFunc-arccosh u))
1749 (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))
1751 (math-defintegral calcFunc-arctanh
1752 (and (equal u math-integ-var)
1753 (math-sub (math-mul u (list 'calcFunc-arctan u))
1754 (math-div (list 'calcFunc-ln
1755 (math-add 1 (math-sqr u)))
1759 (math-defintegral-2 /
1760 (math-integral-rational-funcs u v))
1762 (defun math-integral-rational-funcs (u v)
1763 (let ((pu (math-is-polynomial u math-integ-var 1))
1770 (and (setq pv (math-is-polynomial v math-integ-var 2))
1771 (let ((int (math-mul-thru
1773 (math-integral-q02 (car pv) (nth 1 pv)
1776 (setq int (math-add int
1777 (math-mul-thru
1779 (math-integral-q12
1784 (defun math-integral-q12 (a b c v vpow)
1788 (math-sub (math-div math-integ-var b)
1789 (math-mul (math-div a (math-sqr b))
1792 (math-div (math-add (list 'calcFunc-ln v)
1793 (math-div a v))
1794 (math-sqr b)))
1796 (let ((nm1 (math-sub vpow 1))
1797 (nm2 (math-sub vpow 2)))
1798 (math-div (math-sub
1799 (math-div a (math-mul nm1 (math-pow v nm1)))
1800 (math-div 1 (math-mul nm2 (math-pow v nm2))))
1801 (math-sqr b))))))
1802 ((math-zerop
1803 (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b))))
1804 (let ((part (math-div b (math-mul 2 c))))
1805 (math-mul-thru (math-pow c vpow)
1806 (math-integral-q12 part 1 nil
1807 (math-add math-integ-var part)
1810 (and (math-ratp q) (math-negp q)
1812 (math-ratp (math-sqrt (math-neg q))))
1814 (math-sub (math-div (list 'calcFunc-ln v) (math-mul 2 c))
1815 (math-mul-thru (math-div b (math-mul 2 c))
1816 (math-integral-q02 a b c v 1))))
1819 (math-sub (math-neg (math-div
1820 (math-add (math-mul b math-integ-var)
1821 (math-mul 2 a))
1822 (math-mul n (math-mul q (math-pow v n)))))
1823 (math-mul-thru (math-div (math-mul b (1- (* 2 n)))
1824 (math-mul n q))
1825 (math-integral-q02 a b c v n))))))))
1827 (defun math-integral-q02 (a b c v vpow)
1831 (math-div (list 'calcFunc-ln v) b))
1833 (math-div (math-pow v (- 1 vpow))
1834 (math-mul (- 1 vpow) b)))))
1835 ((math-zerop
1836 (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b))))
1837 (let ((part (math-div b (math-mul 2 c))))
1838 (math-mul-thru (math-pow c vpow)
1839 (math-integral-q02 part 1 nil
1840 (math-add math-integ-var part)
1843 (setq part (math-add (math-mul 2 (math-mul c math-integ-var)) b))
1846 (math-add (math-div part (math-mul n (math-mul q (math-pow v n))))
1847 (math-mul-thru (math-div (math-mul (- (* 4 n) 2) c)
1848 (math-mul n q))
1849 (math-integral-q02 a b c v n)))))
1850 ((math-guess-if-neg q)
1851 (setq rq (list 'calcFunc-sqrt (math-neg q)))
1852 ;;(math-div-thru (list 'calcFunc-ln
1853 ;; (math-div (math-sub part rq)
1854 ;; (math-add part rq)))
1856 (math-div (math-mul -2 (list 'calcFunc-arctanh
1857 (math-div part rq)))
1861 (math-div (math-mul 2 (math-to-radians-2
1863 (math-div part rq))))
1867 (math-defintegral calcFunc-erf
1868 (and (equal u math-integ-var)
1869 (math-add (math-mul u (list 'calcFunc-erf u))
1870 (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u))
1874 (math-defintegral calcFunc-erfc
1875 (and (equal u math-integ-var)
1876 (math-sub (math-mul u (list 'calcFunc-erfc u))
1877 (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u))
1884 (defvar math-tabulate-initial nil)
1885 (defvar math-tabulate-function nil)
1888 ;; but are used by math-scan-for-limits.
1896 (and (or (math-infinitep calc-low) (math-infinitep calc-high))
1898 (math-scan-for-limits expr))
1899 (and step (math-zerop step) (math-reject-arg step 'nonzerop))
1908 (setq count (math-div (math-sub calc-high calc-low) (or step 1)))
1910 (setq count (math-simplify count)))
1912 (setq count (math-trunc count)))))
1917 (vec math-tabulate-initial)
1918 (math-working-step-2 (1+ count))
1919 (math-working-step 0))
1920 (setq expr (math-evaluate-expr
1921 (math-expr-subst expr var '(var DUMMY var-DUMMY))))
1923 (setq math-working-step (1+ math-working-step)
1925 vec (cond ((eq math-tabulate-function 'calcFunc-sum)
1926 (math-add vec (math-evaluate-expr expr)))
1927 ((eq math-tabulate-function 'calcFunc-prod)
1928 (math-mul vec (math-evaluate-expr expr)))
1930 (cons (math-evaluate-expr expr) vec)))
1931 calc-low (math-add calc-low (or step 1))
1933 (if math-tabulate-function
1943 (append (list (or math-tabulate-function 'calcFunc-table)
1950 (defun math-scan-for-limits (x)
1954 (math-expr-contains (nth 2 x) var))
1956 (low-val (math-solve-for (nth 2 x) 1 var nil))
1957 (high-val (math-solve-for (nth 2 x) (1- (length (nth 1 x)))
1960 (and low-val (math-realp low-val)
1961 high-val (math-realp high-val))
1964 (setq calc-low (math-max calc-low (math-ceiling low-val))
1965 calc-high (math-min calc-high (math-floor high-val)))))
1968 (math-scan-for-limits (car x))))))
1971 (defvar math-disable-sums nil)
1973 (if math-disable-sums (math-reject-arg))
1975 (math-sum-rec expr var low high step)))
1976 (math-disable-sums t))
1977 (math-normalize res)))
1979 (defun math-sum-rec (expr var &optional low high step)
1985 ((not (math-expr-contains expr var))
1986 (math-mul expr (math-add (math-div (math-sub high low) (or step 1))
1988 ((and step (not (math-equal-int step 1)))
1989 (if (math-negp step)
1990 (math-sum-rec expr var high low (math-neg step))
1991 (let ((lo (math-simplify (math-div low step))))
1992 (if (math-known-num-integerp lo)
1993 (math-sum-rec (math-normalize
1994 (math-expr-subst expr var
1995 (math-mul step var)))
1996 var lo (math-simplify (math-div high step)))
1997 (math-sum-rec (math-normalize
1998 (math-expr-subst expr var
1999 (math-add (math-mul step var)
2002 (math-simplify (math-div (math-sub high low)
2004 ((memq (setq t1 (math-compare low high)) '(0 1))
2006 (math-expr-subst expr var low)
2008 ((setq t1 (math-is-polynomial expr var 20))
2012 (setq poly (math-poly-mix poly 1
2013 (math-sum-integer-power n) (car t1))
2016 (setq n (math-build-polynomial-expr poly high))
2019 (math-sub n (math-build-polynomial-expr poly
2020 (math-sub low 1))))))
2022 (setq t1 (math-sum-rec (nth 1 expr) var low high)
2023 t2 (math-sum-rec (nth 2 expr) var low high))
2024 (not (and (math-expr-calls t1 '(calcFunc-sum))
2025 (math-expr-calls t2 '(calcFunc-sum)))))
2028 (setq t1 (math-sum-const-factors expr var)))
