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;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; The data in this file contains enhancments. ;;;;;
;;; ;;;;;
;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;;
;;; All rights reserved ;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; (c) Copyright 1982 Massachusetts Institute of Technology ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package :maxima)
(macsyma-module limit)
;;; **************************************************************
;;; ** **
;;; ** LIMIT PACKAGE **
;;; ** **
;;; **************************************************************
;;; I believe a large portion of this file is described in Paul
;;; Wang's thesis, "Evaluation of Definite Integrals by Symbolic
;;; Integration," MIT/LCS/TR-92, Oct. 1971. This can be found at
;;; http://www.lcs.mit.edu/publications/specpub.php?id=660, but some
;;; important pages are black.
;;; TOP LEVEL FUNCTION(S): $LIMIT $LDEFINT
(declare-top (special errorsw errrjfflag raterr origval $lhospitallim low*
*indicator half%pi nn* dn* numer denom exp var val varlist
*zexptsimp? $tlimswitch $logarc taylored logcombed
$exponentialize lhp? lhcount $ratfac genvar
loginprod? $limsubst $logabs a context limit-assumptions
limit-top limitp integer-info old-integer-info $keepfloat $logexpand))
(defconstant +behavior-count+ 4)
(defvar *behavior-count-now*)
(load-macsyma-macros rzmac)
(defmvar infinities '($inf $minf $infinity)
"The types of infinities recognized by Maxima.
INFINITY is complex infinity")
(defmvar real-infinities '($inf $minf)
"The real infinities, `inf' is positive infinity, `minf' negative infinity")
(defmvar infinitesimals '($zeroa $zerob)
"The infinitesimals recognized by Maxima. ZEROA zero from above,
ZEROB zero from below")
(defmvar simplimplus-problems ()
"A list of all problems in the stack of recursive calls to simplimplus.")
(defmvar limit-answers ()
"An association list for storing limit answers.")
(defmvar limit-using-taylor ()
"Is the current limit computation using taylor expansion?")
(defmvar preserve-direction () "Makes `limit' return Direction info.")
(unless (boundp 'integer-info) (setq integer-info ()))
;; This should be made to give more information about the error.
;;(DEFun DISCONT ()
;; (cond (errorsw (throw 'errorsw t))
;; (t (merror "Discontinuity Encountered"))))
;;(DEFUN PUTLIMVAL (E V)
;; (let ((exp (cons '(%limit) (list e var val))))
;; (cond ((not (assolike exp limit-answers))
;; (setq limit-answers (cons (cons exp v) limit-answers))
;; v)
;; (t ()))))
(defun putlimval (e v &aux exp)
(setq exp `((%limit) ,e ,var ,val))
(unless (assolike exp limit-answers)
(push (cons exp v) limit-answers))
v)
(defun getlimval (e)
(let ((exp (cons '(%limit) (list e var val))))
(assolike exp limit-answers)))
(defmacro limit-catch (exp var val)
`(let ((errorsw t))
(let ((ans (catch 'errorsw
(catch 'limit (limit ,exp ,var ,val 'think)))))
(if (or (null ans) (eq ans t))
()
ans))))
(defmfun $limit (&rest args)
(let ((limit-assumptions ())
(old-integer-info ())
($keepfloat t)
(limit-top t))
(declare (special limit-assumptions old-integer-info
$keepfloat limit-top))
(unless limitp
(setq old-integer-info integer-info)
(setq integer-info ()))
(unwind-protect
(let ((exp1 ()) (lhcount $lhospitallim) (*behavior-count-now* 0)
(exp ()) (var ()) (val ()) (dr ())
(*indicator ()) (taylored ()) (origval ())
(logcombed ()) (lhp? ())
(varlist ()) (ans ()) (genvar ()) (loginprod? ())
(limit-answers ()) (limitp t) (simplimplus-problems ())
(lenargs (length args))
(genfoo ()))
(declare (special lhcount *behavior-count-now* exp var val *indicator
taylored origval logcombed lhp?
varlist genvar loginprod? limitp))
(prog ()
(unless (or (= lenargs 3) (= lenargs 4) (= lenargs 1))
(wna-err '$limit))
;; Is it a LIST of Things?
(when (setq ans (apply #'limit-list args))
(return ans))
(setq exp1 (specrepcheck (first args)))
(when (and (atom exp1)
(member exp1 '(nil t)))
;; The expression is 'T or 'NIL. Return immediately.
(return exp1))
(cond ((= lenargs 1)
(setq var (setq genfoo (gensym)) ; Use a gensym. Not foo.
val 0))
(t
(setq var (second args))
(when ($constantp var)
(merror (intl:gettext "limit: second argument must be a variable, not a constant; found: ~M") var))
(unless (or ($subvarp var) (atom var))
(merror (intl:gettext "limit: variable must be a symbol or subscripted symbol; found: ~M") var))
(setq val (infsimp (third args)))
;; infsimp converts -inf to minf. it also converts -infinity to
;; infinity, although perhaps this should generate the error below.
(when (and (not (atom val))
(some #'(lambda (x) (not (freeof x val)))
infinities))
(merror (intl:gettext "limit: third argument must be a finite value or one of: inf, minf, infinity; found: ~M") val))
(when (eq val '$zeroa) (setq dr '$plus))
(when (eq val '$zerob) (setq dr '$minus))))
(cond ((= lenargs 4)
(unless (member (fourth args) '($plus $minus) :test #'eq)
(merror (intl:gettext "limit: direction must be either 'plus' or 'minus'; found: ~M") (fourth args)))
(setq dr (fourth args))))
(if (and (atom var) (not (among var val)))
(setq exp exp1)
(let ((realvar var)) ;; Var is funny so make it a gensym.
(setq var (gensym))
(setq exp (maxima-substitute var realvar exp1))
(putprop var realvar 'limitsub)))
(unless (or $limsubst (eq var genfoo))
(when (limunknown exp)
(return `((%limit) ,@(cons exp1 (cdr args))))))
(setq varlist (ncons var) genvar nil origval val)
;; Transform limits to minf to limits to inf by
;; replacing var with -var everywhere.
(when (eq val '$minf)
(setq val '$inf
origval '$inf
exp (subin (m* -1 var) exp)))
;; Hide noun form of %derivative, %integrate.
(setq exp (hide exp))
;; Transform the limit value.
(unless (infinityp val)
(unless (zerop2 val)
(let ((*atp* t) (realvar var))
;; *atp* prevents substitution from applying to vars
;; bound by %sum, %product, %integrate, %limit
(setq var (gensym))
(putprop var t 'internal)
(setq exp (maxima-substitute (m+ val var) realvar exp))))
(setq val (cond ((eq dr '$plus) '$zeroa)
((eq dr '$minus) '$zerob)
(t 0)))
(setq origval 0))
;; Make assumptions about limit var being very small or very large.
;; Assumptions are forgotten upon exit.
(unless (= lenargs 1)
(limit-context var val dr))
;; Resimplify in light of new assumptions.
(setq exp (resimplify
(factosimp
(tansc
(lfibtophi
(limitsimp ($expand exp 1 0) var))))))
(if (not (or (real-epsilonp val) ;; if direction of limit not specified
(infinityp val)))
(setq ans (both-side exp var val)) ;; compute from both sides
(let ((d (catch 'mabs (mabs-subst exp var val))))
(cond ;; otherwise try to remove absolute value
((eq d '$und) (return '$und))
((eq d 'retn) )
(t (setq exp d)))
(setq ans (limit-catch exp var val))));; and find limit from one side
;; try gruntz
(if (and (not ans)
(or (real-epsilonp val) ;; if direction of limit specified
(real-infinityp val)))
(setq ans (catch 'taylor-catch
(let ((silent-taylor-flag t))
(declare (special silent-taylor-flag))
(gruntz1 exp var val)))))
;; try taylor series expansion if simple limit didn't work
(if (and (null ans) ;; if no limit found and
$tlimswitch ;; user says ok to use taylor and
(not limit-using-taylor));; not already doing taylor
(let ((limit-using-taylor t))
(declare (special limit-using-taylor))
(setq ans (limit-catch exp var val))))
(if ans
(return (clean-limit-exp ans))
(return (cons '(%limit) args))))) ;; failure: return nounform
(restore-assumptions))))
(defun clean-limit-exp (exp)
(setq exp (restorelim exp))
(if preserve-direction exp (ridofab exp)))
(defmfun limit-list (exp1 &rest rest)
(if (mbagp exp1)
`(,(car exp1) ,@(mapcar #'(lambda (x) (apply #'$limit `(,x ,@rest))) (cdr exp1)))
()))
(defun limit-context (var val direction) ;Only works on entry!
(cond (limit-top
(assume '((mgreaterp) lim-epsilon 0))
(assume '((mlessp) lim-epsilon 1e-8))
(assume '((mgreaterp) prin-inf 1e+8))
(setq limit-assumptions (make-limit-assumptions var val direction))
(setq limit-top ()))
(t ()))
limit-assumptions)
(defun make-limit-assumptions (var val direction)
(let ((new-assumptions))
(cond ((or (null var) (null val))
())
((and (not (infinityp val)) (null direction))
())
((eq val '$inf)
`(,(assume `((mgreaterp) ,var 1e+8)) ,@new-assumptions))
((eq val '$minf)
`(,(assume `((mgreaterp) -1e+8 ,var)) ,@new-assumptions))
((eq direction '$plus)
`(,(assume `((mgreaterp) 1e-8 ,var))
,(assume `((mgreaterp) ,var 0)) ,@new-assumptions)) ;All limits around 0
((eq direction '$minus)
`(,(assume `((mgreaterp) ,var -1e-8))
,(assume `((mgreaterp) 0 ,var)) ,@new-assumptions))
(t
()))))
(defun restore-assumptions ()
;;;Hackery until assume and forget take reliable args. Nov. 9 1979.
;;;JIM.
(do ((assumption-list limit-assumptions (cdr assumption-list)))
((null assumption-list) t)
(forget (car assumption-list)))
(forget '((mgreaterp) lim-epsilon 0))
(forget '((mlessp) lim-epsilon 1.0e-8))
(forget '((mgreaterp) prin-inf 1.0e+8))
(cond ((and (not (null integer-info))
(not limitp))
(do ((list integer-info (cdr list)))
((null list) t)
(i-$remove `(,(cadar list) ,(caddar list))))
(setq integer-info old-integer-info))))
;; The optional arg allows the caller to decide on the value of
;; preserve-direction. Default is T, like it used to be.
(defun both-side (exp var val &optional (preserve t))
(let* ((preserve-direction preserve)
(la ($limit exp var val '$plus)) lb)
(when (eq la '$und) (return-from both-side '$und))
(setf lb ($limit exp var val '$minus))
(cond ((alike1 (ridofab la) (ridofab lb)) (ridofab la))
((or (not (free la '%limit))
(not (free lb '%limit))) ())
;; inf + minf => infinity
((and (infinityp la) (infinityp lb)) '$infinity)
(t '$und))))
;; Warning: (CATCH NIL ...) will catch all throws.
;; NIL should not be used as a tag name.
(defun limunknown (f)
(catch 'limunknown (limunknown1 (specrepcheck f))))
(defun limunknown1 (f)
(cond ((mapatom f) nil)
((or (not (safe-get (caar f) 'operators))
(member (caar f) '(%sum %product mncexpt) :test #'eq)
;;Special function code here i.e. for li[2](x).
(and (eq (caar f) 'mqapply)
(not (get (subfunname f) 'specsimp))))
(if (not (free f var)) (throw 'limunknown t)))
(t (mapc #'limunknown1 (cdr f)) nil)))
(defun factosimp(e)
(if (involve e '(%gamma)) (setq e ($makefact e)))
(cond ((involve e '(mfactorial))
(setq e (simplify ($minfactorial e))))
(t e)))
;; returns 1, 0, -1
;; or nil if sign unknown or complex
(defun getsignl (z)
(let ((z (ridofab z)))
(if (not (free z var)) (setq z ($limit z var val)))
(let ((sign ($csign z)))
(cond ((eq sign '$pos) 1)
((eq sign '$neg) -1)
((eq sign '$zero) 0)))))
(defun restorelim (exp)
(cond ((null exp) nil)
((atom exp) (or (and (symbolp exp) (get exp 'limitsub)) exp))
((and (consp (car exp)) (eq (caar exp) 'mrat))
(cons (car exp)
(cons (restorelim (cadr exp))
(restorelim (cddr exp)))))
(t (cons (car exp) (mapcar #'restorelim (cdr exp))))))
(defun mabs-subst (exp var val) ; RETURNS EXP WITH MABS REMOVED, OR THROWS.
(let ((d (involve exp '(mabs)))
arglim)
(cond ((null d) exp)
(t (cond
((not (and (equal ($imagpart (let ((v (limit-catch d var val)))
;; The above call might
;; throw 'limit, so we
;; need to catch it. If
;; we can't find the
;; limit without ABS, we
;; assume the limit is
;; undefined. Is this
;; right? Anyway, this
;; fixes Bug 1548643.
(unless v
(throw 'mabs '$und))
(setq arglim v)))
0)
(equal ($imagpart var) 0)))
(cond ((eq arglim '$infinity)
;; Check for $infinity as limit of argument.
'$inf)
(t
(throw 'mabs 'retn))))
(t (do ((ans d (involve exp '(mabs))) (a () ()))
((null ans) exp)
(setq a (mabs-subst ans var val))
(setq d (limit a var val t))
(cond
((and a d)
(cond ((zerop1 d)
(setq d (behavior a var val))
(if (zerop1 d) (throw 'mabs 'retn))))
(if (eq d '$und)
(throw 'mabs d))
(cond ((or (eq d '$zeroa) (eq d '$inf)
(eq d '$ind)
;; fails on limit(abs(sin(x))/sin(x), x, inf)
(eq ($sign d) '$pos))
(setq exp (maxima-substitute a `((mabs) ,ans) exp)))
((or (eq d '$zerob) (eq d '$minf)
(eq ($sign d) '$neg))
(setq exp (maxima-substitute (m* -1 a) `((mabs) ,ans) exp)))
(t (throw 'mabs 'retn))))
(t (throw 'mabs 'retn))))))))))
;; Called on an expression that might contain $INF, $MINF, $ZEROA, $ZEROB. Tries
;; to simplify it to sort out things like inf^inf or inf+1.
(defun simpinf (exp)
(simpinf-ic exp (count-general-inf exp)))
(defun count-general-inf (expr)
(count-atoms-matching
(lambda (x) (or (infinityp x) (real-epsilonp x))) expr))
(defun count-atoms-matching (predicate expr)
"Count the number of atoms in the Maxima expression EXPR matching PREDICATE,
ignoring dummy variables and array indices."
(cond
((atom expr) (if (funcall predicate expr) 1 0))
;; Don't count atoms that occur as a limit of %integrate, %sum, %product,
;; %limit etc.
((member (caar expr) dummy-variable-operators)
(count-atoms-matching predicate (cadr expr)))
;; Ignore array indices
((member 'array (car expr)) 0)
(t (loop
for arg in (cdr expr)
summing (count-atoms-matching predicate arg)))))
(defun simpinf-ic (exp &optional infinity-count)
(case infinity-count
;; A very slow identity transformation...
(0 exp)
;; If there's only one infinity, we replace it by a variable and take the
;; limit as that variable goes to infinity. Use $gensym in case we can't
;; compute the answer and the limit leaks out.
(1 (let ((limit (or (inf-typep exp) (epsilon-typep exp)))
(var ($gensym)))
($limit (subst var limit exp) var limit)))
;; If more than one infinity, we have to be a bit more careful.
