;;; -*-  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 1981 Massachusetts Institute of Technology         ;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(in-package "MAXIMA")
(macsyma-module laplac)

(DECLARE-TOP(SPECIAL DVAR VAR-LIST VAR-PARM-LIST VAR PARM $SAVEFACTORS
		  CHECKFACTORS $RATFAC $KEEPFLOAT NOUNL NOUNSFLAG)
	 (*EXPR SUBFUNMAKE)
	 (*LEXPR $DIFF $EXPAND $MULTTHRU $RATSIMP)
	 ) 

(DEFUN EXPONENTIATE (POW) 
       ;;;COMPUTES %E**Z WHERE Z IS AN ARBITRARY EXPRESSION TAKING SOME OF THE WORK AWAY FROM SIMPEXPT
       (COND ((ZEROP1 POW) 1)
	     ((EQUAL POW 1) '$%E)
	     (T (POWER '$%E POW)))) 

(DEFUN FIXUPREST (REST) 
       ;;;REST IS A PRODUCT WITHOUT THE MTIMES.FIXUPREST PUTS BACK THE MTIMES
       (COND ((NULL REST) 1)
	     ((CDR REST) (CONS '(MTIMES SIMP) REST))
	     (T (CAR REST)))) 


;(DEFUN POSINT MACRO (X) (SUBST (CADR X) 'Y '(AND (INTEGERP Y) (> Y 0))))
;(DEFUN NEGINT MACRO (X) (SUBST (CADR X) 'Y '(AND (INTEGERP Y) (< Y 0))))

(defmacro posint (x) `(and (integerp ,x) (> ,x 0)))
(defmacro negint (x) `(and (integerp ,x) (< ,x 0)))


(DEFUN ISQUADRATICP (E X)
    ((LAMBDA (B)
	(COND ((ZEROP1 B) (LIST 0 0 E))
	      ((FREEOF X B) (LIST 0 B (MAXIMA-SUBSTITUTE 0 X E)))
	      ((SETQ B (ISLINEAR B X))
		(LIST (DIV* (CAR B) 2) (CDR B) (MAXIMA-SUBSTITUTE 0 X E)))))
	(SDIFF E X)))
 

;;;INITIALIZES SOME GLOBAL VARIABLES THEN CALLS THE DISPATCHING FUNCTION

(DEFMFUN $LAPLACE (FUN VAR PARM) 
       (SETQ FUN (MRATCHECK FUN))
       (COND ((OR NOUNSFLAG (MEMQ '%LAPLACE NOUNL)) (SETQ FUN (REMLAPLACE FUN))))
       (COND ((AND (NULL (ATOM FUN)) (EQ (CAAR FUN) 'MEQUAL))
	      (LIST '(MEQUAL SIMP)
		    (LAPLACE (CADR FUN))
		    (LAPLACE (CADDR FUN))))
	     (T (LAPLACE FUN)))) 

;;;LAMBDA BINDS SOME SPECIAL VARIABLES TO NIL AND DISPATCHES

(DEFUN REMLAPLACE (E)
 (COND ((ATOM E) E)
       (T (CONS (DELQ 'LAPLACE (APPEND (CAR E) NIL) 1) (MAPCAR 'REMLAPLACE (CDR E))))))

(DEFUN LAPLACE (FUN) 
       ((LAMBDA (DVAR VAR-LIST VAR-PARM-LIST) 
	    ;;; Handles easy cases and calls appropriate function on others.
		(COND ((EQUAL FUN 0) 0)
		      ((EQUAL FUN 1)
		       (COND ((ZEROP1 PARM) (SIMPLIFY (LIST '($DELTA) 0)))
			     (T (POWER PARM -1))))
		      ((ALIKE1 FUN VAR) (POWER PARM -2))
		      ((OR (ATOM FUN) (FREEOF VAR FUN))
		       (COND ((ZEROP1 PARM) (MUL2 FUN (SIMPLIFY (LIST '($DELTA) 0))))
			     (T (MUL2 FUN (POWER PARM -1)))))
		      (T ((LAMBDA (OP) 
				  (COND ((EQ OP 'MPLUS)
					 (LAPLUS FUN))
					((EQ OP 'MTIMES)
					 (LAPTIMES (CDR FUN)))
					((EQ OP 'MEXPT)
					 (LAPEXPT FUN NIL))
					((EQ OP '%SIN)
					 (LAPSIN FUN NIL NIL))
					((EQ OP '%COS)
					 (LAPSIN FUN NIL T))
					((EQ OP '%SINH)
					 (LAPSINH FUN NIL NIL))
					((EQ OP '%COSH)
					 (LAPSINH FUN NIL T))
					((EQ OP '%LOG)
					 (LAPLOG FUN))
					((EQ OP '%DERIVATIVE)
					 (LAPDIFF FUN))
					((EQ OP '%INTEGRATE)
					 (LAPINT FUN))
					((EQ OP '%SUM)
					 (LIST '(%SUM SIMP)
					       (LAPLACE (CADR FUN))
					       (CADDR FUN)
					       (CADDDR FUN)
					       (CAR (CDDDDR FUN))))
					((EQ OP '%ERF)
					 (LAPERF FUN))
				((AND (EQ OP '%ILT)(EQ (CADDDR FUN) VAR))
			(COND ((EQ PARM (CADDR FUN))(CADR FUN))
				(T (SUBST PARM (CADDR FUN)(CADR FUN))))
)					((EQ OP '$DELTA)
					 (LAPDELTA FUN NIL))
					((SETQ OP ($GET OP '$LAPLACE))
					 (MCALL OP FUN VAR PARM))
					(T (LAPDEFINT FUN))))
			  (CAAR FUN)))))
	NIL
	NIL
	NIL)) 