2029 (math-mul (car t1) (math-sum-rec (cdr t1) var low high)))
2031 (math-sum-rec (math-add-or-sub (math-mul (nth 1 (nth 1 expr))
2033 (math-mul (nth 2 (nth 1 expr))
2038 (math-sum-rec (math-add-or-sub (math-mul (nth 1 expr)
2040 (math-mul (nth 1 expr)
2045 (not (math-primp (nth 1 expr)))
2046 (setq t1 (math-sum-const-factors (nth 1 expr) var)))
2047 (math-mul (car t1)
2048 (math-sum-rec (math-div (cdr t1) (nth 2 expr))
2051 (setq t1 (math-sum-const-factors (nth 2 expr) var)))
2052 (math-div (math-sum-rec (math-div (nth 1 expr) (cdr t1))
2056 (math-neg (math-sum-rec (nth 1 expr) var low high)))
2058 (not (math-expr-contains (nth 1 expr) var))
2059 (setq t1 (math-is-polynomial (nth 2 expr) var 1)))
2060 (let ((x (math-pow (nth 1 expr) (nth 1 t1))))
2061 (math-div (math-mul (math-sub (math-pow x (math-add 1 high))
2062 (math-pow x low))
2063 (math-pow (nth 1 expr) (car t1)))
2064 (math-sub x 1))))
2065 ((and (setq t1 (math-to-exponentials expr))
2066 (setq t1 (math-sum-rec t1 var low high))
2067 (not (math-expr-calls t1 '(calcFunc-sum))))
2068 (math-to-exps t1))
2073 (not (math-expr-contains (nth 2 expr) var)))
2079 (let* ((math-tabulate-initial 0)
2080 (math-tabulate-function 'calcFunc-sum))
2085 (if (and step (not (math-equal-int step 1)))
2086 (if (math-negp step)
2087 (math-mul (math-pow -1 low)
2088 (calcFunc-asum expr var high low (math-neg step) t))
2089 (let ((lo (math-simplify (math-div low step))))
2090 (if (math-num-integerp lo)
2091 (calcFunc-asum (math-normalize
2092 (math-expr-subst expr var
2093 (math-mul step var)))
2094 var lo (math-simplify (math-div high step)))
2095 (calcFunc-asum (math-normalize
2096 (math-expr-subst expr var
2097 (math-add (math-mul step var)
2100 (math-simplify (math-div (math-sub high low)
2102 (math-mul (if no-mul-flag 1 (math-pow -1 low))
2103 (calcFunc-sum (math-mul (math-pow -1 var) expr) var low high))))
2105 (defun math-sum-const-factors (expr var)
2110 (if (math-expr-contains (nth 1 p) var)
2114 (if (math-expr-contains p var)
2127 (defvar math-sum-int-pow-cache (list '(0 1)))
2129 (defun math-sum-integer-power (pow)
2131 (n (length math-sum-int-pow-cache)))
2135 (pp (cdr (nth (1- n) math-sum-int-pow-cache)))
2140 (setq q (math-div (car pp) p)
2141 new (cons (math-mul q n) new)
2142 sum (math-add sum q)
2145 (setcar lin (math-sub 1 (math-mul n sum)))
2146 (setq math-sum-int-pow-cache
2147 (nconc math-sum-int-pow-cache (list (nreverse new)))
2149 (nth pow math-sum-int-pow-cache)))
2151 (defun math-to-exponentials (expr)
2155 (pi (if calc-symbolic-mode '(var pi var-pi) (math-pi)))
2188 (defun math-to-exps (expr)
2191 (if (equal expr '(var e var-e)) (math-e) expr))
2196 (cons (car expr) (mapcar 'math-to-exps (cdr expr))))))
2199 (defvar math-disable-prods nil)
2201 (if math-disable-prods (math-reject-arg))
2203 (math-prod-rec expr var low high step)))
2204 (math-disable-prods t))
2205 (math-normalize res)))
2207 (defun math-prod-rec (expr var &optional low high step)
2213 ((not (math-expr-contains expr var))
2214 (math-pow expr (math-add (math-div (math-sub high low) (or step 1))
2216 ((and step (not (math-equal-int step 1)))
2217 (if (math-negp step)
2218 (math-prod-rec expr var high low (math-neg step))
2219 (let ((lo (math-simplify (math-div low step))))
2220 (if (math-known-num-integerp lo)
2221 (math-prod-rec (math-normalize
2222 (math-expr-subst expr var
2223 (math-mul step var)))
2224 var lo (math-simplify (math-div high step)))
2225 (math-prod-rec (math-normalize
2226 (math-expr-subst expr var
2227 (math-add (math-mul step
2231 (math-simplify (math-div (math-sub high low)
2234 (setq t1 (math-prod-rec (nth 1 expr) var low high)
2235 t2 (math-prod-rec (nth 2 expr) var low high))
2236 (not (and (math-expr-calls t1 '(calcFunc-prod))
2237 (math-expr-calls t2 '(calcFunc-prod)))))
2240 (not (math-expr-contains (nth 2 expr) var)))
2241 (math-pow (math-prod-rec (nth 1 expr) var low high)
2244 (not (math-expr-contains (nth 1 expr) var)))
2245 (math-pow (nth 1 expr)
2248 (math-normalize (list 'calcFunc-sqrt
2252 (math-mul (math-pow -1 (math-add (math-sub high low) 1))
2253 (math-prod-rec (nth 1 expr) var low high)))
2256 ((and (setq t1 (math-is-polynomial expr var 1))
2259 ((or (and (math-equal-int (nth 1 t1) 1)
2260 (setq low (math-simplify
2261 (math-add low (car t1)))
2262 high (math-simplify
2263 (math-add high (car t1)))))
2264 (and (math-equal-int (nth 1 t1) -1)
2266 low (math-simplify
2267 (math-sub (car t1) high))
2268 high (math-simplify
2269 (math-sub (car t1) t2)))))
2270 (if (or (math-zerop low) (math-zerop high))
2272 (if (and (or (math-negp low) (math-negp high))
2273 (or (math-num-integerp low)
2274 (math-num-integerp high)))
2275 (if (math-posp high)
2277 (math-mul (math-pow -1
2278 (math-add
2279 (math-add low high) 1))
2282 (math-neg low))
2284 (math-sub -1 high)))))
2287 (list 'calcFunc-fact (math-sub low 1))))))
2288 ((and (or (and (math-equal-int (nth 1 t1) 2)
2289 (setq t2 (math-simplify
2290 (math-add (math-mul low 2)
2292 t3 (math-simplify
2293 (math-add (math-mul high 2)
2295 (and (math-equal-int (nth 1 t1) -2)
2296 (setq t2 (math-simplify
2297 (math-sub (car t1)
2298 (math-mul high 2)))
2299 t3 (math-simplify
2300 (math-sub (car t1)
2301 (math-mul low
2303 (or (math-integerp t2)
2304 (and (math-messy-integerp t2)
2305 (setq t2 (math-trunc t2)))
2306 (math-integerp t3)
2307 (and (math-messy-integerp t3)
2308 (setq t3 (math-trunc t3)))))
2309 (if (or (math-zerop t2) (math-zerop t3))
2311 (if (or (math-evenp t2) (math-evenp t3))
2312 (if (or (math-negp t2) (math-negp t3))
2313 (if (math-posp high)
2317 (math-neg t2))
2319 (math-sub -2 t3))))
2323 (math-sub t2 2))))
2324 (if (math-negp t3)
2332 (math-neg t2))
2334 (math-sub -2 t3))))
2335 (if (math-posp t2)
2339 (math-sub t2 2)))
2344 (let* ((math-tabulate-initial 1)
2345 (math-tabulate-function 'calcFunc-prod))
2351 (defvar math-solve-ranges nil)
2352 (defvar math-solve-sign)
2353 ;;; Attempt to reduce math-solve-lhs = math-solve-rhs to
2354 ;;; math-solve-var = math-solve-rhs', where math-solve-var appears
2355 ;;; in math-solve-lhs but not in math-solve-rhs or math-solve-rhs';
2356 ;;; return math-solve-rhs'.
