calcalg2.el 123 KB
Newer Older
1 2
;;; calcalg2.el --- more algebraic functions for Calc

Jay Belanger's avatar
Jay Belanger committed
3
;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2005 Free Software Foundation, Inc.
4 5

;; Author: David Gillespie <daveg@synaptics.com>
6
;; Maintainer: Jay Belanger <belanger@truman.edu>
Eli Zaretskii's avatar
Eli Zaretskii committed
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

;; This file is part of GNU Emacs.

;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY.  No author or distributor
;; accepts responsibility to anyone for the consequences of using it
;; or for whether it serves any particular purpose or works at all,
;; unless he says so in writing.  Refer to the GNU Emacs General Public
;; License for full details.

;; Everyone is granted permission to copy, modify and redistribute
;; GNU Emacs, but only under the conditions described in the
;; GNU Emacs General Public License.   A copy of this license is
;; supposed to have been given to you along with GNU Emacs so you
;; can know your rights and responsibilities.  It should be in a
;; file named COPYING.  Among other things, the copyright notice
;; and this notice must be preserved on all copies.

25
;;; Commentary:
Eli Zaretskii's avatar
Eli Zaretskii committed
26

27
;;; Code:
Eli Zaretskii's avatar
Eli Zaretskii committed
28 29 30

;; This file is autoloaded from calc-ext.el.

Jay Belanger's avatar
Jay Belanger committed
31
(require 'calc-ext)
Eli Zaretskii's avatar
Eli Zaretskii committed
32 33 34 35 36
(require 'calc-macs)

(defun calc-derivative (var num)
  (interactive "sDifferentiate with respect to: \np")
  (calc-slow-wrapper
37 38
   (when (< num 0)
     (error "Order of derivative must be positive"))
Eli Zaretskii's avatar
Eli Zaretskii committed
39 40 41 42 43 44 45
   (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv))
	 n expr)
     (if (or (equal var "") (equal var "$"))
	 (setq n 2
	       expr (calc-top-n 2)
	       var (calc-top-n 1))
       (setq var (math-read-expr var))
46 47
       (when (eq (car-safe var) 'error)
	 (error "Bad format in expression: %s" (nth 1 var)))
Eli Zaretskii's avatar
Eli Zaretskii committed
48 49 50 51
       (setq n 1
	     expr (calc-top-n 1)))
     (while (>= (setq num (1- num)) 0)
       (setq expr (list func expr var)))
52
     (calc-enter-result n "derv" expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
53 54 55 56 57 58 59 60 61 62 63 64 65

(defun calc-integral (var)
  (interactive "sIntegration variable: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$"))
       (calc-enter-result 2 "intg" (list 'calcFunc-integ
					 (calc-top-n 2)
					 (calc-top-n 1)))
     (let ((var (math-read-expr var)))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "intg" (list 'calcFunc-integ
					 (calc-top-n 1)
66
					 var))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
67 68 69 70

(defun calc-num-integral (&optional varname lowname highname)
  (interactive "sIntegration variable: ")
  (calc-tabular-command 'calcFunc-ninteg "Integration" "nint"
71
			nil varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
72 73 74 75

(defun calc-summation (arg &optional varname lowname highname)
  (interactive "P\nsSummation variable: ")
  (calc-tabular-command 'calcFunc-sum "Summation" "sum"
76
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
77 78 79 80

(defun calc-alt-summation (arg &optional varname lowname highname)
  (interactive "P\nsSummation variable: ")
  (calc-tabular-command 'calcFunc-asum "Summation" "asum"
81
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
82 83 84 85

(defun calc-product (arg &optional varname lowname highname)
  (interactive "P\nsIndex variable: ")
  (calc-tabular-command 'calcFunc-prod "Index" "prod"
86
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
87 88 89 90

(defun calc-tabulate (arg &optional varname lowname highname)
  (interactive "P\nsIndex variable: ")
  (calc-tabular-command 'calcFunc-table "Index" "tabl"
91
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148

(defun calc-tabular-command (func prompt prefix arg varname lowname highname)
  (calc-slow-wrapper
   (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr)
     (if (consp arg)
	 (setq stepnum 1)
       (setq stepnum 0))
     (if (or (equal varname "") (equal varname "$") (null varname))
	 (setq high (calc-top-n (+ stepnum 1))
	       low (calc-top-n (+ stepnum 2))
	       var (calc-top-n (+ stepnum 3))
	       num (+ stepnum 4))
       (setq var (if (stringp varname) (math-read-expr varname) varname))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (or lowname
	   (setq lowname (read-string (concat prompt " variable: " varname
					      ", from: "))))
       (if (or (equal lowname "") (equal lowname "$"))
	   (setq high (calc-top-n (+ stepnum 1))
		 low (calc-top-n (+ stepnum 2))
		 num (+ stepnum 3))
	 (setq low (if (stringp lowname) (math-read-expr lowname) lowname))
	 (if (eq (car-safe low) 'error)
	     (error "Bad format in expression: %s" (nth 1 low)))
	 (or highname
	     (setq highname (read-string (concat prompt " variable: " varname
						 ", from: " lowname
						 ", to: "))))
	 (if (or (equal highname "") (equal highname "$"))
	     (setq high (calc-top-n (+ stepnum 1))
		   num (+ stepnum 2))
	   (setq high (if (stringp highname) (math-read-expr highname)
			highname))
	   (if (eq (car-safe high) 'error)
	       (error "Bad format in expression: %s" (nth 1 high)))
	   (if (consp arg)
	       (progn
		 (setq stepname (read-string (concat prompt " variable: "
						     varname
						     ", from: " lowname
						     ", to: " highname
						     ", step: ")))
		 (if (or (equal stepname "") (equal stepname "$"))
		     (setq step (calc-top-n 1)
			   num 2)
		   (setq step (math-read-expr stepname))
		   (if (eq (car-safe step) 'error)
		       (error "Bad format in expression: %s"
			      (nth 1 step)))))))))
     (or step
	 (if (consp arg)
	     (setq step (calc-top-n 1))
	   (if arg
	       (setq step (prefix-numeric-value arg)))))
     (setq expr (calc-top-n num))
     (calc-enter-result num prefix (append (list func expr var low high)
149
					   (and step (list step)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
150 151

(defun calc-solve-for (var)
152
  (interactive "sVariable(s) to solve for: ")
Eli Zaretskii's avatar
Eli Zaretskii committed
153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
  (calc-slow-wrapper
   (let ((func (if (calc-is-inverse)
		   (if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv)
		 (if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve))))
     (if (or (equal var "") (equal var "$"))
	 (calc-enter-result 2 "solv" (list func
					   (calc-top-n 2)
					   (calc-top-n 1)))
       (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
			   (not (string-match "\\[" var)))
		      (math-read-expr (concat "[" var "]"))
		    (math-read-expr var))))
	 (if (eq (car-safe var) 'error)
	     (error "Bad format in expression: %s" (nth 1 var)))
	 (calc-enter-result 1 "solv" (list func
					   (calc-top-n 1)
169
					   var)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185

(defun calc-poly-roots (var)
  (interactive "sVariable to solve for: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$"))
       (calc-enter-result 2 "prts" (list 'calcFunc-roots
					 (calc-top-n 2)
					 (calc-top-n 1)))
     (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
			 (not (string-match "\\[" var)))
		    (math-read-expr (concat "[" var "]"))
		  (math-read-expr var))))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "prts" (list 'calcFunc-roots
					 (calc-top-n 1)
186
					 var))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
187 188 189 190 191 192 193 194 195 196

(defun calc-taylor (var nterms)
  (interactive "sTaylor expansion variable: \nNNumber of terms: ")
  (calc-slow-wrapper
   (let ((var (math-read-expr var)))
     (if (eq (car-safe var) 'error)
	 (error "Bad format in expression: %s" (nth 1 var)))
     (calc-enter-result 1 "tylr" (list 'calcFunc-taylor
				       (calc-top-n 1)
				       var
197
				       (prefix-numeric-value nterms))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
198 199