(otherwise
(let* ((arguments (mapcar 'simpinf (cdr exp)))
(new-expression (cons (list (caar exp)) arguments))
infinities-left)
(cond
;; If any of the arguments are undefined, we are too.
((among '$und arguments) '$und)
;; If we ended up with something indeterminate, we punt and just return
;; the input.
((amongl '(%limit $ind) arguments) exp)
;; Exponentiation & multiplication
((mexptp exp) (simpinf-expt (first arguments) (second arguments)))
((mtimesp exp) (simpinf-times arguments))
;; Down to at most one infinity? We do this after exponentiation to
;; avoid zeroa^zeroa => 0^0, which will raise an error rather than just
;; returning und. We do it after multiplication to avoid zeroa * inf =>
;; 0 * inf => 0.
((<= (setf infinities-left (count-general-inf new-expression)) 1)
(simpinf-ic new-expression infinities-left))
;; Addition
((mplusp exp) (simpinf-plus arguments))
;; Give up!
(t new-expression))))))
(defun simpinf-times (arguments)
(declare (special exp var val))
;; When we have a product, we need to spot that zeroa * zerob = zerob, zeroa *
;; inf = und etc. Note that (SIMPINF '$ZEROA) => 0, so a nonzero atom is not
;; an infinitesimal. Moreover, we can assume that each of ARGUMENTS is either
;; a number, computed successfully by the recursive SIMPINF call, or maybe a
;; %LIMIT noun-form (in which case, we aren't going to be able to tell the
;; answer).
(cond
((member 0 arguments)
(cond
((find-if #'infinityp arguments) '$und)
((every #'atom arguments) 0)
(t exp)))
((member '$infinity arguments)
(if (every #'atom arguments)
'$infinity
exp))
(t (simplimit (cons '(mtimes) arguments) var val))))
(defun simpinf-expt (base exponent)
;; In the comments below, zero* represents one of 0, zeroa, zerob.
;;
;; TODO: In some cases we give up too early. E.g. inf^(2 + 1/inf) => inf^2
;; (which should simplify to inf)
(case base
;; inf^inf = inf
;; inf^minf = 0
;; inf^zero* = und
;; inf^foo = inf^foo
($inf
(case exponent
($inf '$inf)
($minf 0)
((0 $zeroa $zerob) '$und)
(t (list '(mexpt) base exponent))))
;; minf^inf = infinity <== Or should it be und?
;; minf^minf = 0
;; minf^zero* = und
;; minf^foo = minf^foo
($minf
(case exponent
($inf '$infinity)
($minf 0)
((0 $zeroa $zerob) '$und)
(t (list '(mexpt) base exponent))))
;; zero*^inf = 0
;; zero*^minf = und
;; zero*^zero* = und
;; zero*^foo = zero*^foo
((0 $zeroa $zerob)
(case exponent
($inf 0)
($minf '$und)
((0 $zeroa $zerob) '$und)
(t (list '(mexpt) base exponent))))
;; a^b where a is pretty much anything except for a naked
;; inf,minf,zeroa,zerob or 0.
(t
(cond
;; When a isn't crazy, try a^b = e^(b log(a))
((not (amongl (append infinitesimals infinities) base))
(simpinf (m^ '$%e (m* base `((%log) ,exponent)))))
;; No idea. Just return what we've found so far.
(t (list '(mexpt) base exponent))))))
(defun simpinf-plus (arguments)
;; We know that none of the arguments are infinitesimals, since SIMPINF never
;; returns one of them. As such, we partition our arguments into infinities
;; and everything else. The latter won't have any "hidden" infinities like
;; lim(x,x,inf), since SIMPINF gave up on anything containing a %lim already.
(let ((bigs) (others))
(dolist (arg arguments)
(cond ((infinityp arg) (push arg bigs))
(t (push arg others))))
(cond
;; inf + minf or the like
((cdr (setf bigs (delete-duplicates bigs))) '$und)
;; inf + smaller + stuff
(bigs (car bigs))
;; I don't think this can happen, since SIMPINF goes back to the start if
;; there are fewer than two infinities in the arguments, but let's be
;; careful.
(t (cons '(mplus) others)))))
;; Simplify expression with zeroa or zerob.
(defun simpab (small)
(cond ((null small) ())
((member small '($zeroa $zerob $inf $minf $infinity) :test #'eq) small)
((not (free small '$ind)) '$ind) ;Not exactly right but not
((not (free small '$und)) '$und) ;causing trouble now.
((mapatom small) small)
(t (let ((preserve-direction t)
(new-small (subst (m^ '$inf -1) '$zeroa
(subst (m^ '$minf -1) '$zerob small))))
(simpinf new-small)))))
;;;*I* INDICATES: T => USE LIMIT1,THINK, NIL => USE SIMPLIMIT.
(defmfun limit (exp var val *i*)
(cond
((among '$und exp) '$und)
((eq var exp) val)
((atom exp) exp)
((not (among var exp))
(cond ((amongl '($inf $minf $infinity $ind) exp)
(simpinf exp))
((amongl '($zeroa $zerob) exp)
;; Simplify expression with zeroa or zerob.
(simpab exp))
(t exp)))
((getlimval exp))
(t (putlimval exp (cond ((and limit-using-taylor
(null taylored)
(tlimp exp))
(taylim exp *i*))
((ratp exp var) (ratlim exp))
((or (eq *i* t) (radicalp exp var))
(limit1 exp var val))
((eq *i* 'think)
(cond ((or (mtimesp exp) (mexptp exp))
(limit1 exp var val))
(t (simplimit exp var val))))
(t (simplimit exp var val)))))))
(defun limitsimp (exp var)
(limitsimp-expt (sin-sq-cos-sq-sub exp) var))
;;Hack for sin(x)^2+cos(x)^2.
;; if var appears in base and power of expt,
;; push var into power of of expt
(defun limitsimp-expt (exp var)
(cond ((or (atom exp)
(mnump exp)
(freeof var exp)) exp)
((and (mexptp exp)
(not (freeof var (cadr exp)))
(not (freeof var (caddr exp))))
(m^ '$%e (simplify `((%log) ,exp))))
(t (subst0 (cons (cons (caar exp) ())
(mapcar #'(lambda (x)
(limitsimp-expt x var))
(cdr exp)))
exp))))
(defun sin-sq-cos-sq-sub (exp) ;Hack ... Hack
(let ((arg (involve exp '(%sin %cos))))
(cond
((null arg) exp)
(t (let ((new-exp ($substitute (m+t 1 (m- (m^t `((%sin simp) ,arg) 2)))
(m^t `((%cos simp) ,arg) 2)
($substitute
(m+t 1 (m- (m^t `((%cos simp) ,arg) 2)))
(m^t `((%sin simp) ,arg) 2)
exp))))
(cond ((not (involve new-exp '(%sin %cos))) new-exp)
(t exp)))))))
(defun expand-trigs (x var)
(cond ((atom x) x)
((mnump x) x)
((and (or (eq (caar x) '%sin)
(eq (caar x) '%cos))
(not (free (cadr x) var)))
($trigexpand x))
((member 'array (car x))
;; Some kind of array reference. Return it.
x)
(t (simplify (cons (ncons (caar x))
(mapcar #'(lambda (x)
(expand-trigs x var))
(cdr x)))))))
(defun tansc (e)
(cond ((not (involve e
'(%cot %csc %binomial
%sec %coth %sech %csch
%acot %acsc %asec %acoth
%asech %acsch
%jacobi_ns %jacobi_nc %jacobi_cs
%jacobi_ds %jacobi_dc)))
e)
(t ($ratsimp (tansc1 e)))))
(defun tansc1 (e &aux tem)
(cond ((atom e) e)
((and (setq e (cons (car e) (mapcar 'tansc1 (cdr e)))) ()))
((setq tem (assoc (caar e) '((%cot . %tan) (%coth . %tanh)
(%sec . %cos) (%sech . %cosh)
(%csc . %sin) (%csch . %sinh)) :test #'eq))
(tansc1 (m^ (list (ncons (cdr tem)) (cadr e)) -1.)))
((setq tem (assoc (caar e) '((%jacobi_nc . %jacobi_cn)
(%jacobi_ns . %jacobi_sn)
(%jacobi_cs . %jacobi_sc)
(%jacobi_ds . %jacobi_sd)
(%jacobi_dc . %jacobi_cd)) :test #'eq))
;; Converts Jacobi elliptic function to its reciprocal
;; function.
(tansc1 (m^ (list (ncons (cdr tem)) (cadr e) (third e)) -1.)))
((setq tem (member (caar e) '(%sinh %cosh %tanh) :test #'eq))
(let (($exponentialize t))
(resimplify e)))
((setq tem (assoc (caar e) '((%acsc . %asin) (%asec . %acos)
(%acot . %atan) (%acsch . %asinh)
(%asech . %acosh) (%acoth . %atanh)) :test #'eq))
(list (ncons (cdr tem)) (m^t (cadr e) -1.)))
((and (eq (caar e) '%binomial) (among var (cdr e)))
(m// `((mfactorial) ,(cadr e))
(m* `((mfactorial) ,(m+t (cadr e) (m- (caddr e))))
`((mfactorial) ,(caddr e)))))
(t e)))
(defun hyperex (ex)
(cond ((not (involve ex '(%sin %cos %tan %asin %acos %atan
%sinh %cosh %tanh %asinh %acosh %atanh)))
ex)
(t (hyperex0 ex))))
(defun hyperex0 (ex)
(cond ((atom ex) ex)
((eq (caar ex) '%sinh)
(m// (m+ (m^ '$%e (cadr ex)) (m- (m^ '$%e (m- (cadr ex)))))
2))
((eq (caar ex) '%cosh)
(m// (m+ (m^ '$%e (cadr ex)) (m^ '$%e (m- (cadr ex))))
2))
((and (member (caar ex)
'(%sin %cos %tan %asin %acos %atan %sinh
%cosh %tanh %asinh %acosh %atanh) :test #'eq)
(among var ex))
(hyperex1 ex))
(t (cons (car ex) (mapcar #'hyperex0 (cdr ex))))))
(defun hyperex1 (ex)
(resimplify ex))
;;Used by tlimit also.
(defmfun limit1 (exp var val)
(prog ()
(let ((lhprogress? lhp?) (lhp? ()) (ans ()))
(cond ((setq ans (and (not (atom exp))
(getlimval exp)))
(return ans))
((and (not (infinityp val))
(setq ans (simplimsubst val exp)))
(return ans))
(t nil))
;;;NUMDEN* => (numerator . denominator)
(destructuring-let (((n . dn) (numden* exp)))
(cond
((not (among var dn))
(return (simplimit (m// (simplimit n var val) dn)
var
val)))
((not (among var n))
(return (simplimit (m* n
(simplimexpt dn
-1
(simplimit dn var val)
-1))
var
val)))
((and lhprogress?
(/#alike n (car lhprogress?))
(/#alike dn (cdr lhprogress?)))
(throw 'lhospital nil)))
(return (limit2 n dn var val))))))
(defun /#alike (e f)
(cond ((alike1 e f)
t)
(t (let ((deriv (sdiff (m// e f) var)))
(cond ((=0 deriv)
t)
((=0 ($ratsimp deriv))
t)
(t nil))))))
(defun limit2 (n dn var val)
(prog (n1 d1 lim-sign gcp sheur-ans)
(setq n (hyperex n) dn (hyperex dn))
;;;Change to uniform limit call.
(cond ((infinityp val)
(setq d1 (limit dn var val nil))
(setq n1 (limit n var val nil)))
(t (cond ((setq n1 (simplimsubst val n)) nil)
(t (setq n1 (limit n var val nil))))
(cond ((setq d1 (simplimsubst val dn)) nil)
(t (setq d1 (limit dn var val nil))))))
(cond ((or (null n1) (null d1)) (return nil))
(t (setq n1 (sratsimp n1) d1 (sratsimp d1))))
(cond ((or (involve n '(mfactorial)) (involve dn '(mfactorial)))
(let ((ans (limfact2 n dn var val)))
(cond (ans (return ans))))))
(cond ((and (zerop2 n1) (zerop2 d1))
(cond ((not (equal (setq gcp (gcpower n dn)) 1))
(return (colexpt n dn gcp)))
((and (real-epsilonp val)
(not (free n '%log))
(not (free dn '%log)))
(return (liminv (m// n dn))))
((setq n1 (try-lhospital-quit n dn nil))
(return n1))))
((and (zerop2 n1) (not (member d1 '($ind $und) :test #'eq))) (return 0))
((zerop2 d1)
(setq n1 (ridofab n1))
(return (simplimtimes `(,n1 ,(simplimexpt dn -1 d1 -1))))))
(setq n1 (ridofab n1))
(setq d1 (ridofab d1))
(cond ((or (eq d1 '$und)
(and (eq n1 '$und) (not (real-infinityp d1))))
(return '$und))
((eq d1 '$ind)
;; At this point we have n1/$ind. Look if n1 is one of the
;; infinities or zero.
(cond ((and (infinityp n1) (eq ($sign dn) '$pos))
(return n1))
((and (infinityp n1) (eq ($sign dn) '$neg))
(return (simpinf (m* -1 n1))))
((and (not (eq n1 '$ind))
(eq ($csign n1) '$zero))
(return 0))
(t (return '$und))))
((eq n1 '$ind) (return (cond ((infinityp d1) 0)
((equal d1 0) '$und)
(t '$ind)))) ;SET LB
((and (real-infinityp d1) (member n1 '($inf $und $minf) :test #'eq))
(cond ((and (not (atom dn)) (not (atom n))
(cond ((not (equal (setq gcp (gcpower n dn)) 1))
(return (colexpt n dn gcp)))
((and (eq '$inf val)
(or (involve dn '(mfactorial %gamma))
(involve n '(mfactorial %gamma))))
(return (limfact n dn))))))
((eq n1 d1) (setq lim-sign 1) (go cp))
(t (setq lim-sign -1) (go cp))))
((and (infinityp d1) (infinityp n1))
(setq lim-sign (if (or (eq d1 '$minf) (eq n1 '$minf)) -1 1))
(go cp))
(t (return (simplimtimes `(,n1 ,(m^ d1 -1))))))
cp (setq n ($expand n) dn ($expand dn))
(cond ((mplusp n)
(let ((new-n (m+l (maxi (cdr n)))))
(cond ((not (alike1 new-n n))
(return (limit (m// new-n dn) var val 'think))))
(setq n1 new-n)))
(t (setq n1 n)))
(cond ((mplusp dn)
(let ((new-dn (m+l (maxi (cdr dn)))))
(cond ((not (alike1 new-dn dn))
(return (limit (m// n new-dn) var val 'think))))
(setq d1 new-dn)))
(t (setq d1 dn)))
(setq sheur-ans (sheur0 n1 d1))
(cond ((or (member sheur-ans '($inf $zeroa) :test #'eq)
(free sheur-ans var))
(return (simplimtimes `(,lim-sign ,sheur-ans))))
((and (alike1 sheur-ans dn)
(not (mplusp n))))
((member (setq n1 (cond ((expfactorp n1 d1) (expfactor n1 d1 var))
(t ())))
'($inf $zeroa) :test #'eq)
(return n1))
((not (null (setq n1 (cond ((expfactorp n dn) (expfactor n dn var))
(t ())))))
(return n1))
((and (alike1 sheur-ans dn) (not (mplusp n))))
((not (alike1 sheur-ans (m// n dn)))
(return (simplimit (m// ($expand (m// n sheur-ans))
($expand (m// dn sheur-ans)))
var
val))))
(cond ((and (not (and (eq val '$inf) (expp n) (expp dn)))
(setq n1 (try-lhospital-quit n dn nil))
(not (eq n1 '$und)))
(return n1)))
(throw 'limit t)))
;; Test whether both n and dn have form
;; product of poly^poly
(defun expfactorp (n dn)
(do ((llist (append (cond ((mtimesp n) (cdr n))
(t (ncons n)))
(cond ((mtimesp dn) (cdr dn))
(t (ncons dn))))
(cdr llist))
(exp? t) ;IS EVERY ELEMENT SO FAR
(factor nil)) ;A POLY^POLY?