2357 ;;; Uses global values: math-solve-var, math-solve-full.
2358 (defvar math-solve-var)
2359 (defvar math-solve-full)
2361 ;; The variables math-solve-lhs, math-solve-rhs and math-try-solve-sign
2362 ;; are local to math-try-solve-for, but are used by math-try-solve-prod.
2363 ;; (math-solve-lhs and math-solve-rhs are is also local to
2364 ;; math-decompose-poly, but used by math-solve-poly-funny-powers.)
2365 (defvar math-solve-lhs)
2366 (defvar math-solve-rhs)
2367 (defvar math-try-solve-sign)
2369 (defun math-try-solve-for
2370 (math-solve-lhs math-solve-rhs &optional math-try-solve-sign no-poly)
2371 (let (math-t1 math-t2 math-t3)
2372 (cond ((equal math-solve-lhs math-solve-var)
2373 (setq math-solve-sign math-try-solve-sign)
2374 (if (eq math-solve-full 'all)
2375 (let ((vec (list 'vec (math-evaluate-expr math-solve-rhs)))
2377 (while math-solve-ranges
2378 (setq p (car math-solve-ranges)
2383 (cdr (math-expr-subst
2386 math-solve-ranges (cdr math-solve-ranges)))
2387 (math-normalize vec))
2388 math-solve-rhs))
2389 ((Math-primp math-solve-lhs)
2391 ((and (eq (car math-solve-lhs) '-)
2392 (eq (car-safe (nth 1 math-solve-lhs)) (car-safe (nth 2 math-solve-lhs)))
2393 (Math-zerop math-solve-rhs)
2394 (= (length (nth 1 math-solve-lhs)) 2)
2395 (= (length (nth 2 math-solve-lhs)) 2)
2396 (setq math-t1 (get (car (nth 1 math-solve-lhs)) 'math-inverse))
2397 (setq math-t2 (funcall math-t1 '(var SOLVEDUM SOLVEDUM)))
2398 (eq (math-expr-contains-count math-t2 '(var SOLVEDUM SOLVEDUM)) 1)
2399 (setq math-t3 (math-solve-above-dummy math-t2))
2400 (setq math-t1 (math-try-solve-for
2401 (math-sub (nth 1 (nth 1 math-solve-lhs))
2402 (math-expr-subst
2403 math-t2 math-t3
2404 (nth 1 (nth 2 math-solve-lhs))))
2406 math-t1)
2407 ((eq (car math-solve-lhs) 'neg)
2408 (math-try-solve-for (nth 1 math-solve-lhs) (math-neg math-solve-rhs)
2409 (and math-try-solve-sign (- math-try-solve-sign))))
2410 ((and (not (eq math-solve-full 't)) (math-try-solve-prod)))
2412 (setq math-t2
2413 (math-decompose-poly math-solve-lhs
2414 math-solve-var 15 math-solve-rhs)))
2415 (setq math-t1 (cdr (nth 1 math-t2))
2416 math-t1 (let ((math-solve-ranges math-solve-ranges))
2417 (cond ((= (length math-t1) 5)
2418 (apply 'math-solve-quartic (car math-t2) math-t1))
2419 ((= (length math-t1) 4)
2420 (apply 'math-solve-cubic (car math-t2) math-t1))
2421 ((= (length math-t1) 3)
2422 (apply 'math-solve-quadratic (car math-t2) math-t1))
2423 ((= (length math-t1) 2)
2424 (apply 'math-solve-linear
2425 (car math-t2) math-try-solve-sign math-t1))
2426 (math-solve-full
2427 (math-poly-all-roots (car math-t2) math-t1))
2430 (math-try-solve-for
2431 (car math-t2)
2432 (math-poly-any-root (reverse math-t1) 0 t)
2434 (if math-t1
2435 (if (eq (nth 2 math-t2) 1)
2436 math-t1
2437 (math-solve-prod math-t1 (math-try-solve-for (nth 2 math-t2) 0 nil t)))
2440 ((and (math-solve-find-root-term math-solve-lhs nil)
2441 (eq (math-expr-contains-count math-solve-lhs math-t1) 1)) ; just in case
2442 (math-try-solve-for (math-simplify
2443 (math-sub (if (or math-t3 (math-evenp math-t2))
2444 (math-pow math-t1 math-t2)
2445 (math-neg (math-pow math-t1 math-t2)))
2446 (math-expand-power
2447 (math-sub (math-normalize
2448 (math-expr-subst
2449 math-solve-lhs math-t1 0))
2450 math-solve-rhs)
2451 math-t2 math-solve-var)))
2453 ((eq (car math-solve-lhs) '+)
2454 (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2455 (math-try-solve-for (nth 2 math-solve-lhs)
2456 (math-sub math-solve-rhs (nth 1 math-solve-lhs))
2457 math-try-solve-sign))
2458 ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
2459 (math-try-solve-for (nth 1 math-solve-lhs)
2460 (math-sub math-solve-rhs (nth 2 math-solve-lhs))
2461 math-try-solve-sign))))
2462 ((eq (car math-solve-lhs) 'calcFunc-eq)
2463 (math-try-solve-for (math-sub (nth 1 math-solve-lhs) (nth 2 math-solve-lhs))
2464 math-solve-rhs math-try-solve-sign no-poly))
2465 ((eq (car math-solve-lhs) '-)
2466 (cond ((or (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-sin)
2467 (eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-cos))
2468 (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-cos)
2469 (eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-sin)))
2470 (math-try-solve-for (math-sub (nth 1 math-solve-lhs)
2471 (list (car (nth 1 math-solve-lhs))
2472 (math-sub
2473 (math-quarter-circle t)
2474 (nth 1 (nth 2 math-solve-lhs)))))
2475 math-solve-rhs))
2476 ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2477 (math-try-solve-for (nth 2 math-solve-lhs)
2478 (math-sub (nth 1 math-solve-lhs) math-solve-rhs)
2479 (and math-try-solve-sign
2480 (- math-try-solve-sign))))
2481 ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
2482 (math-try-solve-for (nth 1 math-solve-lhs)
2483 (math-add math-solve-rhs (nth 2 math-solve-lhs))
2484 math-try-solve-sign))))
2485 ((and (eq math-solve-full 't) (math-try-solve-prod)))
2486 ((and (eq (car math-solve-lhs) '%)
2487 (not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)))
2488 (math-try-solve-for (nth 1 math-solve-lhs) (math-add math-solve-rhs
2489 (math-solve-get-int
2490 (nth 2 math-solve-lhs)))))
2491 ((eq (car math-solve-lhs) 'calcFunc-log)
2492 (cond ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
2493 (math-try-solve-for (nth 1 math-solve-lhs)
2494 (math-pow (nth 2 math-solve-lhs) math-solve-rhs)))