200 201 202 203 204 205 206 207
;; The following are global variables used by math-derivative and some 
;; related functions
(defvar math-deriv-var)
(defvar math-deriv-total)
(defvar math-deriv-symb)

(defun math-derivative (expr)
  (cond ((equal expr math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
208 209 210 211
	 1)
	((or (Math-scalarp expr)
	     (eq (car expr) 'sdev)
	     (and (eq (car expr) 'var)
212
		  (or (not math-deriv-total)
Eli Zaretskii's avatar
Eli Zaretskii committed
213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283
		      (math-const-var expr)
		      (progn
			(math-setup-declarations)
			(memq 'const (nth 1 (or (assq (nth 2 expr)
						      math-decls-cache)
						math-decls-all)))))))
	 0)
	((eq (car expr) '+)
	 (math-add (math-derivative (nth 1 expr))
		   (math-derivative (nth 2 expr))))
	((eq (car expr) '-)
	 (math-sub (math-derivative (nth 1 expr))
		   (math-derivative (nth 2 expr))))
	((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt
					calcFunc-gt calcFunc-leq calcFunc-geq))
	 (list (car expr)
	       (math-derivative (nth 1 expr))
	       (math-derivative (nth 2 expr))))
	((eq (car expr) 'neg)
	 (math-neg (math-derivative (nth 1 expr))))
	((eq (car expr) '*)
	 (math-add (math-mul (nth 2 expr)
			     (math-derivative (nth 1 expr)))
		   (math-mul (nth 1 expr)
			     (math-derivative (nth 2 expr)))))
	((eq (car expr) '/)
	 (math-sub (math-div (math-derivative (nth 1 expr))
			     (nth 2 expr))
		   (math-div (math-mul (nth 1 expr)
				       (math-derivative (nth 2 expr)))
			     (math-sqr (nth 2 expr)))))
	((eq (car expr) '^)
	 (let ((du (math-derivative (nth 1 expr)))
	       (dv (math-derivative (nth 2 expr))))
	   (or (Math-zerop du)
	       (setq du (math-mul (nth 2 expr)
				  (math-mul (math-normalize
					     (list '^
						   (nth 1 expr)
						   (math-add (nth 2 expr) -1)))
					    du))))
	   (or (Math-zerop dv)
	       (setq dv (math-mul (math-normalize
				   (list 'calcFunc-ln (nth 1 expr)))
				  (math-mul expr dv))))
	   (math-add du dv)))
	((eq (car expr) '%)
	 (math-derivative (nth 1 expr)))   ; a reasonable definition
	((eq (car expr) 'vec)
	 (math-map-vec 'math-derivative expr))
	((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im))
	      (= (length expr) 2))
	 (list (car expr) (math-derivative (nth 1 expr))))
	((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol))
	      (= (length expr) 3))
	 (let ((d (math-derivative (nth 1 expr))))
	   (if (math-numberp d)
	       0    ; assume x and x_1 are independent vars
	     (list (car expr) d (nth 2 expr)))))
	(t (or (and (symbolp (car expr))
		    (if (= (length expr) 2)
			(let ((handler (get (car expr) 'math-derivative)))
			  (and handler
			       (let ((deriv (math-derivative (nth 1 expr))))
				 (if (Math-zerop deriv)
				     deriv
				   (math-mul (funcall handler (nth 1 expr))
					     deriv)))))
		      (let ((handler (get (car expr) 'math-derivative-n)))
			(and handler
			     (funcall handler expr)))))
284
	       (and (not (eq math-deriv-symb 'pre-expand))
Eli Zaretskii's avatar
Eli Zaretskii committed
285 286
		    (let ((exp (math-expand-formula expr)))
		      (and exp
287
			   (or (let ((math-deriv-symb 'pre-expand))
Eli Zaretskii's avatar
Eli Zaretskii committed
288 289 290 291 292
				 (catch 'math-deriv (math-derivative expr)))
			       (math-derivative exp)))))
	       (if (or (Math-objvecp expr)
		       (eq (car expr) 'var)
		       (not (symbolp (car expr))))
293
		   (if math-deriv-symb
Eli Zaretskii's avatar
Eli Zaretskii committed
294
		       (throw 'math-deriv nil)
295
		     (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv)
Eli Zaretskii's avatar
Eli Zaretskii committed
296
			   expr
297
			   math-deriv-var))
Eli Zaretskii's avatar
Eli Zaretskii committed
298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326
		 (let ((accum 0)
		       (arg expr)
		       (n 1)
		       derv)
		   (while (setq arg (cdr arg))
		     (or (Math-zerop (setq derv (math-derivative (car arg))))
			 (let ((func (intern (concat (symbol-name (car expr))
						     "'"
						     (if (> n 1)
							 (int-to-string n)
						       ""))))
			       (prop (cond ((= (length expr) 2)
					    'math-derivative-1)
					   ((= (length expr) 3)
					    'math-derivative-2)
					   ((= (length expr) 4)
					    'math-derivative-3)
					   ((= (length expr) 5)
					    'math-derivative-4)
					   ((= (length expr) 6)
					    'math-derivative-5))))
			   (setq accum
				 (math-add
				  accum
				  (math-mul
				   derv
				   (let ((handler (get func prop)))
				     (or (and prop handler
					      (apply handler (cdr expr)))
327
					 (if (and math-deriv-symb
Eli Zaretskii's avatar
Eli Zaretskii committed
328 329 330 331 332
						  (not (get func
							    'calc-user-defn)))
					     (throw 'math-deriv nil)
					   (cons func (cdr expr))))))))))
		     (setq n (1+ n)))
333
		   accum))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
334

335 336
(defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb)
  (let* ((math-deriv-total nil)
Eli Zaretskii's avatar
Eli Zaretskii committed
337 338 339 340 341 342
	 (res (catch 'math-deriv (math-derivative expr))))
    (or (eq (car-safe res) 'calcFunc-deriv)
	(null res)
	(setq res (math-normalize res)))
    (and res
	 (if deriv-value
343
	     (math-expr-subst res math-deriv-var deriv-value)
344
	   res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
345

346
(defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb)
Eli Zaretskii's avatar
Eli Zaretskii committed
347
  (math-setup-declarations)
348
  (let* ((math-deriv-total t)
Eli Zaretskii's avatar
Eli Zaretskii committed
349 350 351 352 353 354
	 (res (catch 'math-deriv (math-derivative expr))))
    (or (eq (car-safe res) 'calcFunc-tderiv)
	(null res)
	(setq res (math-normalize res)))
    (and res
	 (if deriv-value
355
	     (math-expr-subst res math-deriv-var deriv-value)
356
	   res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410

(put 'calcFunc-inv\' 'math-derivative-1
     (function (lambda (u) (math-neg (math-div 1 (math-sqr u))))))

(put 'calcFunc-sqrt\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u))))))

(put 'calcFunc-deg\' 'math-derivative-1
     (function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))

(put 'calcFunc-rad\' 'math-derivative-1
     (function (lambda (u) (math-pi-over-180))))

(put 'calcFunc-ln\' 'math-derivative-1
     (function (lambda (u) (math-div 1 u))))

(put 'calcFunc-log10\' 'math-derivative-1
     (function (lambda (u)
		 (math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
			   u))))

(put 'calcFunc-lnp1\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-add u 1)))))

(put 'calcFunc-log\' 'math-derivative-2
     (function (lambda (x b)
		 (and (not (Math-zerop b))
		      (let ((lnv (math-normalize
				  (list 'calcFunc-ln b))))
			(math-div 1 (math-mul lnv x)))))))

(put 'calcFunc-log\'2 'math-derivative-2
     (function (lambda (x b)
		 (let ((lnv (list 'calcFunc-ln b)))
		   (math-neg (math-div (list 'calcFunc-log x b)
				       (math-mul lnv b)))))))