((or (null llist)
(not exp?))
exp?)
(setq factor (car llist))
(setq exp? (or (polyinx factor var ())
(and (mexptp factor)
(polyinx (cadr factor) var ())
(polyinx (caddr factor) var ()))))))
(defun expfactor (n dn var) ;Attempts to evaluate limit by grouping
(prog (highest-deg) ; terms with similar exponents.
(let ((new-exp (exppoly n))) ;exppoly unrats expon
(setq n (car new-exp) ;and rtns deg of expons
highest-deg (cdr new-exp)))
(cond ((null n) (return nil))) ;nil means expon is not
(let ((new-exp (exppoly dn))) ;a rat func.
(setq dn (car new-exp)
highest-deg (max highest-deg (cdr new-exp))))
(cond ((or (null dn)
(= highest-deg 0)) ; prevent infinite recursion
(return nil)))
(return
(do ((answer 1)
(degree highest-deg (1- degree))
(numerator n)
(denominator dn)
(numfactors nil)
(denfactors nil))
((= degree -1)
(m* answer
(limit (m// numerator denominator)
var
val
'think)))
(let ((newnumer-factor (get-newexp&factors
numerator
degree
var)))
(setq numerator (car newnumer-factor)
numfactors (cdr newnumer-factor)))
(let ((newdenom-factor (get-newexp&factors
denominator
degree
var)))
(setq denominator (car newdenom-factor)
denfactors (cdr newdenom-factor)))
(setq answer (simplimit (list '(mexpt)
(m* answer
(m// numfactors denfactors))
(cond ((> degree 0) var)
(t 1)))
var
val))
(cond ((member answer '($ind $und) :test #'equal)
(return nil))
((member answer '($inf $minf 0) :test #'equal) ;Really? ZEROA ZEROB?
(return (simplimtimes (list (m// numerator denominator) answer)))))))))
(defun exppoly (exp) ;RETURNS EXPRESSION WITH UNRATTED EXPONENTS
(do ((factor nil)
(highest-deg 0)
(new-exp 1)
(exp (cond ((mtimesp exp)
(cdr exp))
(t (ncons exp)))
(cdr exp)))
((null exp) (cons new-exp highest-deg))
(setq factor (car exp))
(setq new-exp
(m* (cond ((or (not (mexptp factor))
(not (ratp (caddr factor) var)))
factor)
(t (setq highest-deg
(max highest-deg
(ratdegree (caddr factor))))
(m^ (cadr factor) (unrat (caddr factor)))))
new-exp))))
(defun unrat (exp) ;RETURNS UNRATTED EXPRESION
(let ((n-dn (numden* exp)))
(let ((tem ($divide (car n-dn) (cdr n-dn))))
(m+ (cadr tem)
(m// (caddr tem)
(cdr n-dn))))))
(defun get-newexp&factors (exp degree var) ;RETURNS (CONS NEWEXP FACTORS)
(do ((terms (cond ((mtimesp exp)(cdr exp)) ; SUCH THAT
(t (ncons exp))) ; NEWEXP*FACTORS^(VAR^DEGREE)
(cdr terms)) ; IS EQUAL TO EXP.
(factors 1)
(newexp 1)
(factor nil))
((null terms)
(cons newexp
factors))
(setq factor (car terms))
(cond ((not (mexptp factor))
(cond ((= degree 0)
(setq factors (m* factor factors)))
(t (setq newexp (m* factor newexp)))))
((or (= degree -1)
(= (ratdegree (caddr factor))
degree))
(setq factors (m* (m^ (cadr factor)
(leading-coef (caddr factor)))
factors)
newexp (m* (m^ (cadr factor)
(m- (caddr factor)
(m* (leading-coef (caddr factor))
(m^ var degree))))
newexp)))
(t (setq newexp (m* factor newexp))))))
(defun leading-coef (rat)
(ratlim (m// rat (m^ var (ratdegree rat)))))
(defun ratdegree (rat)
(let ((n-dn (numden* rat)))
(- (deg (car n-dn))
(deg (cdr n-dn)))))
(defun limfact2 (n d var val)
(let ((n1 (reflect0 n var val))
(d1 (reflect0 d var val)))
(cond ((and (alike1 n n1)
(alike1 d d1))
nil)
(t (limit (m// n1 d1) var val 'think)))))
;; takes expression and returns operator at front with all flags removed
;; except array flag.
;; array flag must match for alike1 to consider two things to be the same.
;; ((MTIMES SIMP) ... ) => (MTIMES)
;; ((PSI SIMP ARRAY) 0) => (PSI ARRAY)
(defun operator-with-array-flag (exp)
(cond ((member 'array (car exp) :test #'eq)
(list (caar exp) 'array))
(t (list (caar exp)))))
(defun reflect0 (exp var val)
(cond ((atom exp) exp)
((and (eq (caar exp) 'mfactorial)
(let ((argval (limit (cadr exp) var val 'think)))
(or (eq argval '$minf)
(and (numberp argval)
(> 0 argval)))))
(reflect (cadr exp)))
(t (cons (operator-with-array-flag exp)
(mapcar (function
(lambda (term)
(reflect0 term var val)))
(cdr exp))))))
(defun reflect (arg)
(m* -1
'$%pi
(m^ (list (ncons 'mfactorial)
(m+ -1
(m* -1 arg)))
-1)
(m^ (list (ncons '%sin)
(m* '$%pi arg))
-1)))
(defun limfact (n d)
(let ((ans ()))
(setq n (stirling0 n)
d (stirling0 d))
(setq ans ($limit (m// n d) var '$inf))
(cond ((and (atom ans)
(not (member ans '(und ind ) :test #'eq))) ans)
((eq (caar ans) '%limit) ())
(t ans))))
;; substitute asymptotic approximations for gamma, factorial, and
;; polylogarithm
(defun stirling0 (e)
(cond ((atom e) e)
((and (setq e (cons (car e) (mapcar 'stirling0 (cdr e))))
nil))
((and (eq (caar e) '%gamma)
(eq (limit (cadr e) var val 'think) '$inf))
(stirling (cadr e)))
((and (eq (caar e) 'mfactorial)
(eq (limit (cadr e) var val 'think) '$inf))
(m* (cadr e) (stirling (cadr e))))
((and (eq (caar e) 'mqapply) ;; polylogarithm
(eq (subfunname e) '$li)
(integerp (car (subfunsubs e))))
(li-asymptotic-expansion (m- (car (subfunsubs e)) 1)
(car (subfunsubs e))
(car (subfunargs e))))
(t e)))
(defun stirling (x)
(maxima-substitute x '$z
'((mtimes simp)
((mexpt simp) 2 ((rat simp) 1 2))
((mexpt simp) $%pi ((rat simp) 1 2))
((mexpt simp) $z ((mplus simp) ((rat simp) -1 2) $z))
((mexpt simp) $%e ((mtimes simp) -1 $z)))))
(defun no-err-sub (v e &aux ans)
(let ((errorsw t) (errrjfflag t) (*zexptsimp? t)
(errcatch t)
;; Don't print any error messages
($errormsg nil))
(declare (special errcatch))
;; Should we just use IGNORE-ERRORS instead HANDLER-CASE here? I
;; (rtoy) am choosing the latter so that unexpected errors will
;; actually show up instead of being silently discarded.
(handler-case
(setq ans (catch 'errorsw
(catch 'raterr
(sratsimp (subin v e)))))
(maxima-$error ()
(setq ans nil)))
(cond ((null ans) t) ; Ratfun package returns NIL for failure.
(t ans))))
;; substitute value v for var into expression e.
;; if result is defined and e is continuous, we have the limit.
(defun simplimsubst (v e)
(let (ans)
(cond ((involve e '(mfactorial)) nil)
;; functions that are defined at their discontinuities
((amongl '($atan2 $floor $round $ceiling %signum %integrate
%gamma_incomplete)
e) nil)
;; substitute value into expression
((eq (setq ans (no-err-sub (ridofab v) e)) t)
nil)
((and (member v '($zeroa $zerob) :test #'eq) (=0 ($radcan ans)))
(setq ans (behavior e var v))
(cond ((equal ans 1) '$zeroa)
((equal ans -1) '$zerob)
(t nil))) ; behavior can't find direction
(t ans))))
;;;returns (cons numerator denominator)
(defun numden* (e)
(let ((e (factor (simplify e)))
(numer ()) (denom ()))
(cond ((atom e)
(push e numer))
((mtimesp e)
(mapc 'forq (cdr e)))
(t (forq e)))
(cond ((null numer)
(setq numer 1.))
((null (cdr numer))
(setq numer (car numer)))
(t (setq numer (m*l numer))))
(cond ((null denom)
(setq denom 1.))
((null (cdr denom))
(setq denom (car denom)))
(t (setq denom (m*l denom))))
(cons (factor numer) (factor denom))))
;;;FACTOR OR QUOTIENT
;;;Setq's the special vars numer and denom from numden*
(defun forq (e)
(cond ((and (mexptp e)
(not (freeof var e))
(null (pos-neg-p (caddr e))))
(push (m^ (cadr e) (m* -1. (caddr e))) denom))
(t (push e numer))))
;;;Predicate to tell whether an expression is pos,zero or neg as var -> val.
;;;returns T if pos,zero. () if negative or don't know.
(defun pos-neg-p (exp)
(let ((ans (limit exp var val 'think)))
(cond ((and (not (member ans '($und $ind $infinity) :test #'eq))
(equal ($imagpart ans) 0))
(let ((sign (getsignl ans)))
(cond ((or (equal sign 1)
(equal sign 0))
t)
((equal sign -1) nil))))
(t 'unknown))))
(declare-top (unspecial n dn))
(defun expp (e)
(cond ((radicalp e var) nil)
((member (caar e) '(%log %sin %cos %tan %sinh %cosh %tanh mfactorial
%asin %acos %atan %asinh %acosh %atanh) :test #'eq) nil)
((simplexp e) t)
((do ((e (cdr e) (cdr e)))
((null e) nil)
(and (expp (car e)) (return t))))))
(defun simplexp (e)
(and (mexptp e)
(radicalp (cadr e) var)
(among var (caddr e))
(radicalp (caddr e) var)))
(defun gcpower (a b)
($gcd (getexp a) (getexp b)))
(defun getexp (exp)
(cond ((and (mexptp exp)
(free (caddr exp) var)
(eq (ask-integer (caddr exp) '$integer) '$yes))
(caddr exp))
((mtimesp exp) (getexplist (cdr exp)))
(t 1)))
(defun getexplist (list)
(cond ((null (cdr list))
(getexp (car list)))
(t ($gcd (getexp (car list))
(getexplist (cdr list))))))
(defun limroot (exp power)
(cond ((or (atom exp) (not (member (caar exp) '(mtimes mexpt) :test #'eq)))
(limroot (list '(mexpt) exp 1) power)) ;This is strange-JIM.
((mexptp exp) (m^ (cadr exp)
(sratsimp (m* (caddr exp) (m^ power -1.)))))
(t (m*l (mapcar #'(lambda (x)
(limroot x power))
(cdr exp))))))
;;NUMERATOR AND DENOMINATOR HAVE EXPONENTS WITH GCD OF GCP.
;;; Used to call simplimit but some of the transformations used here
;;; were not stable w.r.t. the simplifier, so try keeping exponent separate
;;; from bas.
(defun colexpt (n dn gcp)
(let ((bas (m* (limroot n gcp) (limroot dn (m* -1 gcp))))
(expo gcp)
baslim expolim)
(setq baslim (limit bas var val 'think))
(setq expolim (limit expo var val 'think))
(simplimexpt bas expo baslim expolim)))
;;; This function will transform an expression such that either all logarithms
;;; contain arguments not becoming infinite or are of the form
;;; LOG(LOG( ... LOG(VAR))) This reduction takes place only over the operators
;;; MPLUS, MTIMES, MEXPT, and %LOG.
(defun log-red-contract (facs)
(do ((l facs (cdr l))
(consts ())
(log ()))
((null l)
(if log (cons (cadr log) (m*l consts))
()))
(cond ((freeof var (car l)) (push (car l) consts))
((mlogp (car l))
(if (null log) (setq log (car l))
(return ())))
(t (return ())))))
(defun log-reduce (x)
(cond ((atom x) x)
((freeof var x) x)
((mplusp x)
(do ((l (cdr x) (cdr l))
(sum ())
(weak-logs ())
(strong-logs ())
(temp))
((null l) (m+l `(((%log) ,(m*l strong-logs))
((%log) ,(m*l weak-logs))
,@sum)))
(setq x (log-reduce (car l)))
(cond ((mlogp x)
(if (infinityp (limit (cadr x) var val 'think))
(push (cadr x) strong-logs)
(push (cadr x) weak-logs)))
((and (mtimesp x) (setq temp (log-red-contract (cdr x))))
(if (infinityp (limit (car temp) var val 'think))
(push (m^ (car temp) (cdr temp)) strong-logs)
(push (m^ (car temp) (cdr temp)) weak-logs)))
(t (push x sum)))))
((mtimesp x)
(do ((l (cdr x) (cdr l))
(ans 1))
((null l) ans)
(setq ans ($expand (m* (log-reduce (car l)) ans)))))
((mexptp x) (m^t (log-reduce (cadr x)) (caddr x)))
((mlogp x)
(cond ((not (infinityp (limit (cadr x) var val 'think))) x)
(t
(cond ((eq (cadr x) var) x)
((mplusp (cadr x))
(let ((strongl (maxi (cdadr x))))
(m+ (log-reduce `((%log) ,(car strongl))) `((%log) ,(m// (cadr x) (car strongl))))))
((mtimesp (cadr x))
(do ((l (cdadr x) (cdr l)) (ans 0)) ((null l) ans)
(setq ans (m+ (log-reduce (simplify `((%log) ,(log-reduce (car l))))) ans))))
(t
(let ((red-log (simplify `((%log) ,(log-reduce (cadr x))))))
(if (alike1 red-log x) x (log-reduce red-log))))))))
(t x)))
(defun ratlim (e)
(cond ((member val '($inf $infinity) :test #'eq)
(setq e (maxima-substitute (m^t 'x -1) var e)))
((eq val '$minf)
(setq e (maxima-substitute (m^t -1 (m^t 'x -1)) var e)))
((eq val '$zerob)
(setq e (maxima-substitute (m- 'x) var e)))
((eq val '$zeroa)
(setq e (maxima-substitute 'x var e)))
((setq e (maxima-substitute (m+t 'x val) var e))))
(destructuring-let* ((e (let (($ratfac ()))
($rat (sratsimp e) 'x)))
((h n . d) e)
(g (genfind h 'x))
(nd (lodeg n g))
(dd (lodeg d g)))
(cond ((and (setq e
(subst var
'x
(sratsimp (m// ($ratdisrep `(,h ,(locoef n g) . 1))
($ratdisrep `(,h ,(locoef d g) . 1))))))
(> nd dd))
(cond ((not (member val '($zerob $zeroa $inf $minf) :test #'eq))
0)
((not (equal ($imagpart e) 0))
0)
((null (setq e (getsignl ($realpart e))))
0)
((equal e 1) '$zeroa)
((equal e -1) '$zerob)
(t 0)))
((equal nd dd) e)
((not (member val '($zerob $zeroa $infinity $inf $minf) :test #'eq))
(throw 'limit t))
((eq val '$infinity) '$infinity)
((not (equal ($imagpart e) 0)) '$infinity)
((null (setq e (getsignl ($realpart e)))) '$infinity)
((equal e 1) '$inf)
((equal e -1) '$minf)
(t 0))))
(defun lodeg (n x)
(if (or (atom n) (not (eq (car n) x)))
0
(lowdeg (cdr n))))
(defun locoef (n x)
(if (or (atom n) (not (eq (car n) x)))
n
(car (last n))))
(defun behavior (exp var val) ; returns either -1, 0, 1.