2495 ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2496 (math-try-solve-for (nth 2 math-solve-lhs) (math-pow
2497 (nth 1 math-solve-lhs)
2498 (math-div 1 math-solve-rhs))))))
2499 ((and (= (length math-solve-lhs) 2)
2500 (symbolp (car math-solve-lhs))
2501 (setq math-t1 (get (car math-solve-lhs) 'math-inverse))
2502 (setq math-t2 (funcall math-t1 math-solve-rhs)))
2503 (setq math-t1 (get (car math-solve-lhs) 'math-inverse-sign))
2504 (math-try-solve-for (nth 1 math-solve-lhs) (math-normalize math-t2)
2505 (and math-try-solve-sign math-t1
2506 (if (integerp math-t1)
2507 (* math-t1 math-try-solve-sign)
2508 (funcall math-t1 math-solve-lhs
2509 math-try-solve-sign)))))
2510 ((and (symbolp (car math-solve-lhs))
2511 (setq math-t1 (get (car math-solve-lhs) 'math-inverse-n))
2512 (setq math-t2 (funcall math-t1 math-solve-lhs math-solve-rhs)))
2513 math-t2)
2514 ((setq math-t1 (math-expand-formula math-solve-lhs))
2515 (math-try-solve-for math-t1 math-solve-rhs math-try-solve-sign))
2517 (calc-record-why "*No inverse known" math-solve-lhs)
2521 (defun math-try-solve-prod ()
2522 (cond ((eq (car math-solve-lhs) '*)
2523 (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2524 (math-try-solve-for (nth 2 math-solve-lhs)
2525 (math-div math-solve-rhs (nth 1 math-solve-lhs))
2526 (math-solve-sign math-try-solve-sign
2527 (nth 1 math-solve-lhs))))
2528 ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
2529 (math-try-solve-for (nth 1 math-solve-lhs)
2530 (math-div math-solve-rhs (nth 2 math-solve-lhs))
2531 (math-solve-sign math-try-solve-sign
2532 (nth 2 math-solve-lhs))))
2533 ((Math-zerop math-solve-rhs)
2534 (math-solve-prod (let ((math-solve-ranges math-solve-ranges))
2535 (math-try-solve-for (nth 2 math-solve-lhs) 0))
2536 (math-try-solve-for (nth 1 math-solve-lhs) 0)))))
2537 ((eq (car math-solve-lhs) '/)
2538 (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2539 (math-try-solve-for (nth 2 math-solve-lhs)
2540 (math-div (nth 1 math-solve-lhs) math-solve-rhs)
2541 (math-solve-sign math-try-solve-sign
2542 (nth 1 math-solve-lhs))))
2543 ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
2544 (math-try-solve-for (nth 1 math-solve-lhs)
2545 (math-mul math-solve-rhs (nth 2 math-solve-lhs))
2546 (math-solve-sign math-try-solve-sign
2547 (nth 2 math-solve-lhs))))
2548 ((setq math-t1 (math-try-solve-for (math-sub (nth 1 math-solve-lhs)
2549 (math-mul (nth 2 math-solve-lhs)
2550 math-solve-rhs))
2552 math-t1)))
2553 ((eq (car math-solve-lhs) '^)
2554 (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2555 (math-try-solve-for
2556 (nth 2 math-solve-lhs)
2557 (math-add (math-normalize
2558 (list 'calcFunc-log math-solve-rhs (nth 1 math-solve-lhs)))
2559 (math-div
2560 (math-mul 2
2561 (math-mul '(var pi var-pi)
2562 (math-solve-get-int
2564 (math-normalize
2565 (list 'calcFunc-ln (nth 1 math-solve-lhs)))))))
2566 ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
2567 (cond ((and (integerp (nth 2 math-solve-lhs))
2568 (>= (nth 2 math-solve-lhs) 2)
2569 (setq math-t1 (math-integer-log2 (nth 2 math-solve-lhs))))
2570 (setq math-t2 math-solve-rhs)
2571 (if (and (eq math-solve-full t)
2572 (math-known-realp (nth 1 math-solve-lhs)))
2574 (while (>= (setq math-t1 (1- math-t1)) 0)
2575 (setq math-t2 (list 'calcFunc-sqrt math-t2)))
2576 (setq math-t2 (math-solve-get-sign math-t2)))
2577 (while (>= (setq math-t1 (1- math-t1)) 0)
2578 (setq math-t2 (math-solve-get-sign
2579 (math-normalize
2580 (list 'calcFunc-sqrt math-t2))))))
2581 (math-try-solve-for
2582 (nth 1 math-solve-lhs)
2583 (math-normalize math-t2)))
2584 ((math-looks-negp (nth 2 math-solve-lhs))
2585 (math-try-solve-for
2586 (list '^ (nth 1 math-solve-lhs)
2587 (math-neg (nth 2 math-solve-lhs)))
2588 (math-div 1 math-solve-rhs)))
2589 ((and (eq math-solve-full t)
2590 (Math-integerp (nth 2 math-solve-lhs))
2591 (math-known-realp (nth 1 math-solve-lhs)))
2592 (setq math-t1 (math-normalize
2593 (list 'calcFunc-nroot math-solve-rhs
2594 (nth 2 math-solve-lhs))))
2595 (if (math-evenp (nth 2 math-solve-lhs))
2596 (setq math-t1 (math-solve-get-sign math-t1)))
2597 (math-try-solve-for
2598 (nth 1 math-solve-lhs) math-t1
2599 (and math-try-solve-sign
2600 (math-oddp (nth 2 math-solve-lhs))
2601 (math-solve-sign math-try-solve-sign
2602 (nth 2 math-solve-lhs)))))
2603 (t (math-try-solve-for
2604 (nth 1 math-solve-lhs)
2605 (math-mul
2606 (math-normalize
2608 (if (Math-realp (nth 2 math-solve-lhs))
2609 (math-div (math-mul
2611 (math-solve-get-int
2613 (and (integerp (nth 2 math-solve-lhs))
2614 (math-abs
2615 (nth 2 math-solve-lhs)))))
2616 (math-div (nth 2 math-solve-lhs) 2))
2617 (math-div (math-mul
2619 (math-mul
2621 (math-solve-get-int
2623 (and (integerp (nth 2 math-solve-lhs))
2624 (math-abs
2625 (nth 2 math-solve-lhs))))))
2626 (nth 2 math-solve-lhs)))))
2627 (math-normalize
2629 math-solve-rhs
2630 (nth 2 math-solve-lhs))))
2631 (and math-try-solve-sign
2632 (math-oddp (nth 2 math-solve-lhs))
2633 (math-solve-sign math-try-solve-sign
2634 (nth 2 math-solve-lhs)))))))))
2637 (defun math-solve-prod (lsoln rsoln)
2642 ((eq math-solve-full 'all)
2644 (math-solve-full
2646 (list 'calcFunc-gt (math-solve-get-sign 1) 0)
2652 ;; The variable math-solve-b is local to math-decompose-poly,
2653 ;; but is used by math-solve-poly-funny-powers.