(put 'calcFunc-exp\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-exp u)))))

(put 'calcFunc-expm1\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u)))))

(put 'calcFunc-sin\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2 (math-normalize
					       (list 'calcFunc-cos u))))))

(put 'calcFunc-cos\' 'math-derivative-1
     (function (lambda (u) (math-neg (math-to-radians-2
				      (math-normalize
				       (list 'calcFunc-sin u)))))))

(put 'calcFunc-tan\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2
411 412 413
			    (math-sqr
                             (math-normalize
                              (list 'calcFunc-sec u)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
414

415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434
(put 'calcFunc-sec\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2 
                            (math-mul
                             (math-normalize
                              (list 'calcFunc-sec u))
                             (math-normalize
                              (list 'calcFunc-tan u)))))))

(put 'calcFunc-csc\' 'math-derivative-1
     (function (lambda (u) (math-neg 
                            (math-to-radians-2
                             (math-mul
                              (math-normalize
                               (list 'calcFunc-csc u))
                              (math-normalize
                               (list 'calcFunc-cot u))))))))

(put 'calcFunc-cot\' 'math-derivative-1
     (function (lambda (u) (math-neg
                            (math-to-radians-2
435 436 437
                             (math-sqr
                              (math-normalize
                               (list 'calcFunc-csc u))))))))
438

Eli Zaretskii's avatar
Eli Zaretskii committed
439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463
(put 'calcFunc-arcsin\' 'math-derivative-1
     (function (lambda (u)
		 (math-from-radians-2
		  (math-div 1 (math-normalize
			       (list 'calcFunc-sqrt
				     (math-sub 1 (math-sqr u)))))))))

(put 'calcFunc-arccos\' 'math-derivative-1
     (function (lambda (u)
		 (math-from-radians-2
		  (math-div -1 (math-normalize
				(list 'calcFunc-sqrt
				      (math-sub 1 (math-sqr u)))))))))

(put 'calcFunc-arctan\' 'math-derivative-1
     (function (lambda (u) (math-from-radians-2
			    (math-div 1 (math-add 1 (math-sqr u)))))))

(put 'calcFunc-sinh\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-cosh u)))))

(put 'calcFunc-cosh\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-sinh u)))))

(put 'calcFunc-tanh\' 'math-derivative-1
464 465 466
     (function (lambda (u) (math-sqr
                            (math-normalize
                             (list 'calcFunc-sech u))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
467

468 469 470 471 472 473 474 475 476 477 478 479
(put 'calcFunc-sech\' 'math-derivative-1
     (function (lambda (u) (math-neg
                            (math-mul
                             (math-normalize (list 'calcFunc-sech u))
                             (math-normalize (list 'calcFunc-tanh u)))))))

(put 'calcFunc-csch\' 'math-derivative-1
     (function (lambda (u) (math-neg
                            (math-mul
                             (math-normalize (list 'calcFunc-csch u))
                             (math-normalize (list 'calcFunc-coth u)))))))

480
(put 'calcFunc-coth\' 'math-derivative-1
481
     (function (lambda (u) (math-neg
482 483 484
                            (math-sqr
                             (math-normalize
                              (list 'calcFunc-csch u)))))))
485

Eli Zaretskii's avatar
Eli Zaretskii committed
486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
(put 'calcFunc-arcsinh\' 'math-derivative-1
     (function (lambda (u)
		 (math-div 1 (math-normalize
			      (list 'calcFunc-sqrt
				    (math-add (math-sqr u) 1)))))))

(put 'calcFunc-arccosh\' 'math-derivative-1
     (function (lambda (u)
		  (math-div 1 (math-normalize
			       (list 'calcFunc-sqrt
				     (math-add (math-sqr u) -1)))))))

(put 'calcFunc-arctanh\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u))))))

(put 'calcFunc-bern\'2 'math-derivative-2
     (function (lambda (n x)
		 (math-mul n (list 'calcFunc-bern (math-add n -1) x)))))

(put 'calcFunc-euler\'2 'math-derivative-2
     (function (lambda (n x)
		 (math-mul n (list 'calcFunc-euler (math-add n -1) x)))))

(put 'calcFunc-gammag\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x 1))))

(put 'calcFunc-gammaG\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x -1))))

(put 'calcFunc-gammaP\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x
					       (math-div
						1 (math-normalize
						   (list 'calcFunc-gamma
							 a)))))))

(put 'calcFunc-gammaQ\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x
					       (math-div
						-1 (math-normalize
						    (list 'calcFunc-gamma
							  a)))))))

(defun math-deriv-gamma (a x scale)
  (math-mul scale
	    (math-mul (math-pow x (math-add a -1))
532
		      (list 'calcFunc-exp (math-neg x)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
533 534 535 536 537 538 539 540 541 542 543 544 545

(put 'calcFunc-betaB\' 'math-derivative-3
     (function (lambda (x a b) (math-deriv-beta x a b 1))))

(put 'calcFunc-betaI\' 'math-derivative-3
     (function (lambda (x a b) (math-deriv-beta x a b
						(math-div
						 1 (list 'calcFunc-beta
							 a b))))))

(defun math-deriv-beta (x a b scale)
  (math-mul (math-mul (math-pow x (math-add a -1))
		      (math-pow (math-sub 1 x) (math-add b -1)))
546
	    scale))
Eli Zaretskii's avatar
Eli Zaretskii committed
547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586

(put 'calcFunc-erf\' 'math-derivative-1
     (function (lambda (x) (math-div 2
				     (math-mul (list 'calcFunc-exp
						     (math-sqr x))
					       (if calc-symbolic-mode
						   '(calcFunc-sqrt
						     (var pi var-pi))
						 (math-sqrt-pi)))))))

(put 'calcFunc-erfc\' 'math-derivative-1
     (function (lambda (x) (math-div -2
				     (math-mul (list 'calcFunc-exp
						     (math-sqr x))
					       (if calc-symbolic-mode
						   '(calcFunc-sqrt
						     (var pi var-pi))
						 (math-sqrt-pi)))))))

(put 'calcFunc-besJ\'2 'math-derivative-2
     (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ
						       (math-add v -1)
						       z)
						 (list 'calcFunc-besJ
						       (math-add v 1)
						       z))
				       2))))

(put 'calcFunc-besY\'2 'math-derivative-2
     (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY
						       (math-add v -1)
						       z)
						 (list 'calcFunc-besY
						       (math-add v 1)
						       z))
				       2))))

(put 'calcFunc-sum 'math-derivative-n
     (function
      (lambda (expr)
587
	(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
588 589 590 591 592 593 594 595
	    (throw 'math-deriv nil)
	  (cons 'calcFunc-sum
		(cons (math-derivative (nth 1 expr))
		      (cdr (cdr expr))))))))

(put 'calcFunc-prod 'math-derivative-n
     (function
      (lambda (expr)
596
	(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
597 598 599 600 601 602 603 604 605 606 607
	    (throw 'math-deriv nil)
	  (math-mul expr
		    (cons 'calcFunc-sum
			  (cons (math-div (math-derivative (nth 1 expr))
					  (nth 1 expr))
				(cdr (cdr expr)))))))))

(put 'calcFunc-integ 'math-derivative-n
     (function
      (lambda (expr)
	(if (= (length expr) 3)
608
	    (if (equal (nth 2 expr) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
609 610 611 612 613 614 615 616 617 618 619 620 621 622
		(nth 1 expr)
	      (math-normalize
	       (list 'calcFunc-integ
		     (math-derivative (nth 1 expr))
		     (nth 2 expr))))
	  (if (= (length expr) 5)
	      (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr)
					    (nth 3 expr)))
		    (upper (math-expr-subst (nth 1 expr) (nth 2 expr)
					    (nth 4 expr))))
		(math-add (math-sub (math-mul upper
					      (math-derivative (nth 4 expr)))
				    (math-mul lower
					      (math-derivative (nth 3 expr))))
623
			  (if (equal (nth 2 expr) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645
			      0
			    (math-normalize
			     (list 'calcFunc-integ
				   (math-derivative (nth 1 expr)) (nth 2 expr)
				   (nth 3 expr) (nth 4 expr)))))))))))