(if (= *behavior-count-now* +behavior-count+)
0
(let ((*behavior-count-now* (1+ *behavior-count-now*)) pair sign)
(cond ((real-infinityp val)
(setq val (cond ((eq val '$inf) '$zeroa)
((eq val '$minf) '$zerob)))
(setq exp (sratsimp (subin (m^ var -1) exp)))))
(cond ((eq val '$infinity) 0) ; Needs more hacking for complex.
((and (mtimesp exp)
(prog2 (setq pair (partition exp var 1))
(not (mtimesp (cdr pair)))))
(setq sign (getsignl (car pair)))
(if (not (fixnump sign))
0
(mul sign (behavior (cdr pair) var val))))
((and (=0 (no-err-sub (ridofab val) exp))
(mexptp exp)
(free (caddr exp) var)
(equal (getsignl (caddr exp)) 1))
(let ((bas (cadr exp)) (expo (caddr exp)))
(behavior-expt bas expo)))
(t (behavior-by-diff exp var val))))))
(defun behavior-expt (bas expo)
(let ((behavior (behavior bas var val)))
(cond ((= behavior 1) 1)
((= behavior 0) 0)
((eq (ask-integer expo '$integer) '$yes)
(cond ((eq (ask-integer expo '$even) '$yes) 1)
(t behavior)))
((ratnump expo)
(cond ((evenp (cadr expo)) 1)
((oddp (caddr expo)) behavior)
(t 0)))
(t 0))))
(defun behavior-by-diff (exp var val)
(cond ((not (or (eq val '$zeroa) (eq val '$zerob))) 0)
(t (let ((old-val val) (old-exp exp))
(setq val (ridofab val))
(do ((ct 0 (1+ ct))
(exp (sratsimp (sdiff exp var)) (sratsimp (sdiff exp var)))
(n () (not n))
(ans ())) ; This do wins by a return.
((> ct 0) 0) ; This loop used to run up to 5 times,
;; but the size of some expressions would blow up.
(setq ans (no-err-sub val exp)) ;Why not do an EVENFN and ODDFN
;test here.
(cond ((eq ans t)
(return (behavior-numden old-exp var old-val)))
((=0 ans) ()) ;Do it again.
(t (setq ans (getsignl ans))
(cond (n (return ans))
((equal ans 1)
(return (if (eq old-val '$zeroa) 1 -1)))
((equal ans -1)
(return (if (eq old-val '$zeroa) -1 1)))
(t (return 0))))))))))
(defun behavior-numden (exp var val)
(let ((num ($num exp)) (denom ($denom exp)))
(cond ((equal denom 1) 0) ;Could be hacked more from here.
(t (let ((num-behav (behavior num var val))
(denom-behav (behavior denom var val)))
(cond ((or (= num-behav 0) (= denom-behav 0)) 0)
((= num-behav denom-behav) 1)
(t -1)))))))
(defun try-lhospital (n d ind)
;;Make one catch for the whole bunch of lhospital trials.
(let ((ans (lhospital-catch n d ind)))
(cond ((null ans) ())
((not (free-infp ans)) (simpinf ans))
((not (free-epsilonp ans)) (simpab ans))
(t ans))))
(defun try-lhospital-quit (n d ind)
(let ((ans (or (lhospital-catch n d ind)
(lhospital-catch (m^ d -1) (m^ n -1) ind))))
(cond ((null ans) (throw 'limit t))
((not (free-infp ans)) (simpinf ans))
((not (free-epsilonp ans)) (simpab ans))
(t ans))))
(defun lhospital-catch (n d ind)
(cond ((> 0 lhcount)
(setq lhcount $lhospitallim)
(throw 'lhospital nil))
((equal lhcount $lhospitallim)
(let ((lhcount (m+ lhcount -1)))
(catch 'lhospital (lhospital n d ind))))
(t (setq lhcount (m+ lhcount -1))
(prog1 (lhospital n d ind)
(setq lhcount (m+ lhcount 1))))))
;;If this succeeds then raise LHCOUNT.
(defun lhospital (n d ind)
(declare (special val lhp?))
(if (mtimesp n)
(setq n (m*l (mapcar #'(lambda (term) (lhsimp term var val))
(cdr n)))))
(if (mtimesp d)
(setq d (m*l (mapcar #'(lambda (term) (lhsimp term var val))
(cdr d)))))
(destructuring-let (((n . d) (lhop-numden n d))
const nconst dconst)
(setq lhp? (and (null ind) (cons n d)))
(desetq (nconst . n) (var-or-const n))
(desetq (dconst . d) (var-or-const d))
(setq n (stirling0 n)) ;; replace factorial and %gamma
(setq d (stirling0 d)) ;; with approximations
(setq n (sdiff n var) ;; take derivatives for l'hospital
d (sdiff d var))
(if (or (not (free n '%derivative)) (not (free d '%derivative)))
(throw 'lhospital ()))
(setq n (expand-trigs (tansc n) var))
(setq d (expand-trigs (tansc d) var))
(desetq (const . (n . d)) (remove-singularities n d))
(setq const (m* const (m// nconst dconst)))
(simpinf
(cond (ind (let ((ans (limit2 n d var val)))
(if ans (m* const ans))))
(t (let ((ans (limit
(cond ((mplusp n)
(m+l (mapcar #'(lambda (x)
(sratsimp (m// x d)))
(cdr n))))
(t ($multthru (sratsimp (m// n d)))))
var val 'think)))
(if ans (m* const ans))))))))
;; Heuristics for picking the right way to express a LHOSPITAL problem.
(defun lhop-numden (num denom)
(declare (special var))
(cond ((let ((log-num (involve num '(%log))))
(cond ((null log-num) ())
((lessthan (num-of-logs (factor (sratsimp (sdiff (m^ num -1) var))))
(num-of-logs (factor (sratsimp (sdiff num var)))))
(psetq num (m^ denom -1) denom (m^ num -1))
t)
(t t))))
((let ((log-denom (involve denom '(%log))))
(cond ((null log-denom) ())
((lessthan (num-of-logs (sratsimp (sdiff (m^ denom -1) var)))
(num-of-logs (sratsimp (sdiff denom var))))
(psetq denom (m^ num -1) num (m^ denom -1))
t)
(t t))))
((let ((exp-num (%einvolve num)))
(cond (exp-num
(cond ((%e-right-placep exp-num)
t)
(t (psetq num (m^ denom -1)
denom (m^ num -1)) t)))
(t ()))))
((let ((exp-den (%einvolve denom)))
(cond (exp-den
(cond ((%e-right-placep exp-den)
t)
(t (psetq num (m^ denom -1)
denom (m^ num -1)) t)))
(t ()))))
((let ((scnum (involve num '(%sin))))
(cond (scnum (cond ((trig-right-placep '%sin scnum) t)
(t (psetq num (m^ denom -1)
denom (m^ num -1)) t)))
(t ()))))
((let ((scden (involve denom '(%sin))))
(cond (scden (cond ((trig-right-placep '%sin scden) t)
(t (psetq num (m^ denom -1)
denom (m^ num -1)) t)))
(t ()))))
((let ((scnum (involve num '(%asin %acos %atan))))
;; If the numerator contains an inverse trig and the
;; denominator or reciprocal of denominator is polynomial,
;; leave everthing as is. If the inverse trig is moved to
;; the denominator, things get messy, even if the numerator
;; becomes a polynomial. This is not perfect.
(cond ((and scnum (or (polyinx denom var ())
(polyinx (m^ denom -1) var ())))
t)
(t nil))))
((or (oscip num) (oscip denom)))
((frac num)
(psetq num (m^ denom -1) denom (m^ num -1))))
(cons num denom))
;;i don't know what to do here for some cases, may have to be refined.
(defun num-of-logs (exp)
(cond ((mapatom exp) 0)
((equal (caar exp) '%log)
(m+ 1 (num-of-log-l (cdr exp))))
((and (mexptp exp) (mnump (caddr exp)))
(m* (simplify `((mabs) ,(caddr exp)))
(num-of-logs (cadr exp))))
(t (num-of-log-l (cdr exp)))))
(defun num-of-log-l (llist)
(do ((temp llist (cdr temp)) (ans 0))
((null temp) ans)
(setq ans (m+ ans (num-of-logs (car temp))))))
(defun %e-right-placep (%e-arg)
(let ((%e-arg-diff (sdiff %e-arg var)))
(cond
((free %e-arg-diff var)) ;simple cases
((or (and (mexptp denom)
(equal (cadr denom) -1))
(polyinx (m^ denom -1) var ())) ())
((let ((%e-arg-diff-lim (ridofab (limit %e-arg-diff var val 'think)))
(%e-arg-exp-lim (ridofab (limit (m^ '$%e %e-arg) var val 'think))))
#+nil
(progn
(format t "%e-arg-dif-lim = ~A~%" %e-arg-diff-lim)
(format t "%e-arg-exp-lim = ~A~%" %e-arg-exp-lim))
(cond ((equal %e-arg-diff-lim %e-arg-exp-lim)
t)
((and (mnump %e-arg-diff-lim) (mnump %e-arg-exp-lim))
t)
((and (mnump %e-arg-diff-lim) (infinityp %e-arg-exp-lim))
;; This is meant to make maxima handle bug 1469411
;; correctly. Undoubtedly, this needs work.
t)
(t ())))))))
(defun trig-right-placep (trig-type arg)
(let ((arglim (ridofab (limit arg var val 'think)))
(triglim (ridofab (limit `((,trig-type) ,arg) var val 'think))))
(cond ((and (equal arglim 0) (equal triglim 0)) t)
((and (infinityp arglim) (infinityp triglim)) t)
(t ()))))
;;Takes a numerator and a denominator. If they tries all combinations of
;;products to try and make a simpler set of subproblems for LHOSPITAL.
(defun remove-singularities (numer denom)
(cond
((or (null numer) (null denom)
(atom numer) (atom denom)
(not (mtimesp numer)) ;Leave this here for a while.
(not (mtimesp denom)))
(cons 1 (cons numer denom)))
(t
(destructuring-let (((num-consts . num-vars) (var-or-const numer))
((denom-consts . denom-vars) (var-or-const denom))
(const 1))
(if (not (mtimesp num-vars))
(setq num-vars (list num-vars))
(setq num-vars (cdr num-vars)))
(if (not (mtimesp denom-vars))
(setq denom-vars (list denom-vars))
(setq denom-vars (cdr denom-vars)))
(do ((nl num-vars (cdr nl))
(num-list (copy-list num-vars ))
(den-list denom-vars den-list-temp)
(den-list-temp (copy-list denom-vars )))
((null nl) (cons (m* const (m// num-consts denom-consts))
(cons (m*l num-list) (m*l den-list-temp))))
(do ((dl den-list (cdr dl)))
((null dl) t)
(cond ((or (%einvolve (car nl))
(%einvolve (car nl))) t)
(t (let ((lim (catch 'limit
(simpinf
(simpab (limit (m// (car nl) (car dl))
var val 'think))))))
(cond ((or (eq lim t) (eq lim ())
(equal (ridofab lim) 0)
(infinityp lim)
(not (free lim '$inf))
(not (free lim '$minf))
(not (free lim '$infinity))
(not (free lim '$ind))
(not (free lim '$und)))
())
(t (setq const (m* lim const))
(setq num-list (delete (car nl) num-list :count 1 :test #'equal))
(setq den-list-temp
(delete (car dl) den-list-temp :count 1 :test #'equal))
(return t))))))))))))
;; separate terms that contain var from constant terms
;; returns (const-terms . var-terms)
(defun var-or-const (expr)
(setq expr ($factor expr))
(cond ((atom expr)
(cond ((eq expr var) (cons 1 expr))
(t (cons expr 1))))
((free expr var) (cons expr 1))
((mtimesp expr)
(do ((l (cdr expr) (cdr l))
(const 1) (varl 1))
((null l) (cons const varl))
(cond ((free (car l) var)
(setq const (m* (car l) const)))
(t (setq varl (m* (car l) varl))))))
(t (cons 1 expr))))
;; if term goes to non-zero constant, replace with constant
(defun lhsimp (term var val)
(cond ((atom term) term)
(t
(let ((term-value (ridofab (limit term var val 'think))))
(cond ((not (member term-value
'($inf $minf $und $ind $infinity 0)))
term-value)
(t term))))))
(defun bylog (expo bas)
(simplimexpt '$%e
(setq bas
(try-lhospital-quit (simplify `((%log) ,(tansc bas)))
(m^ expo -1)
nil))
'$%e bas))
(defun simplimexpt (bas expo bl el)
(cond
((or (eq bl '$und) (eq el '$und)) '$und)
((zerop2 bl)
(cond ((eq el '$inf) (if (eq bl '$zeroa) bl 0))
((eq el '$minf) (if (eq bl '$zeroa) '$inf '$infinity))
((eq el '$ind) '$ind)
((eq el '$infinity) '$und)
((zerop2 el) (bylog expo bas))
(t (cond ((equal (getsignl el) -1)
(cond ((eq bl '$zeroa) '$inf)
((eq bl '$zerob)
(cond ((even1 el) '$inf)
((eq (ask-integer el '$integer) '$yes)
(cond ((eq (ask-integer el '$even) '$yes)
'$inf)
(t '$minf))))) ;Gotta be ODD.
(t (setq bas (behavior bas var val))
(cond ((equal bas 1) '$inf)
((equal bas -1) '$minf)
(t (throw 'limit t))))))
((and (mnump el)
(member bl '($zeroa $zerob) :test #'eq))
(cond ((even1 el) '$zeroa)
((and (eq bl '$zerob)
(ratnump el)
(evenp (caddr el))) 0)
(t bl)))
((and (equal (getsignl el) 1)
(eq bl '$zeroa)) bl)
((equal (getsignl el) 0)
1)
(t 0)))))
((eq bl '$infinity)
(cond ((zerop2 el) (bylog expo bas))
((eq el '$minf) 0)
((eq el '$inf) '$infinity)
((member el '($infinity $ind) :test #'eq) '$und)
((equal (setq el (getsignl el)) 1) '$infinity)
((null el) '$und)
((equal el -1) 0)))
((eq bl '$inf)
(cond ((eq el '$inf) '$inf)
((equal el '$minf) 0)
((zerop2 el) (bylog expo bas))
((member el '($infinity $ind) :test #'eq) '$und)
(t (cond ((eql 0 (getsignl el)) 1)
((ratgreaterp 0 el) '$zeroa)
(t '$inf)))))
((eq bl '$minf)
(cond ((zerop2 el) (bylog expo bas))
((eq el '$inf) '$und)
((equal el '$minf) 0)
;;;Why not generalize this. We can ask about the number. -Jim 2/23/81
((mnump el) (cond ((mnegp el)
(cond ((even1 el) '$zeroa)
(t (cond
((eq (ask-integer el '$integer) '$yes)
(cond ((eq (ask-integer el '$even)
'$yes) '$zeroa)
(t '$zerob)))
(t 0)))))
(t (cond
((even1 el) '$inf)
((eq (ask-integer el '$integer) '$yes)
(cond ((eq (ask-integer el '$even) '$yes)
'$inf)
(t '$minf)))
(t '$infinity)))))
(loginprod? (throw 'lip? 'lip!))