2654 (defvar math-solve-b)
2656 (defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2"
2657 (setq math-t1 math-solve-lhs)
2658 (let ((pp math-poly-neg-powers)
2661 (setq fac (math-pow (car pp) (or math-poly-mult-powers 1))
2662 math-t1 (math-mul math-t1 fac)
2663 math-solve-rhs (math-mul math-solve-rhs fac)
2665 (if sub-rhs (setq math-t1 (math-sub math-t1 math-solve-rhs)))
2666 (let ((math-poly-neg-powers nil))
2667 (setq math-t2 (math-mul (or math-poly-mult-powers 1)
2669 (math-div 1 math-poly-frac-powers)))
2670 math-t1 (math-is-polynomial
2671 (math-simplify (calcFunc-expand math-t1)) math-solve-b 50))))
2674 (defun math-solve-crunch-poly (max-degree) ; uses "t1", "t3"
2676 (while (and math-t1 (Math-zerop (car math-t1)))
2677 (setq math-t1 (cdr math-t1)
2679 (and math-t1
2680 (let* ((degree (1- (length math-t1)))
2682 (while (and (> scale 1) (= (car math-t3) 1))
2684 (let ((p math-t1)
2696 (setq math-t3 (cons scale (cdr math-t3))
2697 math-t1 new-t1))))
2699 (setq math-t3 (list (math-mul (car math-t3) math-t2)
2700 (math-mul count math-t2)))
2701 (<= (1- (length math-t1)) max-degree)))))
2705 (or (natnump degree) (math-reject-arg degree 'fixnatnump))
2707 (let ((p (math-is-polynomial expr var degree 'gen)))
2712 (math-reject-arg expr "Expected a polynomial"))))
2716 (or (natnump degree) (math-reject-arg degree 'fixnatnump))
2718 (let* ((math-poly-base-variable var)
2719 (d (math-decompose-poly expr var degree nil)))
2722 (math-reject-arg expr "Expected a polynomial"))))
2724 (defun math-decompose-poly (math-solve-lhs math-solve-var degree sub-rhs)
2725 (let ((math-solve-rhs (or sub-rhs 1))
2726 math-t1 math-t2 math-t3)
2727 (setq math-t2 (math-polynomial-base
2728 math-solve-lhs
2730 (lambda (math-solve-b)
2731 (let ((math-poly-neg-powers '(1))
2732 (math-poly-mult-powers nil)
2733 (math-poly-frac-powers 1)
2734 (math-poly-exp-base t))
2735 (and (not (equal math-solve-b math-solve-lhs))
2736 (or (not (memq (car-safe math-solve-b) '(+ -))) sub-rhs)
2737 (setq math-t3 '(1 0) math-t2 1
2738 math-t1 (math-is-polynomial math-solve-lhs
2739 math-solve-b 50))
2740 (if (and (equal math-poly-neg-powers '(1))
2741 (memq math-poly-mult-powers '(nil 1))
2742 (eq math-poly-frac-powers 1)
2743 sub-rhs)
2744 (setq math-t1 (cons (math-sub (car math-t1) math-solve-rhs)
2745 (cdr math-t1)))
2746 (math-solve-poly-funny-powers sub-rhs))
2747 (math-solve-crunch-poly degree)
2748 (or (math-expr-contains math-solve-b math-solve-var)
2749 (math-expr-contains (car math-t3) math-solve-var))))))))
2750 (if math-t2
2751 (list (math-pow math-t2 (car math-t3))
2752 (cons 'vec math-t1)
2753 (if sub-rhs
2754 (math-pow math-t2 (nth 1 math-t3))
2755 (math-div (math-pow math-t2 (nth 1 math-t3)) math-solve-rhs))))))
2757 (defun math-solve-linear (var sign b a)
2758 (math-try-solve-for var
2759 (math-div (math-neg b) a)
2760 (math-solve-sign sign a)
2763 (defun math-solve-quadratic (var c b a)
2764 (math-try-solve-for
2766 (if (math-looks-evenp b)
2767 (let ((halfb (math-div b 2)))
2768 (math-div
2769 (math-add
2770 (math-neg halfb)
2771 (math-solve-get-sign
2772 (math-normalize
2774 (math-add (math-sqr halfb)
2775 (math-mul (math-neg c) a))))))
2777 (math-div
2778 (math-add
2779 (math-neg b)
2780 (math-solve-get-sign
2781 (math-normalize
2783 (math-add (math-sqr b)
2784 (math-mul 4 (math-mul (math-neg c) a)))))))
2785 (math-mul 2 a)))
2788 (defun math-solve-cubic (var d c b a)
2789 (let* ((p (math-div b a))
2790 (q (math-div c a))
2791 (r (math-div d a))
2792 (psqr (math-sqr p))
2793 (aa (math-sub q (math-div psqr 3)))
2794 (bb (math-add r
2795 (math-div (math-sub (math-mul 2 (math-mul psqr p))
2796 (math-mul 9 (math-mul p q)))
2800 (math-try-solve-for (math-pow (math-add var (math-div p 3)) 3)
2801 (math-neg bb) nil t)
2803 (math-try-solve-for
2804 (math-mul (math-add var (math-div p 3))
2805 (math-add (math-sqr (math-add var (math-div p 3)))
2808 (setq m (math-mul 2 (list 'calcFunc-sqrt (math-div aa -3))))
2809 (math-try-solve-for
2811 (math-sub
2812 (math-normalize
2813 (math-mul
2816 (math-div
2817 (math-sub (list 'calcFunc-arccos
2818 (math-div (math-mul 3 bb)
2819 (math-mul aa m)))
2820 (math-mul 2
2821 (math-mul
2822 (math-add 1 (math-solve-get-int
2824 (math-half-circle
2827 (math-div p 3))
2830 (defun math-solve-quartic (var d c b a aa)
2831 (setq a (math-div a aa))
2832 (setq b (math-div b aa))
2833 (setq c (math-div c aa))
2834 (setq d (math-div d aa))
2835 (math-try-solve-for
2837 (let* ((asqr (math-sqr a))
2838 (asqr4 (math-div asqr 4))
2839 (y (let ((math-solve-full nil)
2841 (math-solve-cubic math-solve-var
2842 (math-sub (math-sub
2843 (math-mul 4 (math-mul b d))
2844 (math-mul asqr d))
2845 (math-sqr c))
2846 (math-sub (math-mul a c)
2847 (math-mul 4 d))
2848 (math-neg b)
2850 (rsqr (math-add (math-sub asqr4 b) y))
2852 (sign1 (math-solve-get-sign 1))
2854 (math-add
2855 (math-sub (math-mul 3 asqr4)
2856 (math-mul 2 b))
2858 (math-mul
2860 (math-mul sign1
2862 (math-sub (math-sqr y)
2863 (math-mul 4 d)))))
2864 (math-sub
2865 (math-mul sign1
2866 (math-div
2867 (math-sub (math-sub
2868 (math-mul 4 (math-mul a b))
2869 (math-mul 8 c))
2870 (math-mul asqr a))
2871 (math-mul 4 r)))
2873 (math-normalize
2874 (math-sub (math-add (math-mul sign1 (math-div r 2))
2875 (math-solve-get-sign (math-div de 2)))
2876 (math-div a 4))))
2879 (defvar math-symbolic-solve nil)
2880 (defvar math-int-coefs nil)
2882 ;; The variable math-int-threshold is local to math-poly-all-roots,
2883 ;; but is used by math-poly-newton-root.