(put 'calcFunc-if 'math-derivative-n
     (function
      (lambda (expr)
	(and (= (length expr) 4)
	     (list 'calcFunc-if (nth 1 expr)
		   (math-derivative (nth 2 expr))
		   (math-derivative (nth 3 expr)))))))

(put 'calcFunc-subscr 'math-derivative-n
     (function
      (lambda (expr)
	(and (= (length expr) 3)
	     (list 'calcFunc-subscr (nth 1 expr)
		   (math-derivative (nth 2 expr)))))))


646 647 648 649 650
(defvar math-integ-var '(var X ---))
(defvar math-integ-var-2 '(var Y ---))
(defvar math-integ-vars (list 'f math-integ-var math-integ-var-2))
(defvar math-integ-var-list (list math-integ-var))
(defvar math-integ-var-list-list (list math-integ-var-list))
Eli Zaretskii's avatar
Eli Zaretskii committed
651

652 653 654 655 656 657 658 659 660 661 662 663 664 665 666
;; math-integ-depth is a local variable for math-try-integral, but is used
;; by math-integral and math-tracing-integral
;; which are called (directly or indirectly) by math-try-integral.
(defvar math-integ-depth)
;; math-integ-level is a local variable for math-try-integral, but is used
;; by math-integral, math-do-integral, math-tracing-integral, 
;; math-sub-integration, math-integrate-by-parts and 
;; math-integrate-by-substitution, which are called (directly or 
;; indirectly) by math-try-integral.
(defvar math-integ-level)
;; math-integral-limit is a local variable for calcFunc-integ, but is
;; used by math-tracing-integral, math-sub-integration and 
;; math-try-integration. 
(defvar math-integral-limit)

Eli Zaretskii's avatar
Eli Zaretskii committed
667 668 669 670 671 672 673 674 675 676 677 678 679 680 681
(defmacro math-tracing-integral (&rest parts)
  (list 'and
	'trace-buffer
	(list 'save-excursion
	      '(set-buffer trace-buffer)
	      '(goto-char (point-max))
	      (list 'and
		    '(bolp)
		    '(insert (make-string (- math-integral-limit
					     math-integ-level) 32)
			     (format "%2d " math-integ-depth)
			     (make-string math-integ-level 32)))
	      ;;(list 'condition-case 'err
		    (cons 'insert parts)
		;;    '(error (insert (prin1-to-string err))))
682
	      '(sit-for 0))))
Eli Zaretskii's avatar
Eli Zaretskii committed
683 684 685 686 687 688 689 690

;;; The following wrapper caches results and avoids infinite recursion.
;;; Each cache entry is: ( A B )          Integral of A is B;
;;;			 ( A N )          Integral of A failed at level N;
;;;			 ( A busy )	  Currently working on integral of A;
;;;			 ( A parts )	  Currently working, integ-by-parts;
;;;			 ( A parts2 )	  Currently working, integ-by-parts;
;;;			 ( A cancelled )  Ignore this cache entry;
691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709
;;;			 ( A [B] )        Same result as for math-cur-record = B.

;; math-cur-record is a local variable for math-try-integral, but is used
;; by math-integral, math-replace-integral-parts and math-integrate-by-parts
;; which are called (directly or indirectly) by math-try-integral, as well as
;; by calc-dump-integral-cache
(defvar math-cur-record)
;; math-enable-subst and math-any-substs are local variables for
;; calcFunc-integ, but are used by math-integral and math-try-integral.
(defvar math-enable-subst)
(defvar math-any-substs)

;; math-integ-msg is a local variable for math-try-integral, but is
;; used (both locally and non-locally) by math-integral.
(defvar math-integ-msg)

(defvar math-integral-cache nil)
(defvar math-integral-cache-state nil)

Eli Zaretskii's avatar
Eli Zaretskii committed
710
(defun math-integral (expr &optional simplify same-as-above)
711 712
  (let* ((simp math-cur-record)
	 (math-cur-record (assoc expr math-integral-cache))
Eli Zaretskii's avatar
Eli Zaretskii committed
713 714 715 716 717
	 (math-integ-depth (1+ math-integ-depth))
	 (val 'cancelled))
    (math-tracing-integral "Integrating "
			   (math-format-value expr 1000)
			   "...\n")
718
    (and math-cur-record
Eli Zaretskii's avatar
Eli Zaretskii committed
719 720
	 (progn
	   (math-tracing-integral "Found "
721 722 723
				  (math-format-value (nth 1 math-cur-record) 1000))
	   (and (consp (nth 1 math-cur-record))
		(math-replace-integral-parts math-cur-record))
Eli Zaretskii's avatar
Eli Zaretskii committed
724
	   (math-tracing-integral " => "
725
				  (math-format-value (nth 1 math-cur-record) 1000)
Eli Zaretskii's avatar
Eli Zaretskii committed
726
				  "\n")))
727 728 729 730
    (or (and math-cur-record
	     (not (eq (nth 1 math-cur-record) 'cancelled))
	     (or (not (integerp (nth 1 math-cur-record)))
		 (>= (nth 1 math-cur-record) math-integ-level)))
Eli Zaretskii's avatar
Eli Zaretskii committed
731 732 733 734 735 736 737 738 739 740 741 742 743 744
	(and (math-integral-contains-parts expr)
	     (progn
	       (setq val nil)
	       t))
	(unwind-protect
	    (progn
	      (let (math-integ-msg)
		(if (eq calc-display-working-message 'lots)
		    (progn
		      (calc-set-command-flag 'clear-message)
		      (setq math-integ-msg (format
					    "Working... Integrating %s"
					    (math-format-flat-expr expr 0)))
		      (message math-integ-msg)))
745 746
		(if math-cur-record
		    (setcar (cdr math-cur-record)
Eli Zaretskii's avatar
Eli Zaretskii committed
747
			    (if same-as-above (vector simp) 'busy))
748
		  (setq math-cur-record
Eli Zaretskii's avatar
Eli Zaretskii committed
749
			(list expr (if same-as-above (vector simp) 'busy))
750
			math-integral-cache (cons math-cur-record
Eli Zaretskii's avatar
Eli Zaretskii committed
751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771
						  math-integral-cache)))
		(if (eq simplify 'yes)
		    (progn
		      (math-tracing-integral "Simplifying...")
		      (setq simp (math-simplify expr))
		      (setq val (if (equal simp expr)
				    (progn
				      (math-tracing-integral " no change\n")
				      (math-do-integral expr))
				  (math-tracing-integral " simplified\n")
				  (math-integral simp 'no t))))
		  (or (setq val (math-do-integral expr))
		      (eq simplify 'no)
		      (let ((simp (math-simplify expr)))
			(or (equal simp expr)
			    (progn
			      (math-tracing-integral "Trying again after "
						     "simplification...\n")
			      (setq val (math-integral simp 'no t))))))))
	      (if (eq calc-display-working-message 'lots)
		  (message math-integ-msg)))
772
	  (setcar (cdr math-cur-record) (or val
Eli Zaretskii's avatar
Eli Zaretskii committed
773 774 775 776
				       (if (or math-enable-subst
					       (not math-any-substs))
					   math-integ-level
					 'cancelled)))))
777
    (setq val math-cur-record)
Eli Zaretskii's avatar
Eli Zaretskii committed
778 779 780 781 782 783 784 785 786 787 788 789 790
    (while (vectorp (nth 1 val))
      (setq val (aref (nth 1 val) 0)))
    (setq val (if (memq (nth 1 val) '(parts parts2))
		  (progn
		    (setcar (cdr val) 'parts2)
		    (list 'var 'PARTS val))
		(and (consp (nth 1 val))
		     (nth 1 val))))
    (math-tracing-integral "Integral of "
			   (math-format-value expr 1000)
			   "  is  "
			   (math-format-value val 1000)
			   "\n")
791
    val))
Eli Zaretskii's avatar
Eli Zaretskii committed
792 793 794 795 796 797 798 799