(t '$und)))
((equal (simplify (ratdisrep (ridofab bl))) 1)
(if (infinityp el) (bylog expo bas) 1))
((and (equal (ridofab bl) -1)
(infinityp el)) '$ind) ;LB
((eq bl '$ind) (cond ((or (zerop2 el) (infinityp el)) '$und)
((not (equal (getsignl el) -1)) '$ind)
(t '$und)))
((eq el '$inf) (cond ((abeq1 bl)
(cond ((equal (getsignl bl) 1) 1)
(t '$ind)))
((abless1 bl)
(cond ((equal (getsignl bl) 1) '$zeroa)
(t 0)))
((equal (getsignl bl) -1) '$infinity)
((equal (getsignl (m1- bl)) 1) '$inf)
(t (throw 'limit t))))
((eq el '$minf) (cond ((abeq1 bl)
(cond ((equal (getsignl bl) 1) 1)
(t '$ind)))
((not (abless1 bl))
(cond ((equal (getsignl bl) 1) '$zeroa)
(t 0)))
((ratgreaterp 0 bl) '$infinity)
(t '$inf)))
((eq el '$infinity)
(if (equal val '$infinity)
'$und ;Not enough info to do anything.
(destructuring-let (((real-el . imag-el) (trisplit expo)))
(setq real-el (limit real-el var origval nil))
(cond ((eq real-el '$minf) 0)
((and (eq real-el '$inf)
(not (equal (ridofab (limit imag-el var origval nil))
0))) '$infinity)
((eq real-el '$infinity)
(throw 'limit t)) ;; don't really know real component
(t '$ind)))))
((eq el '$ind) '$ind)
((zerop2 el) 1)
(t (m^ bl el))))
(defun even1 (x)
(cond ((numberp x) (and (integerp x) (evenp x)))
((and (mnump x) (evenp (cadr x))))))
;; is absolute value less than one?
(defun abless1 (bl)
(setq bl (nmr bl))
(cond ((mnump bl)
(and (ratgreaterp 1. bl) (ratgreaterp bl -1.)))
(t (equal (getsignl (m1- `((mabs) ,bl))) -1.))))
;; is absolute value equal to one?
(defun abeq1 (bl)
(setq bl (nmr bl))
(cond ((mnump bl)
(or (equal 1. bl) (equal bl -1.)))
(t (equal (getsignl (m1- `((mabs) ,bl))) 0))))
(defmfun simplimit (exp var val &aux op)
(cond
((eq var exp) val)
((or (atom exp) (mnump exp)) exp)
((and (not (infinityp val))
(not (amongl '(%sin %cos %atanh %cosh %sinh %tanh mfactorial %log)
exp))
(not (inf-typep exp))
(simplimsubst val exp)))
((eq (caar exp) '%limit) (throw 'limit t))
((mplusp exp) (simplimplus exp))
((mtimesp exp) (simplimtimes (cdr exp)))
((mexptp exp) (simplimexpt (cadr exp) (caddr exp)
(limit (cadr exp) var val 'think)
(limit (caddr exp) var val 'think)))
((mlogp exp) (simplimln exp var val))
((member (caar exp) '(%sin %cos) :test #'eq)
(simplimsc exp (caar exp) (limit (cadr exp) var val 'think)))
((eq (caar exp) '%tan) (simplim%tan (cadr exp)))
((eq (caar exp) '%atan) (simplim%atan (limit (cadr exp) var val 'think)))
((eq (caar exp) '$atan2) (simplim%atan2 exp))
((member (caar exp) '(%sinh %cosh) :test #'eq)
(simplimsch (caar exp) (limit (cadr exp) var val 'think)))
((eq (caar exp) 'mfactorial)
(simplimfact exp var val))
((member (caar exp) '(%erf %tanh) :test #'eq)
(simplim%erf-%tanh (caar exp) (cadr exp)))
((member (caar exp) '(%acos %asin) :test #'eq)
(simplim%asin-%acos (caar exp) (limit (cadr exp) var val 'think)))
((eq (caar exp) '%atanh)
(simplim%atanh (limit (cadr exp) var val 'think) val))
((eq (caar exp) '%acosh)
(simplim%acosh (limit (cadr exp) var val 'think)))
((eq (caar exp) '%asinh)
(simplim%asinh (limit (cadr exp) var val 'think)))
((eq (caar exp) '%inverse_jacobi_ns)
(simplim%inverse_jacobi_ns (limit (cadr exp) var val 'think) (third exp)))
((eq (caar exp) '%inverse_jacobi_nc)
(simplim%inverse_jacobi_nc (limit (cadr exp) var val 'think) (third exp)))
((eq (caar exp) '%inverse_jacobi_sc)
(simplim%inverse_jacobi_sc (limit (cadr exp) var val 'think) (third exp)))
((eq (caar exp) '%inverse_jacobi_cs)
(simplim%inverse_jacobi_cs (limit (cadr exp) var val 'think) (third exp)))
((eq (caar exp) '%inverse_jacobi_dc)
(simplim%inverse_jacobi_dc (limit (cadr exp) var val 'think) (third exp)))
((eq (caar exp) '%inverse_jacobi_ds)
(simplim%inverse_jacobi_ds (limit (cadr exp) var val 'think) (third exp)))
((and (eq (caar exp) 'mqapply)
(eq (subfunname exp) '$li))
(simplim$li (subfunsubs exp) (subfunargs exp) val))
((and (eq (caar exp) 'mqapply)
(eq (subfunname exp) '$psi))
(simplim$psi (subfunsubs exp) (subfunargs exp) val))
((and (eq (caar exp) var)
(member 'array (car exp) :test #'eq)
(every #'(lambda (sub-exp)
(free sub-exp var))
(cdr exp)))
exp) ;LIMIT(B[I],B,INF); -> B[I]
((setq op (safe-get (mop exp) 'simplim%function))
;; Lookup a simplim%function from the property list
(funcall op exp var val))
(t (if $limsubst
(simplify (cons (operator-with-array-flag exp)
(mapcar #'(lambda (a)
(limit a var val 'think))
(cdr exp))))
(throw 'limit t)))))
(defun liminv (e)
(setq e (resimplify (subst (m// 1 var) var e)))
(let ((new-val (cond ((eq val '$zeroa) '$inf)
((eq val '$zerob) '$minf))))
(if new-val (let ((preserve-direction t))
($limit e var new-val)) (throw 'limit t))))
(defun simplimtimes (exp)
;; The following test
;; handles (-1)^x * 2^x => (-2)^x => $infinity
;; wants to avoid (-1)^x * 2^x => $ind * $inf => $und
(let ((try
(and (expfactorp (cons '(mtimes) exp) 1)
(expfactor (cons '(mtimes) exp) 1 var))))
(when try (return-from simplimtimes try)))
(let ((prod 1) (num 1) (denom 1)
(zf nil) (ind-flag nil) (inf-type nil)
(constant-zero nil) (constant-infty nil))
(dolist (term exp)
(let* ((loginprod? (involve term '(%log)))
(y (catch 'lip? (limit term var val 'think))))
(cond
;; limit failed due to log in product
((eq y 'lip!)
(return-from simplimtimes (liminv (cons '(mtimes simp) exp))))
;; If the limit is infinitesimal or zero
((zerop2 y)
(setf num (m* num term)
constant-zero (or constant-zero (not (among var term))))
(case y
($zeroa
(unless zf (setf zf 1)))
($zerob
(setf zf (* -1 (or zf 1))))))
;; If the limit is not some form of infinity or
;; undefined/indeterminate.
((not (member y '($inf $minf $infinity $ind $und) :test #'eq))
(setq prod (m* prod y)))
((eq y '$und) (return-from simplimtimes '$und))
((eq y '$ind) (setq ind-flag t))
;; Some form of infinity
(t
(setf denom (m* denom term)
constant-infty (or constant-infty (not (among var term))))
(unless (eq inf-type 'infinity)
(cond
((eq y '$infinity) (setq inf-type '$infinity))
((null inf-type) (setf inf-type y))
;; minf * minf or inf * inf
((eq y inf-type) (setf inf-type '$inf))
;; minf * inf
(t (setf inf-type '$minf))))))))
(cond
;; If there are zeros and infinities among the terms that are free of
;; VAR, then we have an expression like "inf * zeroa * f(x)" or
;; similar. That gives an undefined result. Note that we don't
;; necessarily have something undefined if only the zeros have a term
;; free of VAR. For example "zeroa * exp(-1/x) * 1/x" as x -> 0. And
;; similarly for the infinities.
((and constant-zero constant-infty) '$und)
;; If num=denom=1, we didn't find any explicit infinities or zeros, so we
;; either return the simplified product or ind
((and (eql num 1) (eql denom 1))
(if ind-flag '$ind prod))
;; If denom=1 (and so num != 1), we have some form of zero
((equal denom 1)
(if (null zf)
0
(let ((sign (getsignl prod)))
(if (or (not sign) (eq sign 'complex))
0
(ecase (* zf sign)
(1 '$zeroa)
(-1 '$zerob))))))
;; If num=1 (and so denom != 1), we have some form of infinity
((equal num 1)
(let ((sign ($csign prod)))
(cond
(ind-flag '$und)
((eq sign '$pos) inf-type)
((eq sign '$neg) (case inf-type
($inf '$minf)
($minf '$inf)
(t '$infinity)))
((member sign '($complex $imaginary)) '$infinity)
; sign is '$zero, $pnz, $pz, etc
(t (throw 'limit t)))))
;; Both zeros and infinities
(t
;; All bets off if there are some infinities or some zeros, but it
;; needn't be undefined (see above)
(when (or constant-zero constant-infty) (throw 'limit t))
(let ((ans (limit2 num (m^ denom -1) var val)))
(if ans
(simplimtimes (list prod ans))
(throw 'limit t)))))))
;;;PUT CODE HERE TO ELIMINATE FAKE SINGULARITIES??
(defun simplimplus (exp)
(cond ((memalike exp simplimplus-problems)
(throw 'limit t))
(t (unwind-protect
(progn (push exp simplimplus-problems)
(let ((ans (catch 'limit (simplimplus1 exp))))
(cond ((or (eq ans ())
(eq ans t)
(among '%limit ans))
(let ((new-exp (sratsimp exp)))
(cond ((not (alike1 exp new-exp))
(setq ans
(limit new-exp var val 'think))))
(cond ((or (eq ans ())
(eq ans t))
(throw 'limit t))
(t ans))))
(t ans))))
(pop simplimplus-problems)))))
(defun simplimplus1 (exp)
(prog (sum y infl infinityl minfl indl)
(setq sum 0.)
(do ((exp (cdr exp) (cdr exp)) (f))
((or y (null exp)) nil)
(setq f (limit (car exp) var val 'think))
(cond ((null f)
(throw 'limit t))
((eq f '$und) (setq y t))
((not (member f '($inf $minf $infinity $ind) :test #'eq))
(setq sum (m+ sum f)))
((eq f '$ind) (push (car exp) indl))
(infinityl (throw 'limit t))
;;;Don't know what to do with an '$infinity and an $inf or $minf
((eq f '$inf) (push (car exp) infl))
((eq f '$minf) (push (car exp) minfl))
((eq f '$infinity)
(cond ((or infl minfl)
(throw 'limit t))
(t (push (car exp) infinityl))))))
(cond (y (return '$und))
((not (or infl minfl indl infinityl))
(return (cond ((atom sum) sum)
((or (not (free sum '$zeroa))
(not (free sum '$zerob)))
(simpab sum))
(t sum))))
(t (cond ((null infinityl)
(cond (infl (cond ((null minfl) (return '$inf))
(t (go oon))))
(minfl (return '$minf))
((> (length indl) 1)
;; At this point we have a sum of '$ind. We factor
;; the sum and try again. This way we get the limit
;; of expressions like (a-b)*ind, where (a-b)--> 0.
(cond ((not (alike1 (setq y ($factorsum exp)) exp))
(return (limit y var val 'think)))
(t
(return '$ind))))
(t (return '$ind))))
(t (setq infl (append infl infinityl))))))
oon (setq y (m+l (append minfl infl)))
(cond ((alike1 exp (setq y (sratsimp (log-reduce (hyperex y)))))
(cond ((not (infinityp val))
(setq infl (cnv infl val)) ;THIS IS HORRIBLE!!!!
(setq minfl (cnv minfl val))))
(let ((val '$inf))
(cond ((every #'(lambda (j) (radicalp j var))
(append infl minfl))
(setq y (rheur infl minfl)))
(t (setq y (sheur infl minfl))))))
(t (setq y (limit y var val 'think))))
(cond ((or (eq y ())
(eq y t)) (return ()))
((infinityp y) (return y))
(t (return (m+ sum y))))))
;; Limit n/d, using heuristics on the order of growth.
(defun sheur0 (n d)
(let ((orig-n n))
(cond ((and (free n var)
(free d var))
(m// n d))
(t (setq n (cpa n d nil))
(cond ((equal n 1)
(cond ((oscip orig-n) '$und)
(t '$inf)))
((equal n -1) '$zeroa)
((equal n 0) (m// orig-n d)))))))
;;;L1 is a list of INF's and L2 is a list of MINF's. Added together
;;;it is indeterminate.
(defun sheur (l1 l2)
(let ((term (sheur1 l1 l2)))
(cond ((equal term '$inf) '$inf)
((equal term '$minf) '$minf)
(t (let ((new-num (m+l (mapcar #'(lambda (num-term)
(m// num-term (car l1)))
(append l1 l2)))))
(cond ((limit2 new-num (m// 1 (car l1)) var val))))))))
(defun frac (exp)
(cond ((atom exp) nil)
(t (setq exp (nformat exp))
(cond ((and (eq (caar exp) 'mquotient)
(among var (caddr exp)))
t)))))
(defun zerop2 (z) (=0 (ridofab z)))
(defun raise (a) (m+ a '$zeroa))
(defun lower (a) (m+ a '$zerob))
(defun sincoshk (exp1 l sc)
(cond ((equal l 1) (lower l))
((equal l -1) (raise l))
((among sc l) l)
((member val '($zeroa $zerob) :test #'eq) (spangside exp1 l))
(t l)))
(defun spangside (e l)
(setq e (behavior e var val))
(cond ((equal e 1) (raise l))
((equal e -1) (lower l))
(t l)))
;; get rid of zeroa and zerob
(defmfun ridofab (e)
(if (among '$zeroa e) (setq e (maxima-substitute 0 '$zeroa e)))
(if (among '$zerob e) (setq e (maxima-substitute 0 '$zerob e)))
e)
;; simple radical
;; returns true if exp is a polynomial raised to a numeric power
(defun simplerd (exp)
(and (mexptp exp)
(mnump (caddr exp)) ;; exponent must be a number - no variables
(polyp (cadr exp))))
(defun branch1 (exp val)
(cond ((polyp exp) nil)
((simplerd exp) (zerop2 (subin val (cadr exp))))
(t
(loop for v on (cdr exp)
when (branch1 (car v) val)
do (return v)))))
(defun branch (exp val)
(cond ((polyp exp) nil)
((or (simplerd exp) (mtimesp exp))
(branch1 exp val))
((mplusp exp)
(every #'(lambda (j) (branch j val)) (the list (cdr exp))))))
(defun ser0 (e n d val)
(cond ((and (branch n val) (branch d val))
(setq nn* nil)
(setq n (ser1 n))
(setq d (ser1 d))
;;;NN* gets set by POFX, called by SER1, to get a list of exponents.