2884 (defvar math-int-threshold)
2885 ;; The variables math-int-scale, math-int-factors and math-double-roots
2886 ;; are local to math-poly-all-roots, but are used by math-poly-integer-root.
2887 (defvar math-int-scale)
2888 (defvar math-int-factors)
2889 (defvar math-double-roots)
2891 (defun math-poly-all-roots (var p &optional math-factoring)
2893 (let* ((math-symbolic-solve calc-symbolic-mode)
2897 (math-int-coefs nil)
2898 (math-int-scale nil)
2899 (math-double-roots nil)
2900 (math-int-factors nil)
2901 (math-int-threshold nil)
2907 (if (or math-factoring math-symbolic-solve)
2911 (scale (apply 'math-lcm-denoms p)))
2912 (setq math-int-scale (math-abs (math-mul scale lead))
2913 math-int-threshold (math-div '(float 5 -2) math-int-scale)
2914 math-int-coefs (cdr (math-div (cons 'vec orig-p) lead)))))
2924 (while (> deg (if math-symbolic-solve 2 4))
2925 (let* ((x (math-poly-any-root def-p '(float 0 0) nil))
2928 (math-nearly-zerop (nth 2 x) (nth 1 x)))
2930 (or math-factoring
2932 (or (math-numberp x)
2933 (setq x (math-evaluate-expr x)))
2939 (setq b (math-add (math-mul x b) c)))
2944 (let ((math-solve-var '(var DUMMY var-DUMMY))
2945 (math-solve-sign nil)
2946 (math-solve-ranges nil)
2947 (math-solve-full 'all))
2948 (if (= (length p) (length math-int-coefs))
2949 (setq p (reverse math-int-coefs)))
2951 'math-solve-quadratic)
2953 'math-solve-cubic)
2955 'math-solve-quartic))
2956 math-solve-var p))
2959 (setq roots (cons (math-div (math-neg (car p)) (nth 1 p))
2961 (if math-factoring
2964 (math-poly-integer-root (car roots))
2966 (list math-int-factors (nreverse math-int-coefs) math-int-scale))
2970 (math-solve-full (and math-solve-full 'all)))
2971 (if (math-floatp root)
2972 (setq root (math-poly-any-root orig-p root t)))
2974 (cdr (or (math-try-solve-for var root nil t)
2978 (if math-symbolic-solve
2979 (setq vec (math-normalize vec)))
2980 (if (eq math-solve-full t)
2983 (math-solve-get-int 1 (1- (length orig-p)) 1))
2986 (defun math-lcm-denoms (&rest fracs)
2994 (defun math-poly-any-root (p x polish) ; p is a reverse poly coeff list
2995 (let* ((newt (if (math-zerop x)
2996 (math-poly-newton-root
2998 (math-poly-newton-root p x 4)))
2999 (res (if (math-zerop (cdr newt))
3001 (if (and (math-lessp (cdr newt) '(float 1 -3)) (not polish))
3002 (setq newt (math-poly-newton-root p (car newt) 30)))
3003 (if (math-zerop (cdr newt))
3005 (math-poly-laguerre-root p x polish)))))
3006 (and math-symbolic-solve (math-floatp res)
3010 (defun math-poly-newton-root (p x iters)
3013 (try-integer math-int-coefs)
3017 (math-working "newton" x)
3021 (setq d (math-add (math-mul x d) b)
3022 b (math-add (math-mul x b) (car pp))))
3023 (not (math-zerop d)))
3025 (setq dx (math-div b d)
3026 x (math-sub x dx))
3028 (let ((adx (math-abs-approx dx)))
3029 (and (math-lessp adx math-int-threshold)
3030 (let ((iroot (math-poly-integer-root x)))
3035 (math-nearly-zerop dx (math-abs-approx x))))
3037 (cons x (if (math-zerop x)
3038 1 (math-div (math-abs-approx dx) (math-abs-approx x))))))
3040 (defun math-poly-integer-root (x)
3041 (and (math-lessp (calcFunc-xpon (math-abs-approx x)) calc-internal-prec)
3042 math-int-coefs
3046 (xresq (math-sqr xre))
3047 (ximsq (math-sqr xim)))
3048 (if (math-lessp ximsq (calcFunc-scf xresq -1))
3050 (let* ((rnd (math-div (math-round (math-mul xre math-int-scale))
3051 math-int-scale))
3052 (icp math-int-coefs)
3057 rem (math-add (car icp)
3058 (math-mul rem rnd))))
3059 (and (math-zerop rem)
3061 (setq math-int-coefs (nreverse newcoef)
3062 math-int-factors (cons (list (math-neg rnd))
3063 math-int-factors))
3066 (let* ((rnd1 (math-div (math-round
3067 (math-mul xre (math-mul -2 math-int-scale)))
3068 math-int-scale))
3069 (sqscale (math-sqr math-int-scale))
3070 (rnd0 (math-div (math-round (math-mul (math-add xresq ximsq)
3073 (rem1 (car math-int-coefs))
3074 (icp (cdr math-int-coefs))
3077 (found (assoc (list rnd0 rnd1 (math-posp xim))
3078 math-double-roots))
3081 (setq math-double-roots (delq found math-double-roots)
3086 rem1 (math-sub rem0 (math-mul this rnd1))
3087 rem0 (math-sub (car icp) (math-mul this rnd0)))))
3088 (and (math-zerop rem0)
3089 (math-zerop rem1)
3090 (let ((aa (math-div rnd1 -2)))
3091 (or found (setq math-int-coefs (reverse newcoef)
3092 math-double-roots (cons (list
3095 (math-negp xim)))
3096 math-double-roots)
3097 math-int-factors (cons (cons rnd0 rnd1)
3098 math-int-factors)))
3099 (math-add aa
3100 (let ((calc-symbolic-mode math-symbolic-solve))
3101 (math-mul (math-sqrt (math-sub (math-sqr aa)
3103 (if (math-negp xim) -1 1)))))))))))
3106 (defun math-poly-laguerre-root (p x polish)
3116 (math-reject-arg x "*Laguerre's method failed to converge"))
3117 (let ((err (math-abs-approx (car p)))
3118 (abx (math-abs-approx x))
3123 (setq f (math-add (math-mul x f) d)
3124 d (math-add (math-mul x d) b)
3125 b (math-add (math-mul x b) (car pp))
3126 err (math-add (math-abs-approx b) (math-mul abx err))))
3127 (math-lessp (calcFunc-scf err (- -2 calc-internal-prec))
3128 (math-abs-approx b)))
3129 (or (not (math-zerop d))
3130 (not (math-zerop f))
3132 (setq x (math-pow (math-neg b) (list 'frac 1 m)))
3134 (let* ((g (math-div d b))