(defun math-integral-contains-parts (expr)
  (if (Math-primp expr)
      (and (eq (car-safe expr) 'var)
	   (eq (nth 1 expr) 'PARTS)
	   (listp (nth 2 expr)))
    (while (and (setq expr (cdr expr))
		(not (math-integral-contains-parts (car expr)))))
800
    expr))
Eli Zaretskii's avatar
Eli Zaretskii committed
801 802 803 804 805 806 807 808 809 810 811 812

(defun math-replace-integral-parts (expr)
  (or (Math-primp expr)
      (while (setq expr (cdr expr))
	(and (consp (car expr))
	     (if (eq (car (car expr)) 'var)
		 (and (eq (nth 1 (car expr)) 'PARTS)
		      (consp (nth 2 (car expr)))
		      (if (listp (nth 1 (nth 2 (car expr))))
			  (progn
			    (setcar expr (nth 1 (nth 2 (car expr))))
			    (math-replace-integral-parts (cons 'foo expr)))
813
			(setcar (cdr math-cur-record) 'cancelled)))
814
	       (math-replace-integral-parts (car expr)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
815

816 817 818
(defvar math-linear-subst-tried t
  "Non-nil means that a linear substitution has been tried.")

819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839
;; The variable math-has-rules is a local variable for math-try-integral,
;; but is used by math-do-integral, which is called (non-directly) by
;; math-try-integral.
(defvar math-has-rules)

;; math-old-integ is a local variable for math-do-integral, but is
;; used by math-sub-integration.
(defvar math-old-integ)

;; The variables math-t1, math-t2 and math-t3 are local to 
;; math-do-integral, math-try-solve-for and math-decompose-poly, but
;; are used by functions they call (directly or indirectly); 
;; math-do-integral calls math-do-integral-methods;
;; math-try-solve-for calls math-try-solve-prod, 
;; math-solve-find-root-term and math-solve-find-root-in-prod;
;; math-decompose-poly calls math-solve-poly-funny-powers and
;; math-solve-crunch-poly.
(defvar math-t1)
(defvar math-t2)
(defvar math-t3)

Eli Zaretskii's avatar
Eli Zaretskii committed
840
(defun math-do-integral (expr)
841
  (let ((math-linear-subst-tried nil)
842
        math-t1 math-t2)
Eli Zaretskii's avatar
Eli Zaretskii committed
843 844 845 846 847
    (or (cond ((not (math-expr-contains expr math-integ-var))
	       (math-mul expr math-integ-var))
	      ((equal expr math-integ-var)
	       (math-div (math-sqr expr) 2))
	      ((eq (car expr) '+)
848 849 850
	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (setq math-t2 (math-integral (nth 2 expr)))
		    (math-add math-t1 math-t2)))
Eli Zaretskii's avatar
Eli Zaretskii committed
851
	      ((eq (car expr) '-)
852 853 854
	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (setq math-t2 (math-integral (nth 2 expr)))
		    (math-sub math-t1 math-t2)))
Eli Zaretskii's avatar
Eli Zaretskii committed
855
	      ((eq (car expr) 'neg)
856 857
	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (math-neg math-t1)))
Eli Zaretskii's avatar
Eli Zaretskii committed
858 859
	      ((eq (car expr) '*)
	       (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
860 861
		      (and (setq math-t1 (math-integral (nth 2 expr)))
			   (math-mul (nth 1 expr) math-t1)))
Eli Zaretskii's avatar
Eli Zaretskii committed
862
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
863 864
		      (and (setq math-t1 (math-integral (nth 1 expr)))
			   (math-mul math-t1 (nth 2 expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882
		     ((memq (car-safe (nth 1 expr)) '(+ -))
		      (math-integral (list (car (nth 1 expr))
					   (math-mul (nth 1 (nth 1 expr))
						     (nth 2 expr))
					   (math-mul (nth 2 (nth 1 expr))
						     (nth 2 expr)))
				     'yes t))
		     ((memq (car-safe (nth 2 expr)) '(+ -))
		      (math-integral (list (car (nth 2 expr))
					   (math-mul (nth 1 (nth 2 expr))
						     (nth 1 expr))
					   (math-mul (nth 2 (nth 2 expr))
						     (nth 1 expr)))
				     'yes t))))
	      ((eq (car expr) '/)
	       (cond ((and (not (math-expr-contains (nth 1 expr)
						    math-integ-var))
			   (not (math-equal-int (nth 1 expr) 1)))
883 884
		      (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr))))
			   (math-mul (nth 1 expr) math-t1)))
Eli Zaretskii's avatar
Eli Zaretskii committed
885
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
886 887
		      (and (setq math-t1 (math-integral (nth 1 expr)))
			   (math-div math-t1 (nth 2 expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
888 889 890
		     ((and (eq (car-safe (nth 1 expr)) '*)
			   (not (math-expr-contains (nth 1 (nth 1 expr))
						    math-integ-var)))
891
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
892 893
				     (math-div (nth 2 (nth 1 expr))
					       (nth 2 expr))))
894
			   (math-mul math-t1 (nth 1 (nth 1 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
895 896 897
		     ((and (eq (car-safe (nth 1 expr)) '*)
			   (not (math-expr-contains (nth 2 (nth 1 expr))
						    math-integ-var)))
898
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
899 900
				     (math-div (nth 1 (nth 1 expr))
					       (nth 2 expr))))
901
			   (math-mul math-t1 (nth 2 (nth 1 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
902 903 904
		     ((and (eq (car-safe (nth 2 expr)) '*)
			   (not (math-expr-contains (nth 1 (nth 2 expr))
						    math-integ-var)))
905
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
906 907
				     (math-div (nth 1 expr)
					       (nth 2 (nth 2 expr)))))
908
			   (math-div math-t1 (nth 1 (nth 2 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
909 910 911
		     ((and (eq (car-safe (nth 2 expr)) '*)
			   (not (math-expr-contains (nth 2 (nth 2 expr))
						    math-integ-var)))
912
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
913 914
				     (math-div (nth 1 expr)
					       (nth 1 (nth 2 expr)))))
915
			   (math-div math-t1 (nth 2 (nth 2 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
916 917 918 919 920 921 922
		     ((eq (car-safe (nth 2 expr)) 'calcFunc-exp)
		      (math-integral
		       (math-mul (nth 1 expr)
				 (list 'calcFunc-exp
				       (math-neg (nth 1 (nth 2 expr)))))))))
	      ((eq (car expr) '^)
	       (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
923
		      (or (and (setq math-t1 (math-is-polynomial (nth 2 expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
924 925
							    math-integ-var 1))
			       (math-div expr
926
					 (math-mul (nth 1 math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
927 928 929 930 931 932 933 934 935 936 937 938 939 940 941
						   (math-normalize
						    (list 'calcFunc-ln
							  (nth 1 expr))))))
			  (math-integral
			   (list 'calcFunc-exp
				 (math-mul (nth 2 expr)
					   (math-normalize
					    (list 'calcFunc-ln
						  (nth 1 expr)))))
			   'yes t)))
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
		      (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0))
			  (math-integral
			   (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr))))
			   nil t)
942
			(or (and (setq math-t1 (math-is-polynomial (nth 1 expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
943 944
							      math-integ-var
							      1))
945 946 947
				 (setq math-t2 (math-add (nth 2 expr) 1))
				 (math-div (math-pow (nth 1 expr) math-t2)
					   (math-mul math-t2 (nth 1 math-t1))))
Eli Zaretskii's avatar
Eli Zaretskii committed
948 949 950 951 952 953 954 955 956 957
			    (and (Math-negp (nth 2 expr))
				 (math-integral
				  (math-div 1
					    (math-pow (nth 1 expr)
						      (math-neg
						       (nth 2 expr))))
				  nil t))
			    nil))))))