(setq nn* (ratmin nn*))
(setq n (sratsimp (m^ n (m^ var nn*))))
(setq d (sratsimp (m^ d (m^ var nn*))))
(cond ((member val '($zeroa $zerob) :test #'eq) nil)
(t (setq val 0.)))
(radlim e n d))
(t (try-lhospital-quit n d nil))))
(defun rheur (l1 l2)
(prog (ans m1 m2)
(setq m1 (mapcar (function asymredu) l1))
(setq m2 (mapcar (function asymredu) l2))
(setq ans (m+l (append m1 m2)))
(cond ((rptrouble (m+l (append l1 l2)))
(return (limit (simplify (rdsget (m+l (append l1 l2))))
var
val
nil)))
((mplusp ans) (return (sheur m1 m2)))
(t (return (limit ans var val t))))))
(defun rptrouble (rp)
(not (equal (rddeg rp nil) (rddeg (asymredu rp) nil))))
(defun radicalp (exp var)
(cond ((polyinx exp var ()))
((mexptp exp) (cond ((equal (caddr exp) -1.)
(radicalp (cadr exp) var))
((simplerd exp))))
((member (caar exp) '(mplus mtimes) :test #'eq)
(every #'(lambda (j) (radicalp j var))
(cdr exp)))))
(defun involve (e nn*)
(declare (special var))
(cond ((atom e) nil)
((mnump e) nil)
((and (member (caar e) nn* :test #'eq) (among var (cdr e))) (cadr e))
(t (some #'(lambda (j) (involve j nn*)) (cdr e)))))
(defun notinvolve (exp nn*)
(cond ((atom exp) t)
((mnump exp) t)
((member (caar exp) nn* :test #'eq) (not (among var (cdr exp))))
((every #'(lambda (j) (notinvolve j nn*))
(cdr exp)))))
(defun sheur1 (l1 l2)
(prog (ans)
(setq l1 (m+l (maxi l1)))
(setq l2 (m+l (maxi l2)))
(setq ans (cpa l1 l2 t))
(return (cond ((=0 ans) (m+ l1 l2))
((equal ans 1.) '$inf)
(t '$minf)))))
(defun zero-lim (cpa-list)
(do ((l cpa-list (cdr l)))
((null l) ())
(and (eq (caar l) 'gen)
(zerop2 (limit (cadar l) var val 'think))
(return t))))
;; Compare order of growth for R1 and R2. The result is 0, -1, +1
;; depending on the relative order of growth. 0 is returned if R1 and
;; R2 have the same growth; -1 if R1 grows much more slowly than R2;
;; +1 if R1 grows much more quickly than R2.
(defun cpa (r1 r2 flag)
(let ((t1 r1)
(t2 r2))
(cond ((alike1 t1 t2) 0.)
((free t1 var)
(cond ((free t2 var) 0.)
(t (let ((lim-ans (limit1 t2 var val)))
(cond ((not (member lim-ans '($inf $minf $und $ind) :test #'eq)) 0.)
(t -1.))))))
((free t2 var)
(let ((lim-ans (limit1 t1 var val)))
(cond ((not (member lim-ans '($inf $minf $und $ind) :test #'eq)) 0.)
(t 1.))))
(t
;; Make T1 and T2 be a list of terms that are multipled
;; together.
(cond ((mtimesp t1) (setq t1 (cdr t1)))
(t (setq t1 (list t1))))
(cond ((mtimesp t2) (setq t2 (cdr t2)))
(t (setq t2 (list t2))))
;; Find the strengths of each term of T1 and T2
(setq t1 (mapcar (function istrength) t1))
(setq t2 (mapcar (function istrength) t2))
;; Compute the max of the strengths of the terms.
(let ((ans (ismax t1))
(d (ismax t2)))
(cond ((or (null ans) (null d)
(eq (car ans) 'gen) (eq (car d) 'gen)) 0.))
(if (eq (car ans) 'var) (setq ans (add-up-deg t1)))
(if (eq (car d) 'var) (setq d (add-up-deg t2)))
;; Cant just just compare dominating terms if there are
;; indeterm-inates present; e.g. X-X^2*LOG(1+1/X). So
;; check for this.
(cond ((or (zero-lim t1)
(zero-lim t2))
(cpa-indeterm ans d t1 t2 flag))
((isgreaterp ans d) 1.)
((isgreaterp d ans) -1.)
(t 0)))))))
(defun cpa-indeterm (ans d t1 t2 flag)
(cond ((not (eq (car ans) 'var))
(setq ans (gather ans t1) d (gather d t2))))
(let ((*indicator (and (eq (car ans) 'exp)
flag))
(test ()))
(setq test (cpa1 ans d))
(cond ((and (zerop1 test)
(or (equal ($radcan (m// (cadr ans) (cadr d))) 1.)
(and (polyp (cadr ans))
(polyp (cadr d))
(equal (limit (m// (cadr ans) (cadr d)) var val 'think)
1.))))
(let ((new-term1 (m// t1 (cadr ans)))
(new-term2 (m// t2 (cadr d))))
(cpa new-term1 new-term2 flag)))
(t 0))))
(defun add-up-deg (strengthl)
(do ((stl strengthl (cdr stl))
(poxl)
(degl))
((null stl) (list 'var (m*l poxl) (m+l degl)))
(cond ((eq (caar stl) 'var)
(push (cadar stl) poxl)
(push (caddar stl) degl)))))
(defun cpa1 (p1 p2)
(prog (flag s1 s2)
(cond ((eq (car p1) 'gen) (return 0.)))
(setq flag (car p1))
(setq p1 (cadr p1))
(setq p2 (cadr p2))
(cond
((eq flag 'var)
(setq s1 (istrength p1))
(setq s2 (istrength p2))
(return
(cond
((isgreaterp s1 s2) 1.)
((isgreaterp s2 s1) -1.)
(*indicator
(setq *indicator nil)
(cond
((and (poly? p1 var) (poly? p2 var))
(setq p1 (m- p1 p2))
(cond ((zerop1 p1) 0.)
(t (getsignl (hot-coef p1)))))
(t
(setq s1
(rheur (list p1)
(list (m*t -1 p2))))
(cond ((zerop2 s1) 0.)
((ratgreaterp s1 0.) 1.)
(t -1.)))))
(t 0.))))
((eq flag 'exp)
(setq p1 (caddr p1))
(setq p2 (caddr p2))
(cond ((and (poly? p1 var) (poly? p2 var))
(setq p1 (m- p1 p2))
(return (cond ((or (zerop1 p1)
(not (among var p1)))
0.)
(t (getsignl (hot-coef p1))))))
((and (radicalp p1 var) (radicalp p2 var))
(setq s1
(rheur (list p1)
(list (m*t -1 p2))))
(return (cond ((eq s1 '$inf) 1.)
((eq s1 '$minf) -1.)
((mnump s1)
(cond ((ratgreaterp s1 0.) 1.)
((ratgreaterp 0. s1) -1.)
(t 0.)))
(t 0.))))
(t (return (cpa p1 p2 t)))))
((eq flag 'log)
(setq p1 (try-lhospital (asymredu p1) (asymredu p2) nil))
(return (cond ((zerop2 p1) -1.)
((real-infinityp p1) 1.)
(t 0.)))))))
;;;EXPRESSIONS TO ISGREATERP ARE OF THE FOLLOWING FORMS
;;; ("VAR" POLY DEG)
;;; ("EXP" %E^EXP)
;;; ("LOG" LOG(EXP))
;;; ("FACT" <A FACTORIAL EXPRESSION>)
;;; ("GEN" <ANY OTHER TYPE OF EXPRESSION>)
(defun isgreaterp (a b)
(let ((ta (car a))
(tb (car b)))
(cond ((or (eq ta 'gen)
(eq tb 'gen)) ())
((and (eq ta tb) (eq ta 'var))
(ratgreaterp (caddr a) (caddr b)))
((and (eq ta tb) (eq ta 'exp))
;; Both are exponential order of infinity. Check the
;; exponents to determine which exponent is bigger.
(eq (limit (m- `((%log) ,(second a)) `((%log) ,(second b)))
var val 'think)
'$inf))
((member ta (cdr (member tb '(num log var exp fact gen) :test #'eq)) :test #'eq)))))
(defun ismax (l)
;; Preprocess the list of products. Separate the terms that
;; exponentials and those that don't. Actually multiply the
;; exponential terms together to form a single term. Pass this and
;; the rest to ismax-core to find the max.
(let (exp-terms non-exp-terms)
(dolist (term l)
(if (eq 'exp (car term))
(push term exp-terms)
(push term non-exp-terms)))
;; Multiply the exp-terms together
(if exp-terms
(let ((product 1))
;;(format t "exp-terms = ~A~%" exp-terms)
(dolist (term exp-terms)
(setf product (simplify (mul product (second term)))))
;;(format t "product = ~A~%" product)
(setf product `(exp ,($logcontract product)))
;;(format t "product = ~A~%" product)
(ismax-core (cons product non-exp-terms)))
(ismax-core l))))
(defun ismax-core (l)
(cond ((null l) ())
((atom l) ())
((= (length l) 1) (car l)) ;If there is only 1 thing give it back.
((every #'(lambda (x)
(not (eq (car x) 'gen))) l)
(do ((l1 (cdr l) (cdr l1))
(temp-ans (car l))
(ans ()))
((null l1) ans)
(cond ((isgreaterp temp-ans (car l1))
(setq ans temp-ans))
((isgreaterp (car l1) temp-ans)
(setq temp-ans (car l1))
(setq ans temp-ans))
(t (setq ans ())))))
(t ())))
;RETURNS LIST OF HIGH TERMS
(defun maxi (all)
(cond ((atom all) nil)
(t (do ((l (cdr all) (cdr l))
(hi-term (car all))
(total 1) ; running total constant factor of hi-term
(hi-terms (ncons (car all)))
(compare nil))
((null l) (if (zerop2 total) ; if high-order terms cancel each other
all ; keep everything
hi-terms)) ; otherwise return list of high terms
(setq compare (limit (m// (car l) hi-term) var val 'think))
(cond
((or (infinityp compare)
(and (eq compare '$und)
(zerop2 (limit (m// hi-term (car l)) var val 'think))))
(setq total 1) ; have found new high term
(setq hi-terms (ncons (setq hi-term (car l)))))
((zerop2 compare) nil)
;; COMPARE IS IND, FINITE-VALUED, or und in both directions
(t ; add to list of high terms
(setq total (m+ total compare))
(setq hi-terms (append hi-terms (ncons (car l))))))))))
(defun ratmax (l)
(prog (ans)
(cond ((atom l) (return nil)))
l1 (setq ans (car l))
l2 (setq l (cdr l))
(cond ((null l) (return ans))
((ratgreaterp ans (car l)) (go l2))
(t (go l1)))))
(defun ratmin (l)
(prog (ans)
(cond ((atom l) (return nil)))
l1 (setq ans (car l))
l2 (setq l (cdr l))
(cond ((null l) (return ans))
((ratgreaterp (car l) ans) (go l2))
(t (go l1)))))
(defun pofx (e)
(cond ((atom e)
(cond ((eq e var)
(push 1 nn*))
(t ())))
((or (mnump e) (not (among var e))) nil)
((and (mexptp e) (eq (cadr e) var))
(push (caddr e) nn*))
((simplerd e) nil)
(t (mapc #'pofx (cdr e)))))
(defun ser1 (e)
(cond ((member val '($zeroa $zerob) :test #'eq) nil)
(t (setq e (subin (m+ var val) e))))
(setq e (rdfact e))
(cond ((pofx e) e)))
(defun gather (ind l)
(prog (ans)
(setq ind (car ind))
loop (cond ((null l)
(return (list ind (m*l ans))))
((equal (caar l) ind)
(push (cadar l) ans)))
(setq l (cdr l))
(go loop)))
; returns rough class-of-growth of term
(defun istrength (term)
(cond ((mnump term) (list 'num term))
((atom term) (cond ((eq term var)
(list 'var var 1.))
(t (list 'num term))))
((not (among var term)) (list 'num term))
((radicalp term var) (list 'var term (rddeg term nil)))
((mplusp term)
(let ((temp (ismax (mapcar #'istrength (cdr term)))))
(cond ((not (null temp)) temp)
(t `(gen ,term)))))
((mtimesp term)
(let ((temp (mapcar #'istrength (cdr term)))
(temp1 ()))
(setq temp1 (ismax temp))
(cond ((null temp1) `(gen ,term))
((eq (car temp1) 'log) `(log ,temp))
((eq (car temp1) 'var) (add-up-deg temp))
(t `(gen ,temp)))))
((and (mexptp term)
(real-infinityp (limit term var val t)))
(let ((logterm (logred term)))
(cond ((and (among var (caddr term))
(member (car (istrength logterm))
'(var exp fact) :test #'eq)
(real-infinityp (limit logterm var val t)))
(list 'exp (m^ '$%e logterm)))
((not (among var (caddr term)))
(let ((temp (istrength (cadr term))))
(cond ((not (alike1 temp term))
(rplaca (cdr temp) term)
(and (eq (car temp) 'var)
(rplaca (cddr temp)
(m* (caddr temp) (caddr term))))
temp)
(t `(gen ,term)))))
(t `(gen ,term)))))
((and (eq (caar term) '%log)
(real-infinityp (limit term var val t)))
(let ((stren (istrength (cadr term))))
(cond ((member (car stren) '(log var) :test #'eq)
`(log ,term))
((and (eq (car stren) 'exp)
(eq (caar (second stren)) 'mexpt))
(istrength (logred (second stren))))
(t `(gen ,term)))))
((eq (caar term) 'mfactorial)
(list 'fact term))
((let ((temp (hyperex term)))
(and (not (alike1 term temp))
(istrength temp))))
(t (list 'gen term))))
;; log reduce - returns log of s1
(defun logred (s1)
(or (and (eq (cadr s1) '$%e) (caddr s1))
(m* (caddr s1) `((%log) ,(cadr s1)))))
(defun asymredu (rd)
(cond ((atom rd) rd)
((mnump rd) rd)
((not (among var rd)) rd)
((polyinx rd var t))
((simplerd rd)
(cond ((eq (cadr rd) var) rd)
(t (mabs-subst
(factor ($expand (m^ (polyinx (cadr rd) var t)
(caddr rd))))
var
val))))
(t (simplify (cons (list (caar rd))
(mapcar #'asymredu (cdr rd)))))))
(defun rdfact (rd)
(let ((dn** ()) (nn** ()))
(cond ((atom rd) rd)
((mnump rd) rd)
((not (among var rd)) rd)
((polyp rd)
(factor rd))
((simplerd rd)
(cond ((eq (cadr rd) var) rd)
(t (setq dn** (caddr rd))
(setq nn** (factor (cadr rd)))
(cond ((mtimesp nn**)
(m*l (mapcar #'(lambda (j) (m^ j dn**))
(cdr nn**))))
(t rd)))))
(t (simplify (cons (ncons (caar rd))
(mapcar #'rdfact (cdr rd))))))))
(defun cnv (expl val)
(mapcar #'(lambda (e)
(maxima-substitute (cond ((eq val '$zerob)
(m* -1 (m^ var -1)))
((eq val '$zeroa)
(m^ var -1))
((eq val '$minf)
(m* -1 var))
(t (m^ (m+ var (m* -1 val)) -1.)))
var
e))
expl))
(defun pwtaylor (exp var l terms)
(prog (coef ans c mc)
(cond ((=0 terms) (return nil)) ((=0 l) (setq mc t)))
(setq c 0.)