3135 (g2 (math-sqr g))
3136 (h (math-sub g2 (math-mul 2 (math-div f b))))
3137 (sq (math-sqrt
3138 (math-mul (1- m) (math-sub (math-mul m h) g2))))
3139 (gp (math-add g sq))
3140 (gm (math-sub g sq)))
3141 (if (math-lessp (calcFunc-abssqr gp) (calcFunc-abssqr gm))
3143 (setq dx (math-div m gp)
3144 x1 (math-sub x dx))
3146 (math-lessp (math-abs-approx dx)
3147 (calcFunc-scf (math-abs-approx x) -3)))
3148 (let ((newt (math-poly-newton-root p x1 7)))
3151 (if (math-zerop (cdr newt))
3153 (if (math-lessp (cdr newt) '(float 1 -6))
3154 (let ((newt2 (math-poly-newton-root
3156 (if (math-zerop (cdr newt2))
3160 (math-nearly-equal x x1))))
3161 (let ((cdx (math-abs-approx dx)))
3166 (math-lessp cdx dxold)
3170 (math-div (math-abs-approx x) cdx))))
3175 (setq x (math-normalize x))))))
3179 (math-lessp (calcFunc-scf (math-abs-approx x)
3182 (or (and (math-floatp x)
3183 (math-poly-integer-root x))
3186 (defun math-solve-above-dummy (x)
3193 (not (setq res (math-solve-above-dummy (car x))))))
3196 (defun math-solve-find-root-term (x neg) ; sets "t2", "t3"
3197 (if (math-solve-find-root-in-prod x)
3198 (setq math-t3 neg
3199 math-t1 x)
3201 (or (math-solve-find-root-term (nth 1 x) neg)
3202 (math-solve-find-root-term (nth 2 x)
3205 (defun math-solve-find-root-in-prod (x)
3207 (math-expr-contains x math-solve-var)
3209 (setq math-t2 2))
3211 (or (and (memq (math-quarter-integer (nth 2 x)) '(1 2 3))
3212 (setq math-t2 2))
3215 (setq math-t2 3))))
3217 (or (and (not (math-expr-contains (nth 1 x) math-solve-var))
3218 (math-solve-find-root-in-prod (nth 2 x)))
3219 (and (not (math-expr-contains (nth 2 x) math-solve-var))
3220 (math-solve-find-root-in-prod (nth 1 x))))))))
3222 ;; The variable math-solve-vars is local to math-solve-system,
3223 ;; but is used by math-solve-system-rec.
3224 (defvar math-solve-vars)
3226 ;; The variable math-solve-simplifying is local to math-solve-system
3227 ;; and math-solve-system-rec, but is used by math-solve-system-subst.
3228 (defvar math-solve-simplifying)
3230 (defun math-solve-system (exprs math-solve-vars math-solve-full)
3234 math-solve-vars (if (Math-vectorp math-solve-vars)
3235 (cdr math-solve-vars)
3236 (list math-solve-vars)))
3237 (or (let ((math-solve-simplifying nil))
3238 (math-solve-system-rec exprs math-solve-vars nil))
3239 (let ((math-solve-simplifying t))
3240 (math-solve-system-rec exprs math-solve-vars nil))))
3253 ;; The variables math-solve-system-res and math-solve-system-vv are
3254 ;; local to math-solve-system-rec, but are used by math-solve-system-subst.
3255 (defvar math-solve-system-vv)
3256 (defvar math-solve-system-res)
3259 (defun math-solve-system-rec (eqn-list var-list solns)
3262 (math-solve-system-res nil))
3268 (let* ((math-solve-system-vv (car v))
3270 (elim (eq (car-safe math-solve-system-vv) 'calcFunc-elim)))
3272 (setq math-solve-system-vv (nth 1 math-solve-system-vv)))
3281 (setq math-solve-system-res nil)
3283 ;; Try to solve for math-solve-system-vv the list of equations e2.
3287 (math-solve-for (car e2) 0
3288 math-solve-system-vv
3289 math-solve-full))))
3291 math-solve-system-res (cons (if (eq math-solve-full 'all)
3294 math-solve-system-res)
3297 (setq math-solve-system-res nil)
3300 (setq math-solve-system-res (nreverse math-solve-system-res)
3301 math-solve-system-res (math-solve-system-rec
3303 'math-solve-system-subst
3307 (let ((math-solve-simplifying nil)
3313 (math-solve-system-subst
3319 math-solve-system-vv
3320 (apply 'append math-solve-system-res))
3322 (not math-solve-system-res))))
3324 (not math-solve-system-res)))
3326 math-solve-system-res)
3332 (not (memq (car x) (memq (car y) math-solve-vars)))))))
3333 (if (eq math-solve-full 'all)
3334 (math-transpose
3335 (math-normalize
3340 (math-normalize
3346 (defun math-solve-system-subst (x) ; uses "res" and "v"
3348 (res2 math-solve-system-res))
3353 (if math-solve-simplifying
3354 (math-simplify
3355 (math-expr-subst
3356 (car x) math-solve-system-vv r))
3357 (math-expr-subst
3358 (car x) math-solve-system-vv r))))
3368 (defun math-get-from-counter (name)
3378 (defun math-solve-get-sign (val)
3379 (setq val (math-simplify val))
3382 (list '* (nth 1 val) (math-solve-get-sign (nth 2 val)))
3385 (setq val (math-normalize (list '^
3387 (math-div (nth 2 (nth 1 val)) 2)))))
3388 (if math-solve-full
3391 (not (eq math-solve-full 'all)))
3393 (math-mul (list 'calcFunc-as var-GenCount) val)
3394 (setq var-GenCount (math-add var-GenCount 1))
3396 (let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign))))
3398 (if (eq math-solve-full 'all)
3399 (setq math-solve-ranges (cons (list var2 1 -1)
3400 math-solve-ranges)))
3401 (math-mul var2 val)))
3405 (defun math-solve-get-int (val &optional range first)
3406 (if math-solve-full
3409 (not (eq math-solve-full 'all)))
3411 (math-mul val (list 'calcFunc-an var-GenCount))
3412 (setq var-GenCount (math-add var-GenCount 1))
3415 (math-get-from-counter 'solve-int))))
3417 (if (and range (eq math-solve-full 'all))
3418 (setq math-solve-ranges (cons (cons var2
3421 math-solve-ranges)))
3422 (math-mul val var2)))
3426 (defun math-solve-sign (sign expr)
3428 (let ((s1 (math-possible-signs expr)))