	;; Integral of a polynomial.
958
	(and (setq math-t1 (math-is-polynomial expr math-integ-var 20))
Eli Zaretskii's avatar
Eli Zaretskii committed
959 960
	     (let ((accum 0)
		   (n 1))
961
	       (while math-t1
Eli Zaretskii's avatar
Eli Zaretskii committed
962
		 (if (setq accum (math-add accum
963
					   (math-div (math-mul (car math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
964 965 966 967
							       (math-pow
								math-integ-var
								n))
						     n))
968
			   math-t1 (cdr math-t1))
Eli Zaretskii's avatar
Eli Zaretskii committed
969 970 971 972 973 974
		     (setq n (1+ n))))
	       accum))

	;; Try looking it up!
	(cond ((= (length expr) 2)
	       (and (symbolp (car expr))
975
		    (setq math-t1 (get (car expr) 'math-integral))
Eli Zaretskii's avatar
Eli Zaretskii committed
976
		    (progn
977 978
		      (while (and math-t1
				  (not (setq math-t2 (funcall (car math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
979
							 (nth 1 expr)))))
980 981
			(setq math-t1 (cdr math-t1)))
		      (and math-t2 (math-normalize math-t2)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
982 983
	      ((= (length expr) 3)
	       (and (symbolp (car expr))
984
		    (setq math-t1 (get (car expr) 'math-integral-2))
Eli Zaretskii's avatar
Eli Zaretskii committed
985
		    (progn
986 987
		      (while (and math-t1
				  (not (setq math-t2 (funcall (car math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
988 989
							 (nth 1 expr)
							 (nth 2 expr)))))
990 991
			(setq math-t1 (cdr math-t1)))
		      (and math-t2 (math-normalize math-t2))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
992 993 994

	;; Integral of a rational function.
	(and (math-ratpoly-p expr math-integ-var)
995 996 997
	     (setq math-t1 (calcFunc-apart expr math-integ-var))
	     (not (equal math-t1 expr))
	     (math-integral math-t1))
Eli Zaretskii's avatar
Eli Zaretskii committed
998 999

	;; Try user-defined integration rules.
1000
	(and math-has-rules
Eli Zaretskii's avatar
Eli Zaretskii committed
1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060
	     (let ((math-old-integ (symbol-function 'calcFunc-integ))
		   (input (list 'calcFunc-integtry expr math-integ-var))
		   res part)
	       (unwind-protect
		   (progn
		     (fset 'calcFunc-integ 'math-sub-integration)
		     (setq res (math-rewrite input
					     '(var IntegRules var-IntegRules)
					     1))
		     (fset 'calcFunc-integ math-old-integ)
		     (and (not (equal res input))
			  (if (setq part (math-expr-calls
					  res '(calcFunc-integsubst)))
			      (and (memq (length part) '(3 4 5))
				   (let ((parts (mapcar
						 (function
						  (lambda (x)
						    (math-expr-subst
						     x (nth 2 part)
						     math-integ-var)))
						 (cdr part))))
				     (math-integrate-by-substitution
				      expr (car parts) t
				      (or (nth 2 parts)
					  (list 'calcFunc-integfailed
						math-integ-var))
				      (nth 3 parts))))
			    (if (not (math-expr-calls res
						      '(calcFunc-integtry
							calcFunc-integfailed)))
				res))))
		 (fset 'calcFunc-integ math-old-integ))))

	;; See if the function is a symbolic derivative.
	(and (string-match "'" (symbol-name (car expr)))
	     (let ((name (symbol-name (car expr)))
		   (p expr) (n 0) (which nil) (bad nil))
	       (while (setq n (1+ n) p (cdr p))
		 (if (equal (car p) math-integ-var)
		     (if which (setq bad t) (setq which n))
		   (if (math-expr-contains (car p) math-integ-var)
		       (setq bad t))))
	       (and which (not bad)
		    (let ((prime (if (= which 1) "'" (format "'%d" which))))
		      (and (string-match (concat prime "\\('['0-9]*\\|$\\)")
					 name)
			   (cons (intern
				  (concat
				   (substring name 0 (match-beginning 0))
				   (substring name (+ (match-beginning 0)
						      (length prime)))))
				 (cdr expr)))))))

	;; Try transformation methods (parts, substitutions).
	(and (> math-integ-level 0)
	     (math-do-integral-methods expr))

	;; Try expanding the function's definition.
	(let ((res (math-expand-formula expr)))
	  (and res
1061
	       (math-integral res))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1062 1063 1064 1065 1066 1067 1068 1069 1070 1071

(defun math-sub-integration (expr &rest rest)
  (or (if (or (not rest)
	      (and (< math-integ-level math-integral-limit)
		   (eq (car rest) math-integ-var)))
	  (math-integral expr)
	(let ((res (apply math-old-integ expr rest)))
	  (and (or (= math-integ-level math-integral-limit)
		   (not (math-expr-calls res 'calcFunc-integ)))
	       res)))
1072
      (list 'calcFunc-integfailed expr)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1073

1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085
;; math-so-far is a local variable for math-do-integral-methods, but
;; is used by math-integ-try-linear-substitutions and 
;; math-integ-try-substitutions.
(defvar math-so-far)

;; math-integ-expr is a local variable for math-do-integral-methods,
;; but is used by math-integ-try-linear-substitutions and 
;; math-integ-try-substitutions.
(defvar math-integ-expr)

(defun math-do-integral-methods (math-integ-expr)
  (let ((math-so-far math-integ-var-list-list)
Eli Zaretskii's avatar
Eli Zaretskii committed
1086 1087 1088 1089
	rat-in)

    ;; Integration by substitution, for various likely sub-expressions.
    ;; (In first pass, we look only for sub-exprs that are linear in X.)
1090 1091
    (or (math-integ-try-linear-substitutions math-integ-expr)
        (math-integ-try-substitutions math-integ-expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
1092 1093

	;; If function has sines and cosines, try tan(x/2) substitution.
1094
	(and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1095 1096 1097
	       (while (and p
			   (memq (car (car p)) '(calcFunc-sin
						 calcFunc-cos
1098 1099 1100 1101
						 calcFunc-tan
                                                 calcFunc-sec
                                                 calcFunc-csc
                                                 calcFunc-cot))
Eli Zaretskii's avatar
Eli Zaretskii committed
1102 1103 1104
			   (equal (nth 1 (car p)) math-integ-var))
		 (setq p (cdr p)))
	       (null p))
1105 1106
	     (or (and (math-integ-parts-easy math-integ-expr)
		      (math-integ-try-parts math-integ-expr t))
Eli Zaretskii's avatar
Eli Zaretskii committed
1107
		 (math-integrate-by-good-substitution
1108
		  math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1109 1110 1111 1112 1113 1114 1115

	;; If function has sinh and cosh, try tanh(x/2) substitution.
	(and (let ((p rat-in))
	       (while (and p
			   (memq (car (car p)) '(calcFunc-sinh
						 calcFunc-cosh
						 calcFunc-tanh
1116 1117 1118
                                                 calcFunc-sech
                                                 calcFunc-csch
                                                 calcFunc-coth
Eli Zaretskii's avatar
Eli Zaretskii committed
1119 1120 1121 1122
						 calcFunc-exp))
			   (equal (nth 1 (car p)) math-integ-var))
		 (setq p (cdr p)))
	       (null p))
1123 1124
	     (or (and (math-integ-parts-easy math-integ-expr)
		      (math-integ-try-parts math-integ-expr t))
Eli Zaretskii's avatar
Eli Zaretskii committed
1125
		 (math-integrate-by-good-substitution
1126
		  math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1127 1128 1129