(go tag1)
loop (setq c (1+ c))
(cond ((or (> c 10.) (equal c terms))
(return (m+l ans)))
(t (setq exp (sdiff exp var))))
tag1 (setq coef ($radcan (subin l exp)))
(cond ((=0 coef) (setq terms (1+ terms)) (go loop)))
(setq
ans
(append
ans
(list
(m* coef
(m^ `((mfactorial) ,c) -1)
(m^ (if mc var (m+t (m*t -1 l) var)) c)))))
(go loop)))
(defun rdsget (e)
(cond ((polyp e) e)
((simplerd e) (rdtay e))
(t (cons (list (caar e))
(mapcar #'rdsget (cdr e))))))
(defun rdtay (rd)
(cond (limit-using-taylor ($ratdisrep ($taylor rd var val 1.)))
(t (lrdtay rd))))
(defun lrdtay (rd)
(prog (varlist p c e d $ratfac)
(setq varlist (ncons var))
(setq p (ratnumerator (cdr (ratrep* (cadr rd)))))
(cond ((< (length p) 3.) (return rd)))
(setq e (caddr rd))
(setq d (pdegr p))
(setq c (m^ var (m* d e)))
(setq d ($ratsimp (varinvert (m* (pdis p) (m^ var (m- d)))
var)))
(setq d (pwtaylor (m^ d e) var 0. 3.))
(return (m* c (varinvert d var)))))
(defun varinvert (e var) (subin (m^t var -1.) e))
(defun deg (p)
(prog ((varlist (list var)))
(return (let (($ratfac nil))
(newvar p)
(pdegr (cadr (ratrep* p)))))))
(defun rat-no-ratfac (e)
(let (($ratfac nil))
(newvar e)
(ratrep* e)))
(setq low* nil)
(defun rddeg (rd low*)
(cond ((or (mnump rd)
(not (among var rd)))
0)
((polyp rd)
(deg rd))
((simplerd rd)
(m* (deg (cadr rd)) (caddr rd)))
((mtimesp rd)
(addn (mapcar #'(lambda (j)
(rddeg j low*))
(cdr rd)) nil))
((and (mplusp rd)
(setq rd (andmapcar #'(lambda (j) (rddeg j low*))
(cdr rd))))
(cond (low* (ratmin rd))
(t (ratmax rd))))))
(defun pdegr (pf)
(cond ((or (atom pf) (not (eq (caadr (ratf var)) (car pf))))
0)
(low* (cadr (reverse pf)))
(t (cadr pf))))
;;There is some confusion here. We need to be aware of Branch cuts etc....
;;when doing this section of code. It is not very carefully done so there
;;are bugs still lurking. Another misfortune is that LIMIT or its inferiors
;;somtimes decides to change the limit VAL in midstream. This must be corrected
;;since LIMIT's interaction with the data base environment must be maintained.
;;I'm not sure that this code can ever be called with VAL other than $INF but
;;there is a hook in the first important cond clause to cathc them anyway.
(defun asy (n d)
(let ((num-power (rddeg n nil))
(den-power (rddeg d nil))
(coef ()) (coef-sign ()) (power ()))
(setq coef (m// ($ratcoef ($expand n) var num-power)
($ratcoef ($expand d) var den-power)))
(setq coef-sign (getsignl coef))
(setq power (m// num-power den-power))
(cond ((eq (ask-integer power '$integer) '$integer)
(cond ((eq (ask-integer power '$even) '$even) '$even)
(t '$odd)))) ;Can be extended from here.
(cond ((or (eq val '$minf)
(eq val '$zerob)
(eq val '$zeroa)
(equal val 0)) ()) ;Can be extended to cover some these.
((ratgreaterp den-power num-power)
(cond ((equal coef-sign 1.) '$zeroa)
((equal coef-sign -1) '$zerob)
((equal coef-sign 0) 0)
(t 0)))
((ratgreaterp num-power den-power)
(cond ((equal coef-sign 1.) '$inf)
((equal coef-sign -1) '$minf)
((equal coef-sign 0) nil) ; should never be zero
((null coef-sign) '$infinity)))
(t coef))))
(defun radlim (e n d)
(prog (nl dl)
(cond ((eq val '$infinity) (throw 'limit nil))
((eq val '$minf)
(setq nl (m* var -1))
(setq n (subin nl n))
(setq d (subin nl d))
(setq val '$inf))) ;This is the Culprit. Doesn't tell the DATABASE.
(cond ((eq val '$inf)
(setq nl (asymredu n))
(setq dl (asymredu d))
(cond
((or (rptrouble n) (rptrouble d))
(return (limit (m* (rdsget n) (m^ (rdsget d) -1.)) var val t)))
(t (return (asy nl dl))))))
(setq nl (limit n var val t))
(setq dl (limit d var val t))
(cond ((and (zerop2 nl) (zerop2 dl))
(cond ((or (polyp n) (polyp d))
(return (try-lhospital-quit n d t)))
(t (return (ser0 e n d val)))))
(t (return ($radcan (ratrad (m// n d) n d nl dl)))))))
(defun ratrad (e n d nl dl)
(prog (n1 d1)
(cond ((equal nl 0) (return 0))
((zerop2 dl)
(setq n1 nl)
(cond ((equal dl 0) (setq d1 '$infinity)) ;No direction Info.
((eq dl '$zeroa)
(setq d1 '$inf))
((equal (setq d (behavior d var val)) 1)
(setq d1 '$inf))
((equal d -1) (setq d1 '$minf))
(t (throw 'limit nil))))
((zerop2 nl)
(setq d1 dl)
(cond ((equal (setq n (behavior n var val)) 1)
(setq n1 '$zeroa))
((equal n -1) (setq n1 '$zerob))
(t (setq n1 0))))
(t (return ($radcan (ridofab (subin val e))))))
(return (simplimtimes (list n1 d1)))))
;;; Limit of the Logarithm function
(defun simplimln (expr var val)
;; We need to be careful with log because of the branch cut on the
;; negative real axis. So we look at the imagpart of the argument. If
;; it's not identically zero, we compute the limit of the real and
;; imaginary parts and combine them. Otherwise, we can use the
;; original method for real limits.
(let ((arglim (limit (cadr expr) var val 'think)))
(cond ((eq arglim '$inf) '$inf)
((member arglim '($minf $infinity) :test #'eq)
'$infinity)
((member arglim '($ind $und) :test #'eq) '$und)
((equal ($imagpart (cadr expr)) 0)
;; argument is real.
(let* ((real-lim (ridofab arglim)))
(if (=0 real-lim)
(cond ((eq arglim '$zeroa) '$minf)
((eq arglim '$zerob) '$infinity)
(t (let ((dir (behavior (cadr expr) var val)))
(cond ((equal dir 1) '$minf)
((equal dir -1) '$infinity)
(t (throw 'limit t))))))
(cond ((equal arglim 1)
(let ((dir (behavior (cadr expr) var val)))
(if (equal dir 1) '$zeroa 0)))
(t
;; Call simplifier to get value at the limit
;; of the argument.
(simplify `((%log) ,real-lim)))))))
(t
;; argument is complex.
(destructuring-let* (((rp . ip) (trisplit expr)))
(if (eq (setq rp (limit rp var val 'think)) '$minf)
;; Realpart is minf, do not return minf+%i*ip but infinity.
'$infinity
;; Return a complex limit value.
(add rp (mul '$%i (limit ip var val 'think)))))))))
;;; Limit of the Factorial function
(defun simplimfact (expr var val)
(let* ((arglim (limit (cadr expr) var val 'think)) ; Limit of the argument.
(arg2 arglim))
(cond ((eq arglim '$inf) '$inf)
((member arglim '($minf $infinity $und $ind) :test #'eq) '$und)
((and (or (maxima-integerp arglim)
(setq arg2 (integer-representation-p arglim)))
(eq ($sign arg2) '$neg))
;; A negative integer or float or bigfloat representation.
(let ((dir (limit (add (cadr expr) (mul -1 arg2)) var val 'think))
(even (mevenp arg2)))
(cond ((or (and even
(eq dir '$zeroa))
(and (not even)
(eq dir '$zerob)))
'$minf)
((or (and even
(eq dir '$zerob))
(and (not even)
(eq dir '$zeroa)))
'$inf)
(t (throw 'limit nil)))))
(t
;; Call simplifier to get value at the limit of the argument.
(simplify (list '(mfactorial) arglim))))))
(defun simplim%erf-%tanh (fn arg)
(let ((arglim (limit arg var val 'think)))
(cond ((eq arglim '$inf) 1)
((eq arglim '$minf) -1)
((eq arglim '$infinity)
(destructuring-let (((rpart . ipart) (trisplit arg))
(ans ()) (rlim ()))
(setq rlim (limit rpart var origval 'think))
(cond ((eq fn '%tanh)
(cond ((equal rlim '$inf) 1)
((equal rlim '$minf) -1)))
((eq fn '%erf)
(setq ans
(limit (m* rpart (m^t ipart -1)) var origval 'think))
(setq ans ($asksign (m+ `((mabs) ,ans) -1)))
(cond ((or (eq ans '$pos) (eq ans '$zero))
(cond ((eq rlim '$inf) 1)
((eq rlim '$minf) -1)
(t '$und)))
(t '$und))))))
((eq arglim '$und) '$und)
((member arglim '($zeroa $zerob $ind) :test #'eq) arglim)
;;;Ignore tanh(%pi/2*%I) and multiples of the argument.
(t
;; erf (or tanh) of a known value is just erf(arglim).
(simplify (list (ncons fn) arglim))))))
(defun simplim%atan (exp1)
(cond ((zerop2 exp1) exp1)
((member exp1 '($und $infinity) :test #'eq)
(throw 'limit ()))
((eq exp1 '$inf) half%pi)
((eq exp1 '$minf)
(m*t -1. half%pi))
(t `((%atan) ,exp1))))
;; Most instances of atan2 are simplified to expressions in atan
;; by simpatan2 before we get to this point. This routine handles
;; tricky cases such as limit(atan2((x^2-2), x^3-2*x), x, sqrt(2), minus).
;; Taylor and Gruntz cannot handle the discontinuity at atan(0, -1)
(defun simplim%atan2 (exp)
(let* ((exp1 (cadr exp))
(exp2 (caddr exp))
(lim1 (limit (cadr exp) var val 'think))
(lim2 (limit (caddr exp) var val 'think))
(sign2 ($csign lim2)))
(cond ((and (zerop2 lim1) ;; atan2( 0+, + )
(eq sign2 '$pos))
lim1) ;; result is zeroa or zerob
((and (eq lim1 '$zeroa)
(eq sign2 '$neg))
'$%pi)
((and (eq lim1 '$zerob) ;; atan2( 0-, - )
(eq sign2 '$neg))
(m- '$%pi))
((and (eq lim1 '$zeroa) ;; atan2( 0+, 0 )
(zerop2 lim2))
(simplim%atan (limit (m// exp1 exp2) var val 'think)))
((and (eq lim1 '$zerob) ;; atan2( 0-, 0 )
(zerop2 lim2))
(m+ (porm (eq lim2 '$zeroa) '$%pi)
(simplim%atan (limit (m// exp1 exp2) var val 'think))))
((member lim1 '($und $infinity) :test #'eq)
(throw 'limit ()))
((eq lim1 '$inf) half%pi)
((eq lim1 '$minf)
(m*t -1. half%pi))
(t (take '($atan2) lim1 lim2)))))
(defun simplimsch (sch arg)
(cond ((real-infinityp arg)
(cond ((eq sch '%sinh) arg) (t '$inf)))
((eq arg '$infinity) '$infinity)
((eq arg '$ind) '$ind)
((eq arg '$und) '$und)
(t (let (($exponentialize t))
(resimplify (list (ncons sch) (ridofab arg)))))))
;; simple limit of sin and cos
(defun simplimsc (exp fn arg)
(cond ((member arg '($inf $minf $ind) :test #'eq) '$ind)
((member arg '($und $infinity) :test #'eq)
(throw 'limit ()))
((member arg '($zeroa $zerob) :test #'eq)
(cond ((eq fn '%sin) arg)
(t (m+ 1 '$zerob))))
((sincoshk exp
(simplify (list (ncons fn) (ridofab arg)))
fn))))
(defun simplim%tan (arg)
(let ((arg1 (ridofab (limit arg var val 'think))))
(cond
((member arg1 '($inf $minf $infinity $ind $und) :test #'eq) '$und)
((pip arg1)
(let ((c (trigred (pip arg1))))
(cond ((not (equal ($imagpart arg1) 0)) '$infinity)
((and (eq (caar c) 'rat)
(equal (caddr c) 2)
(> (cadr c) 0))
(setq arg1 (behavior arg var val))
(cond ((= arg1 1) '$inf)
((= arg1 -1) '$minf)
(t '$und)))
((and (eq (caar c) 'rat)
(equal (caddr c) 2)
(< (cadr c) 0))
(setq arg1 (behavior arg var val))
(cond ((= arg1 1) '$minf)
((= arg1 -1) '$inf)
(t '$und)))
(t (throw 'limit ())))))
((equal arg1 0)
(setq arg1 (behavior arg var val))
(cond ((equal arg1 1) '$zeroa)
((equal arg1 -1) '$zerob)
(t 0)))
(t (simp-%tan (list '(%tan) arg1) 1 nil)))))
(defun simplim%asinh (arg)
(cond ((member arg '($inf $minf $zeroa $zerob $ind $und) :test #'eq)
arg)
((eq arg '$infinity) '$und)
(t (simplify (list '(%asinh) (ridofab arg))))))
(defun simplim%acosh (arg)
(cond ((equal (ridofab arg) 1) '$zeroa)
((eq arg '$inf) arg)
((eq arg '$minf) '$infinity)
((member arg '($und $ind $infinity) :test #'eq) '$und)
(t (simplify (list '(%acosh) (ridofab arg))))))
(defun simplim%atanh (arg dir)
;; Compute limit(atanh(x),x,arg). If ARG is +/-1, we need to take
;; into account which direction we're approaching ARG.
(cond ((zerop2 arg) arg)
((member arg '($ind $und $infinity $minf $inf) :test #'eq)
'$und)
((equal (setq arg (ridofab arg)) 1.)
;; The limit at 1 should be complex infinity because atanh(x)
;; is complex for x > 1, but inf if we're approaching 1 from
;; below.
(if (eq dir '$zerob)
'$inf
'$infinity))
((equal arg -1.)
;; Same as above, except for the limit is at -1.