3434 (defun math-looks-evenp (expr)
3436 (math-evenp expr)
3438 (math-looks-evenp (nth 1 expr)))))
3440 (defun math-solve-for (lhs rhs math-solve-var math-solve-full &optional sign)
3441 (if (math-expr-contains rhs math-solve-var)
3442 (math-solve-for (math-sub lhs rhs) 0 math-solve-var math-solve-full)
3443 (and (math-expr-contains lhs math-solve-var)
3444 (math-with-extra-prec 1
3445 (let* ((math-poly-base-variable math-solve-var)
3446 (res (math-try-solve-for lhs rhs sign)))
3447 (if (and (eq math-solve-full 'all)
3448 (math-known-realp math-solve-var))
3467 (defun math-solve-eqn (expr var full)
3470 (let ((res (math-solve-for (cons '- (cdr expr))
3474 (if (eq math-solve-sign 1)
3476 (if (eq math-solve-sign -1)
3483 (let ((res (math-solve-for expr 0 var full)))
3487 (defun math-reject-solution (expr var func)
3488 (if (math-expr-contains expr var)
3495 (math-solve-system expr var nil)
3496 (math-solve-eqn expr var nil))
3497 (math-reject-solution expr var 'calcFunc-solve)))
3501 (math-solve-system expr var t)
3502 (math-solve-eqn expr var t))
3503 (math-reject-solution expr var 'calcFunc-fsolve)))
3506 (let ((math-solve-ranges nil))
3508 (math-solve-system expr var 'all)
3509 (math-solve-for expr 0 var 'all))
3510 (math-reject-solution expr var 'calcFunc-roots))))
3513 (let ((res (math-solve-for expr math-integ-var var nil)))
3515 (math-normalize (math-expr-subst res math-integ-var var))
3516 (math-reject-solution expr var 'calcFunc-finv))))
3519 (let ((res (math-solve-for expr math-integ-var var t)))
3521 (math-normalize (math-expr-subst res math-integ-var var))
3522 (math-reject-solution expr var 'calcFunc-finv))))
3525 (put 'calcFunc-inv 'math-inverse
3526 (function (lambda (x) (math-div 1 x))))
3527 (put 'calcFunc-inv 'math-inverse-sign -1)
3529 (put 'calcFunc-sqrt 'math-inverse
3530 (function (lambda (x) (math-sqr x))))
3532 (put 'calcFunc-conj 'math-inverse
3535 (put 'calcFunc-abs 'math-inverse
3536 (function (lambda (x) (math-solve-get-sign x))))
3538 (put 'calcFunc-deg 'math-inverse
3540 (put 'calcFunc-deg 'math-inverse-sign 1)
3542 (put 'calcFunc-rad 'math-inverse
3544 (put 'calcFunc-rad 'math-inverse-sign 1)
3546 (put 'calcFunc-ln 'math-inverse
3548 (put 'calcFunc-ln 'math-inverse-sign 1)
3550 (put 'calcFunc-log10 'math-inverse
3552 (put 'calcFunc-log10 'math-inverse-sign 1)
3554 (put 'calcFunc-lnp1 'math-inverse
3556 (put 'calcFunc-lnp1 'math-inverse-sign 1)
3558 (put 'calcFunc-exp 'math-inverse
3559 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x))
3560 (math-mul 2
3561 (math-mul '(var pi var-pi)
3562 (math-solve-get-int
3564 (put 'calcFunc-exp 'math-inverse-sign 1)
3566 (put 'calcFunc-expm1 'math-inverse
3567 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x))
3568 (math-mul 2
3569 (math-mul '(var pi var-pi)
3570 (math-solve-get-int
3572 (put 'calcFunc-expm1 'math-inverse-sign 1)
3574 (put 'calcFunc-sin 'math-inverse
3575 (function (lambda (x) (let ((n (math-solve-get-int 1)))
3576 (math-add (math-mul (math-normalize
3578 (math-pow -1 n))
3579 (math-mul (math-half-circle t)
3582 (put 'calcFunc-cos 'math-inverse
3583 (function (lambda (x) (math-add (math-solve-get-sign
3584 (math-normalize
3586 (math-solve-get-int
3587 (math-full-circle t))))))
3589 (put 'calcFunc-tan 'math-inverse
3590 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x))
3591 (math-solve-get-int
3592 (math-half-circle t))))))
3594 (put 'calcFunc-arcsin 'math-inverse
3595 (function (lambda (x) (math-normalize (list 'calcFunc-sin x)))))
3597 (put 'calcFunc-arccos 'math-inverse
3598 (function (lambda (x) (math-normalize (list 'calcFunc-cos x)))))
3600 (put 'calcFunc-arctan 'math-inverse
3601 (function (lambda (x) (math-normalize (list 'calcFunc-tan x)))))
3603 (put 'calcFunc-sinh 'math-inverse
3604 (function (lambda (x) (let ((n (math-solve-get-int 1)))
3605 (math-add (math-mul (math-normalize
3607 (math-pow -1 n))
3608 (math-mul (math-half-circle t)
3609 (math-mul
3612 (put 'calcFunc-sinh 'math-inverse-sign 1)
3614 (put 'calcFunc-cosh 'math-inverse
3615 (function (lambda (x) (math-add (math-solve-get-sign
3616 (math-normalize
3618 (math-mul (math-full-circle t)
3619 (math-solve-get-int
3622 (put 'calcFunc-tanh 'math-inverse
3623 (function (lambda (x) (math-add (math-normalize
3625 (math-mul (math-half-circle t)
3626 (math-solve-get-int
3628 (put 'calcFunc-tanh 'math-inverse-sign 1)
3630 (put 'calcFunc-arcsinh 'math-inverse
3631 (function (lambda (x) (math-normalize (list 'calcFunc-sinh x)))))
3632 (put 'calcFunc-arcsinh 'math-inverse-sign 1)
3634 (put 'calcFunc-arccosh 'math-inverse
3635 (function (lambda (x) (math-normalize (list 'calcFunc-cosh x)))))
3637 (put 'calcFunc-arctanh 'math-inverse
3638 (function (lambda (x) (math-normalize (list 'calcFunc-tanh x)))))
3639 (put 'calcFunc-arctanh 'math-inverse-sign 1)
3646 (setq x0 (if (eq (car var) '+) (math-neg (nth 2 var)) (nth 2 var))
3649 (math-expr-contains expr v)
3651 (let ((accum (math-expr-subst expr v x0))
3660 (setq fprime (math-simplify fprime)
3661 nfac (math-mul nfac n)
3662 accum (math-add accum
3663 (math-div (math-mul (math-pow var2 n)
3664 (math-expr-subst
3668 (math-normalize accum))))