	;; If function has square roots, try sin, tan, or sec substitution.
	(and (let ((p rat-in))
1130
	       (setq math-t1 nil)
Eli Zaretskii's avatar
Eli Zaretskii committed
1131 1132 1133
	       (while (and p
			   (or (equal (car p) math-integ-var)
			       (and (eq (car (car p)) 'calcFunc-sqrt)
1134 1135
				    (setq math-t1 (math-is-polynomial
					      (nth 1 (setq math-t2 (car p)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1136 1137
					      math-integ-var 2)))))
		 (setq p (cdr p)))
1138 1139 1140 1141 1142 1143
	       (and (null p) math-t1))
	     (if (cdr (cdr math-t1))
		 (if (math-guess-if-neg (nth 2 math-t1))
		     (let* ((c (math-sqrt (math-neg (nth 2 math-t1))))
			    (d (math-div (nth 1 math-t1) (math-mul -2 c)))
			    (a (math-sqrt (math-add (car math-t1) (math-sqr d)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1144
		       (math-integrate-by-good-substitution
1145
			math-integ-expr (list 'calcFunc-arcsin
Eli Zaretskii's avatar
Eli Zaretskii committed
1146 1147 1148
				   (math-div-thru
				    (math-add (math-mul c math-integ-var) d)
				    a))))
1149 1150 1151
		   (let* ((c (math-sqrt (nth 2 math-t1)))
			  (d (math-div (nth 1 math-t1) (math-mul 2 c)))
			  (aa (math-sub (car math-t1) (math-sqr d))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1152 1153
		     (if (and nil (not (and (eq d 0) (eq c 1))))
			 (math-integrate-by-good-substitution
1154
			  math-integ-expr (math-add (math-mul c math-integ-var) d))
Eli Zaretskii's avatar
Eli Zaretskii committed
1155 1156
		       (if (math-guess-if-neg aa)
			   (math-integrate-by-good-substitution
1157
			    math-integ-expr (list 'calcFunc-arccosh
Eli Zaretskii's avatar
Eli Zaretskii committed
1158 1159 1160 1161 1162
				       (math-div-thru
					(math-add (math-mul c math-integ-var)
						  d)
					(math-sqrt (math-neg aa)))))
			 (math-integrate-by-good-substitution
1163
			  math-integ-expr (list 'calcFunc-arcsinh
Eli Zaretskii's avatar
Eli Zaretskii committed
1164 1165 1166 1167
				     (math-div-thru
				      (math-add (math-mul c math-integ-var)
						d)
				      (math-sqrt aa))))))))
1168
	       (math-integrate-by-good-substitution math-integ-expr math-t2)) )
Eli Zaretskii's avatar
Eli Zaretskii committed
1169 1170

	;; Try integration by parts.
1171
	(math-integ-try-parts math-integ-expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
1172 1173

	;; Give up.
1174
	nil)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188

(defun math-integ-parts-easy (expr)
  (cond ((Math-primp expr) t)
	((memq (car expr) '(+ - *))
	 (and (math-integ-parts-easy (nth 1 expr))
	      (math-integ-parts-easy (nth 2 expr))))
	((eq (car expr) '/)
	 (and (math-integ-parts-easy (nth 1 expr))
	      (math-atomic-factorp (nth 2 expr))))
	((eq (car expr) '^)
	 (and (natnump (nth 2 expr))
	      (math-integ-parts-easy (nth 1 expr))))
	((eq (car expr) 'neg)
	 (math-integ-parts-easy (nth 1 expr)))
1189
	(t t)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1190

1191 1192 1193 1194 1195 1196 1197 1198 1199
;; math-prev-parts-v is local to calcFunc-integ (as well as
;; math-integrate-by-parts), but is used by math-integ-try-parts.
(defvar math-prev-parts-v)

;; math-good-parts is local to calcFunc-integ (as well as
;; math-integ-try-parts), but is used by math-integrate-by-parts.
(defvar math-good-parts)


Eli Zaretskii's avatar
Eli Zaretskii committed
1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223
(defun math-integ-try-parts (expr &optional math-good-parts)
  ;; Integration by parts:
  ;;   integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x)
  ;;     where h(x) = integ(g(x),x).
  (or (let ((exp (calcFunc-expand expr)))
	(and (not (equal exp expr))
	     (math-integral exp)))
      (and (eq (car expr) '*)
	   (let ((first-bad (or (math-polynomial-p (nth 1 expr)
						   math-integ-var)
				(equal (nth 2 expr) math-prev-parts-v))))
	     (or (and first-bad   ; so try this one first
		      (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))
		 (math-integrate-by-parts (nth 2 expr) (nth 1 expr))
		 (and (not first-bad)
		      (math-integrate-by-parts (nth 1 expr) (nth 2 expr))))))
      (and (eq (car expr) '/)
	   (math-expr-contains (nth 1 expr) math-integ-var)
	   (let ((recip (math-div 1 (nth 2 expr))))
	     (or (math-integrate-by-parts (nth 1 expr) recip)
		 (math-integrate-by-parts recip (nth 1 expr)))))
      (and (eq (car expr) '^)
	   (math-integrate-by-parts (math-pow (nth 1 expr)
					      (math-sub (nth 2 expr) 1))
1224
				    (nth 1 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235

(defun math-integrate-by-parts (u vprime)
  (let ((math-integ-level (if (or math-good-parts
				  (math-polynomial-p u math-integ-var))
			      math-integ-level
			    (1- math-integ-level)))
	(math-doing-parts t)
	v temp)
    (and (>= math-integ-level 0)
	 (unwind-protect
	     (progn
1236
	       (setcar (cdr math-cur-record) 'parts)
Eli Zaretskii's avatar
Eli Zaretskii committed
1237 1238 1239 1240 1241 1242 1243 1244 1245 1246
	       (math-tracing-integral "Integrating by parts, u = "
				      (math-format-value u 1000)
				      ", v' = "
				      (math-format-value vprime 1000)
				      "\n")
	       (and (setq v (math-integral vprime))
		    (setq temp (calcFunc-deriv u math-integ-var nil t))
		    (setq temp (let ((math-prev-parts-v v))
				 (math-integral (math-mul v temp) 'yes)))
		    (setq temp (math-sub (math-mul u v) temp))
1247
		    (if (eq (nth 1 math-cur-record) 'parts)
Eli Zaretskii's avatar
Eli Zaretskii committed
1248
			(calcFunc-expand temp)
1249
		      (setq v (list 'var 'PARTS math-cur-record)
Eli Zaretskii's avatar
Eli Zaretskii committed
1250 1251 1252 1253
			    temp (let (calc-next-why)
				   (math-solve-for (math-sub v temp) 0 v nil)))
		      (and temp (not (integerp temp))
			   (math-simplify-extended temp)))))
1254
	   (setcar (cdr math-cur-record) 'busy)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1255 1256 1257 1258 1259 1260

;;; This tries two different formulations, hoping the algebraic simplifier
;;; will be strong enough to handle at least one.
(defun math-integrate-by-substitution (expr u &optional user uinv uinvprime)
  (and (> math-integ-level 0)
       (let ((math-integ-level (max (- math-integ-level 2) 0)))
1261
	 (math-integrate-by-good-substitution expr u user uinv uinvprime))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310