(if (eq dir '$zeroa)
'$minf
'$infinity))
(t (simplify (list '(%atanh) arg)))))
(defun simplim%asin-%acos (fn arg)
(cond ((member arg '($und $ind $inf $minf $infinity) :test #'eq)
'$und)
((and (eq fn '%asin)
(member arg '($zeroa $zerob) :test #'eq))
arg)
(t (simplify (list (ncons fn) (ridofab arg))))))
(defun simplim$li (order arg val)
(cond ((and (not (equal (length order) 1))
(not (equal (length arg) 1))) (throw 'limit ()))
(t (setq order (car order) arg (car arg))))
(cond ((not (equal order 2)) (throw 'limit ()))
(t (destructuring-let (((rpart . ipart) (trisplit arg)))
(cond ((not (equal ipart 0)) (throw 'limit ()))
(t (setq rpart (limit rpart var val 'think))
(cond ((eq rpart '$zeroa) '$zeroa)
((eq rpart '$zerob) '$zerob)
((eq rpart '$minf) '$minf)
((eq rpart '$inf) '$infinity)
(t (simplify (subfunmake '$li (list order)
(list rpart)))))))))))
(defun simplim$psi (order arg val)
(cond ((and (not (equal (length order) 1))
(not (equal (length arg) 1))) (throw 'limit ()))
(t (setq order (car order) arg (car arg))))
(cond ((equal order 0)
(destructuring-let (((rpart . ipart) (trisplit arg)))
(cond ((not (equal ipart 0)) (throw 'limit ()))
(t (setq rpart (limit rpart var val 'think))
(cond ((eq rpart '$zeroa) '$minf)
((eq rpart '$zerob) '$inf)
((eq rpart '$inf) '$inf)
((eq rpart '$minf) '$und)
((equal (getsignl rpart) -1) (throw 'limit ()))
(t (simplify (subfunmake '$psi (list order)
(list rpart)))))))))
((and (integerp order) (> order 0)
(equal (limit arg var val 'think) '$inf))
(cond ((mevenp order) '$zerob)
((moddp order) '$zeroa)
(t (throw 'limit ()))))
(t (throw 'limit ()))))
(defun simplim%inverse_jacobi_ns (arg m)
(if (or (eq arg '$inf) (eq arg '$minf))
0
`((%inverse_jacobi_ns) ,arg ,m)))
(defun simplim%inverse_jacobi_nc (arg m)
(if (or (eq arg '$inf) (eq arg '$minf))
`((%elliptic_kc) ,m)
`((%inverse_jacobi_nc) ,arg ,m)))
(defun simplim%inverse_jacobi_sc (arg m)
(if (or (eq arg '$inf) (eq arg '$minf))
`((%elliptic_kc) ,m)
`((%inverse_jacobi_sc) ,arg ,m)))
(defun simplim%inverse_jacobi_dc (arg m)
(if (or (eq arg '$inf) (eq arg '$minf))
`((%elliptic_kc) ,m)
`((%inverse_jacobi_dc) ,arg ,m)))
(defun simplim%inverse_jacobi_cs (arg m)
(if (or (eq arg '$inf) (eq arg '$minf))
0
`((%inverse_jacobi_cs) ,arg ,m)))
(defun simplim%inverse_jacobi_ds (arg m)
(if (or (eq arg '$inf) (eq arg '$minf))
0
`((%inverse_jacobi_ds) ,arg ,m)))
(setf (get '%signum 'simplim%function) 'simplim%signum)
(defun simplim%signum (e x pt)
(let* ((e (limit (cadr e) x pt 'think)) (sgn (mnqp e 0)))
(cond ((eq t sgn) (take '(%signum) e)) ;; limit of argument of signum is not zero
((eq nil sgn) '$und) ;; limit of argument of signum is zero (noncontinuous)
(t (throw 'limit nil))))) ;; don't know
;; more functions for limit to handle
(defun lfibtophi (e)
(cond ((not (involve e '($fib))) e)
((eq (caar e) '$fib)
(let ((lnorecurse t))
($fibtophi (list '($fib) (lfibtophi (cadr e))) lnorecurse)))
(t (cons (car e)
(mapcar #'lfibtophi (cdr e))))))
;;; FOLLOWING CODE MAKES $LDEFINT WORK
(defmfun $ldefint (exp var ll ul &aux $logabs ans a1 a2)
(setq $logabs t ans (sinint exp var)
a1 ($limit ans var ul '$minus)
a2 ($limit ans var ll '$plus))
(and (member a1 '($inf $minf $infinity $und $ind) :test #'eq)
(setq a1 (nounlimit ans var ul)))
(and (member a2 '($inf $minf $infinity $und $ind) :test #'eq)
(setq a2 (nounlimit ans var ll)))
($expand (m- a1 a2)))
(defun nounlimit (exp var val)
(setq exp (restorelim exp))
(nconc (list '(%limit) exp var (ridofab val))
(cond ((eq val '$zeroa) '($plus))
((eq val '$zerob) '($minus)))))
;; replace noun form of %derivative and indefinite %integrate with gensym.
;; prevents substitution x -> x+1 for limit('diff(x+2,x), x, 1)
;;
;; however, this doesn't work for limit('diff(x+2,x)/x, x, inf)
;; because the rest of the limit code thinks the gensym is const wrt x.
(defun hide (exp)
(cond ((atom exp) exp)
((or (eq '%derivative (caar exp))
(and (eq '%integrate (caar exp)) ; indefinite integral
(null (cdddr exp))))
(hidelim exp (caar exp)))
(t (cons (car exp) (mapcar 'hide (cdr exp))))))
(defun hidelim (exp func)
(setq func (gensym))
(putprop func
(hidelima exp)
'limitsub)
func)
(defun hidelima (e)
(if (among var e)
(nounlimit e var val)
e))
;;;Used by Defint also.
(defun oscip (e)
(or (involve e '(%sin %cos %tan))
(among '$%i (%einvolve e))))
(defun %einvolve (e)
(when (among '$%e e) (%einvolve01 e)))
(defun %einvolve01 (e)
(cond ((atom e) nil)
((mnump e) nil)
((and (mexptp e)
(eq (cadr e) '$%e)
(among var (caddr e)))
(caddr e))
(t (some #'%einvolve (cdr e)))))
(declare-top (unspecial *indicator nn* dn* exp var val origval taylored
$tlimswitch logcombed lhp? lhcount $ratfac))
;; GRUNTZ ALGORITHM
;; Dominik Gruntz
;; "On Computing Limits in a Symbolic Manipulation System"
;; PhD Dissertation ETH Zurich 1996
;; The algorithm identifies the most rapidly varying (MRV) subexpression,
;; replaces it with a new variable w, rewrites the expression in terms
;; of the new variable, and then repeats.
;; The algorithm doesn't handle oscillating functions, so it can't do things like
;; limit(sin(x)/x, x, inf).
;; To handle limits involving functions like gamma(x) and erf(x), the
;; gruntz algorithm requires them to be written in terms of asymptotic
;; expansions, which maxima cannot currently do.
;; The algorithm assumes that everything is real, so it can't
;; currently handle limit((-2)^x, x, inf).
;; This is one of the methods used by maxima's $limit.
;; It is also directly available to the user as $gruntz.
;; most rapidly varying subexpression of expression exp with respect to limit variable var.
;; returns a list of subexpressions which are in the same MRV equivalence class.
(defun mrv (exp var)
(cond ((freeof var exp)
nil)
((eq var exp)
(list var))
((mtimesp exp)
(mrv-max (mrv (cadr exp) var)
(mrv (m*l (cddr exp)) var)
var))
((mplusp exp)
(mrv-max (mrv (cadr exp) var)
(mrv (m+l (cddr exp)) var)
var))
((mexptp exp)
(cond ((freeof var (caddr exp))
(mrv (cadr exp) var))
((member (limitinf (logred exp) var) '($inf $minf) :test #'eq)
(mrv-max (list exp) (mrv (caddr exp) var) var))
(t (mrv-max (mrv (cadr exp) var) (mrv (caddr exp) var) var))))
((mlogp exp)
(mrv (cadr exp) var))
((equal (length (cdr exp)) 1)
(mrv (cadr exp) var))
((equal (length (cdr exp)) 2)
(mrv-max (mrv (cadr exp) var)
(mrv (caddr exp) var)
var))
(t (tay-error "mrv not implemented" exp))))
;; takes two lists of expresions, f and g, and limit variable var.
;; members in each list are assumed to be in same MRV equivalence
;; class. returns MRV set of the union of the inputs - either f or g
;; or the union of f and g.
(defun mrv-max (f g var)
(prog ()
(cond ((not f)
(return g))
((not g)
(return f))
((intersection f g)
(return (union f g))))
(let ((c (mrv-compare (car f) (car g) var)))
(cond ((eq c '>)
(return f))
((eq c '<)
(return g))
((eq c '=)
(return (union f g)))
(t (merror "MRV-MAX: expected '>' '<' or '='; found: ~M" c))))))
(defun mrv-compare (a b var)
(let ((c (limitinf (m// `((%log) ,a) `((%log) ,b)) var)))
(cond ((equal c 0)
'<)
((member c '($inf $minf) :test #'eq)
'>)
(t '=))))
;; rewrite expression exp by replacing members of MRV set omega with
;; expressions in terms of new variable wsym. return cons pair of new
;; version of exp and the log of the new variable wsym.
(defun mrv-rewrite (exp omega var wsym)
(setq omega (sort omega (lambda (x y) (> (length (mrv x var))
(length (mrv y var))))))
(let* ((g (car (last omega)))
(logg (logred g))
(sig (equal (mrv-sign logg var) 1))
(w (if sig (m// 1 wsym) wsym))
(logw (if sig (m* -1 logg) logg)))
(mapcar (lambda (x y)
;;(mtell "y:~M x:~M exp:~M~%" y x exp)
(setq exp (syntactic-substitute y x exp)))
omega
(mapcar (lambda (f) ;; rewrite each element of omega
(let* ((logf (logred f))
(c (mrv-leadterm (m// logf logg) var nil)))
(cond ((not (equal (cadr c) 0))
(merror "MRV-REWRITE: expected leading term to be constant in ~M" c)))
;;(mtell "logg: ~M logf: ~M~%" logg logf)
(m* (m^ w (car c))
(m^ '$%e (m- logf
(m* (car c) logg))))))
omega))
(cons exp logw)))
;; returns list of two elements: coeff and exponent of leading term of exp,
;; after rewriting exp in term of its MRV set omega.
(defun mrv-leadterm (exp var omega)
(prog ((new-omega ()))
(cond ((freeof var exp)
(return (list exp 0))))
(dolist (term omega)
(cond ((subexp exp term)
(push term new-omega))))
(setq omega new-omega)
(cond ((not omega)
(setq omega (mrv exp var))))
(cond ((member var omega :test #'eq)
(let* ((omega-up (mrv-moveup omega var))
(e-up (car (mrv-moveup (list exp) var)))
(mrv-leadterm-up (mrv-leadterm e-up var omega-up)))
(return (mrv-movedown mrv-leadterm-up var)))))
(destructuring-let* ((wsym (gensym "w"))
lo
coef
((f . logw) (mrv-rewrite exp omega var wsym))
(series (calculate-series f wsym)))
(setq series (maxima-substitute logw `((%log) ,wsym) series))
(setq lo ($lopow series wsym))
(when (or (not ($constantp lo))
(not (free series '%derivative)))
;; (mtell "series: ~M lo: ~M~%" series lo)
(tay-error "error in series expansion" f))
(setq coef ($coeff series wsym lo))
;;(mtell "exp: ~M f: ~M~%" exp f)
;;(mtell "series: ~M~%coeff: ~M~%pow: ~M~%" series coef lo)
(return (list coef lo)))))
(defun mrv-moveup (l var)
(mapcar (lambda (exp)
(simplify-log-of-exp
(syntactic-substitute `((mexpt) $%e ,var) var exp)))
l))
(defun mrv-movedown (l var)
(mapcar (lambda (exp) (syntactic-substitute `((%log simp) ,var) var exp))
l))
;; test whether sub is a subexpression of exp
(defun subexp (exp sub)
(not (equal (maxima-substitute 'dummy
sub
exp)
exp)))
;; Generate $lhospitallim terms of taylor expansion.
;; Ideally we would use a lazy series representation that generates
;; more terms as higher order terms cancel.
(defun calculate-series (exp var)
(assume `((mgreaterp) ,var 0))
(putprop var t 'internal);; keep var from appearing in questions to user
(let ((series ($taylor exp var 0 $lhospitallim)))
(forget `((mgreaterp) ,var 0))
series))
(defun mrv-sign (exp var)
(cond ((freeof var exp)
(cond ((eq ($sign exp) '$zero)
0)
((eq ($sign exp) '$pos)
1)
((eq ($sign exp) '$neg)
-1)
(t (tay-error " cannot determine mrv-sign" exp))))
((eq exp var)
1)
((mtimesp exp)
(* (mrv-sign (cadr exp) var)
(mrv-sign (m*l (cddr exp)) var)))
((and (mexptp exp)
(equal (mrv-sign (cadr exp) var) 1))
1)
((mlogp exp)
(cond ((equal (mrv-sign (cadr exp) var) -1)
(tay-error " complex expression in gruntz limit" exp)))
(mrv-sign (m+ -1 (cadr exp)) var))
((mplusp exp)
(mrv-sign (limitinf exp var) var))
(t (tay-error " cannot determine mrv-sign" exp))))
;; gruntz algorithm for limit of exp as var goes to positive infinity
(defun limitinf (exp var)
(prog (($exptsubst nil))
(cond ((freeof var exp)
(return exp)))
(destructuring-let* ((c0-e0 (mrv-leadterm exp var nil))
(c0 (car c0-e0))
(e0 (cadr c0-e0))
(sig (mrv-sign e0 var)))
(cond ((equal sig 1)
(return 0))
((equal sig -1)
(cond ((equal (mrv-sign c0 var) 1)
(return '$inf))
((equal (mrv-sign c0 var) -1)
(return '$minf))))
((equal sig 0)
(return (limitinf c0 var)))))))
;; user-level function equivalent to $limit.
;; direction must be specified if limit point is not infinite
;; The arguments are checked and a failure of taylor is catched.
(defmfun $gruntz (expr var val &rest rest)
(let (ans dir)
(when (> (length rest) 1)
(merror
(intl:gettext "gruntz: too many arguments; expected just 3 or 4")))
(setq dir (car rest))
(when (and (not (member val '($inf $minf $zeroa $zerob)))
(not (member dir '($plus $minus))))
(merror
(intl:gettext "gruntz: direction must be 'plus' or 'minus'")))
(setq ans
(catch 'taylor-catch
(let ((silent-taylor-flag t))
(declare (special silent-taylor-flag))
(gruntz1 expr var val dir))))
(if (or (null ans) (eq ans t))
(if dir
`(($gruntz simp) ,expr ,var, val ,dir)
`(($gruntz simp) ,expr ,var ,val))
ans)))
;; This function is for internal use in $limit.
(defun gruntz1 (exp var val &rest rest)
(cond ((> (length rest) 1)
(merror (intl:gettext "gruntz: too many arguments; expected just 3 or 4"))))
(let (($logexpand t) ; gruntz needs $logexpand T
(newvar (gensym "w"))
(dir (car rest)))
(cond ((eq val '$inf)
(setq newvar var))
((eq val '$minf)
(setq exp (maxima-substitute (m* -1 newvar) var exp)))
((eq val '$zeroa)
(setq exp (maxima-substitute (m// 1 newvar) var exp)))
((eq val '$zerob)
(setq exp (maxima-substitute (m// -1 newvar) var exp)))
((eq dir '$plus)
(setq exp (maxima-substitute (m+ val (m// 1 newvar)) var exp)))
((eq dir '$minus)
(setq exp (maxima-substitute (m+ val (m// -1 newvar)) var exp)))
(t (merror (intl:gettext "gruntz: direction must be 'plus' or 'minus'; found: ~M") dir)))
(limitinf exp newvar)))
;; substitute y for x in exp
;; similar to maxima-substitute but does not simplify result
(defun syntactic-substitute (y x exp)
(cond ((alike1 x exp) y)
((atom exp) exp)
(t (cons (car exp)
(mapcar (lambda (exp)
(syntactic-substitute y x exp))
(cdr exp))))))
;; log(exp(subexpr)) -> subexpr
;; without simplifying entire exp
(defun simplify-log-of-exp (exp)
(cond ((atom exp) exp)
((and (mlogp exp)
(mexptp (cadr exp))
(eq '$%e (cadadr exp)))
(caddr (cadr exp)))
(t (cons (car exp)
(mapcar #'simplify-log-of-exp
(cdr exp))))))