(defun math-integrate-by-good-substitution (expr u &optional user
						 uinv uinvprime)
  (let ((math-living-dangerously t)
	deriv temp)
    (and (setq uinv (if uinv
			(math-expr-subst uinv math-integ-var
					 math-integ-var-2)
		      (let (calc-next-why)
			(math-solve-for u
					math-integ-var-2
					math-integ-var nil))))
	 (progn
	   (math-tracing-integral "Integrating by substitution, u = "
				  (math-format-value u 1000)
				  "\n")
	   (or (and (setq deriv (calcFunc-deriv u
						math-integ-var nil
						(not user)))
		    (setq temp (math-integral (math-expr-subst
					       (math-expr-subst
						(math-expr-subst
						 (math-div expr deriv)
						 u
						 math-integ-var-2)
						math-integ-var
						uinv)
					       math-integ-var-2
					       math-integ-var)
					      'yes)))
	       (and (setq deriv (or uinvprime
				    (calcFunc-deriv uinv
						    math-integ-var-2
						    math-integ-var
						    (not user))))
		    (setq temp (math-integral (math-mul
					       (math-expr-subst
						(math-expr-subst
						 (math-expr-subst
						  expr
						  u
						  math-integ-var-2)
						 math-integ-var
						 uinv)
						math-integ-var-2
						math-integ-var)
					       deriv)
					      'yes)))))
	 (math-simplify-extended
1311
	  (math-expr-subst temp math-integ-var u)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1312 1313 1314

;;; Look for substitutions of the form u = a x + b.
(defun math-integ-try-linear-substitutions (sub-expr)
1315
  (setq math-linear-subst-tried t)
Eli Zaretskii's avatar
Eli Zaretskii committed
1316 1317 1318 1319 1320 1321 1322 1323 1324
  (and (not (Math-primp sub-expr))
       (or (and (not (memq (car sub-expr) '(+ - * / neg)))
		(not (and (eq (car sub-expr) '^)
			  (integerp (nth 2 sub-expr))))
		(math-expr-contains sub-expr math-integ-var)
		(let ((res nil))
		  (while (and (setq sub-expr (cdr sub-expr))
			      (or (not (math-linear-in (car sub-expr)
						       math-integ-var))
1325
				  (assoc (car sub-expr) math-so-far)
Eli Zaretskii's avatar
Eli Zaretskii committed
1326
				  (progn
1327 1328
				    (setq math-so-far (cons (list (car sub-expr))
						       math-so-far))
Eli Zaretskii's avatar
Eli Zaretskii committed
1329 1330
				    (not (setq res
					       (math-integrate-by-substitution
1331
						math-integ-expr (car sub-expr))))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1332 1333 1334 1335 1336
		  res))
	   (let ((res nil))
	     (while (and (setq sub-expr (cdr sub-expr))
			 (not (setq res (math-integ-try-linear-substitutions
					 (car sub-expr))))))
1337
	     res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1338 1339 1340 1341

;;; Recursively try different substitutions based on various sub-expressions.
(defun math-integ-try-substitutions (sub-expr &optional allow-rat)
  (and (not (Math-primp sub-expr))
1342
       (not (assoc sub-expr math-so-far))
Eli Zaretskii's avatar
Eli Zaretskii committed
1343 1344 1345 1346 1347 1348
       (math-expr-contains sub-expr math-integ-var)
       (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg)))
			 (not (and (eq (car sub-expr) '^)
				   (integerp (nth 2 sub-expr)))))
		    (setq allow-rat t)
		  (prog1 allow-rat (setq allow-rat nil)))
1349 1350
		(not (eq sub-expr math-integ-expr))
		(or (math-integrate-by-substitution math-integ-expr sub-expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
1351 1352 1353 1354 1355 1356 1357
		    (and (eq (car sub-expr) '^)
			 (integerp (nth 2 sub-expr))
			 (< (nth 2 sub-expr) 0)
			 (math-integ-try-substitutions
			  (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr)))
			  t))))
	   (let ((res nil))
1358
	     (setq math-so-far (cons (list sub-expr) math-so-far))
Eli Zaretskii's avatar
Eli Zaretskii committed
1359 1360 1361
	     (while (and (setq sub-expr (cdr sub-expr))
			 (not (setq res (math-integ-try-substitutions
					 (car sub-expr) allow-rat)))))
1362
	     res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1363

1364 1365
;; The variable math-expr-parts is local to math-expr-rational-in,
;; but is used by math-expr-rational-in-rec
1366
(defvar math-expr-parts)
1367

Eli Zaretskii's avatar
Eli Zaretskii committed
1368
(defun math-expr-rational-in (expr)
1369
  (let ((math-expr-parts nil))
Eli Zaretskii's avatar
Eli Zaretskii committed
1370
    (math-expr-rational-in-rec expr)
1371
    (mapcar 'car math-expr-parts)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1372 1373 1374 1375

(defun math-expr-rational-in-rec (expr)
  (cond ((Math-primp expr)
	 (and (equal expr math-integ-var)
1376 1377
	      (not (assoc expr math-expr-parts))
	      (setq math-expr-parts (cons (list expr) math-expr-parts))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1378 1379 1380 1381 1382 1383 1384 1385
	((or (memq (car expr) '(+ - * / neg))
	     (and (eq (car expr) '^) (integerp (nth 2 expr))))
	 (math-expr-rational-in-rec (nth 1 expr))
	 (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr))))
	((and (eq (car expr) '^)
	      (eq (math-quarter-integer (nth 2 expr)) 2))
	 (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr))))
	(t
1386
	 (and (not (assoc expr math-expr-parts))
Eli Zaretskii's avatar
Eli Zaretskii committed
1387
	      (math-expr-contains expr math-integ-var)
1388
	      (setq math-expr-parts (cons (list expr) math-expr-parts))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402

(defun math-expr-calls (expr funcs &optional arg-contains)
  (if (consp expr)
      (if (or (memq (car expr) funcs)
	      (and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt)
		   (eq (math-quarter-integer (nth 2 expr)) 2)))
	  (and (or (not arg-contains)
		   (math-expr-contains expr arg-contains))
	       expr)
	(and (not (Math-primp expr))
	     (let ((res nil))
	       (while (and (setq expr (cdr expr))
			   (not (setq res (math-expr-calls
					   (car expr) funcs arg-contains)))))
1403
	       res)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1404 1405 1406 1407 1408 1409 1410 1411 1412 1413

(defun math-fix-const-terms (expr except-vars)
  (cond ((not (math-expr-depends expr except-vars)) 0)
	((Math-primp expr) expr)
	((eq (car expr) '+)
	 (math-add (math-fix-const-terms (nth 1 expr) except-vars)
		   (math-fix-const-terms (nth 2 expr) except-vars)))
	((eq (car expr) '-)
	 (math-sub (math-fix-const-terms (nth 1 expr) except-vars)
		   (math-fix-const-terms (nth 2 expr) except-vars)))
1414
	(t expr)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1415 1416 1417 1418 1419 1420 1421

;; Command for debugging the Calculator's symbolic integrator.
(defun calc-dump-integral-cache (&optional arg)
  (interactive "P")
  (let ((buf (current-buffer)))
    (unwind-protect
	(let ((p math-integral-cache)
1422
	      math-cur-record)
1423
	  (display-buffer (get-buffer-create "*Integral Cache*"))
Eli Zaretskii's avatar
Eli Zaretskii committed
1424 1425 1426
	  (set-buffer (get-buffer "*Integral Cache*"))
	  (erase-buffer)
	  (while p
1427 1428 1429
	    (setq math-cur-record (car p))
	    (or arg (math-replace-integral-parts math-cur-record))
	    (insert (math-format-flat-expr (car math-cur-record) 0)
Eli Zaretskii's avatar
Eli Zaretskii committed
1430
		    " --> "
1431 1432 1433
		    (if (symbolp (nth 1 math-cur-record))
			(concat "(" (symbol-name (nth 1 math-cur-record)) ")")
		      (math-format-flat-expr (nth 1 math-cur-record) 0))
Eli Zaretskii's avatar
Eli Zaretskii committed
1434 1435 1436
		    "\n")
	    (setq p (cdr p)))
	  (goto-char (point-min)))
1437
      (set-buffer buf))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1438

1439 1440 1441 1442
;; The variable math-max-integral-limit is local to calcFunc-integ,
;; but is used by math-try-integral.
(defvar math-max-integral-limit)