(DEFUN LAPLUS (FUN) 
       (SIMPLUS (CONS '(MPLUS)
		      (MAPCAR (FUNCTION LAPLACE) (CDR FUN)))
		1.
		T)) 

(DEFUN LAPTIMES (FUN) 
       ;;;EXPECTS A LIST (PERHAPS EMPTY) OF FUNCTIONS MULTIPLIED TOGETHER WITHOUT THE MTIMES
       ;;;SEES IF IT CAN APPLY THE FIRST AS A TRANSFORMATION ON THE REST OF THE FUNCTIONS
       (COND ((NULL FUN) (LIST '(MEXPT SIMP) PARM -1.))
	     ((NULL (CDR FUN)) (LAPLACE (CAR FUN)))
	     ((FREEOF VAR (CAR FUN))
	      (SIMPTIMES (LIST '(MTIMES)
			       (CAR FUN)
			       (LAPTIMES (CDR FUN)))
			 1.
			 T))
	     ((EQ (CAR FUN) VAR)
	      (SIMPTIMES (LIST '(MTIMES)
			       -1.
			       (SDIFF (LAPTIMES (CDR FUN)) PARM))
			 1.
			 T))
	     (T ((LAMBDA (OP) 
			 (COND ((EQ OP 'MEXPT)
				(LAPEXPT (CAR FUN) (CDR FUN)))
			       ((EQ OP 'MPLUS)
				(LAPLUS ($MULTTHRU (FIXUPREST (CDR FUN)) (CAR FUN))))
			       ((EQ OP '%SIN)
				(LAPSIN (CAR FUN) (CDR FUN) NIL))
			       ((EQ OP '%COS)
				(LAPSIN (CAR FUN) (CDR FUN) T))
			       ((EQ OP '%SINH)
				(LAPSINH (CAR FUN) (CDR FUN) NIL))
			       ((EQ OP '%COSH)
				(LAPSINH (CAR FUN) (CDR FUN) T))
			       ((EQ OP '$DELTA)
				(LAPDELTA (CAR FUN) (CDR FUN)))

			       (T (LAPSHIFT (CAR FUN) (CDR FUN)))))
		 (CAAAR FUN))))) 

(DEFUN LAPEXPT (FUN REST) 
       ;;;HANDLES %E**(A*T+B)*REST(T), %E**(A*T**2+B*T+C),
       ;;; 1/SQRT(A*T+B), OR T**K*REST(T)
       (PROG (AB BASE-OF-FUN POWER RESULT) 
	     (SETQ BASE-OF-FUN (CADR FUN) POWER (CADDR FUN))
	     (COND
	      ((AND
		(FREEOF VAR BASE-OF-FUN)
		(SETQ 
		 AB
		 (ISQUADRATICP
		  (COND ((EQ BASE-OF-FUN '$%E) POWER)
			(T (SIMPTIMES (LIST '(MTIMES)
					    POWER
					    (LIST '(%LOG)
						  BASE-OF-FUN))
				      1.
				      NIL)))
		  VAR)))
	       (COND ((EQUAL (CAR AB) 0.) (GO %E-CASE-LIN))
		     ((NULL REST) (GO %E-CASE-QUAD))
		     (T (GO NOLUCK))))
	      ((AND (EQ BASE-OF-FUN VAR) (FREEOF VAR POWER))
	       (GO VAR-CASE))
	      ((AND (ALIKE1 '((RAT) -1. 2.) POWER) (NULL REST)
		    (SETQ AB (ISLINEAR BASE-OF-FUN VAR)))
	       (SETQ RESULT (DIV* (CDR AB) (CAR AB)))
	       (RETURN (SIMPTIMES
		(LIST '(MTIMES)
		      (LIST '(MEXPT)
			    (DIV* '$%PI
				  (LIST '(MTIMES)
					(CAR AB)
					PARM))
			    '((RAT) 1. 2.))
		      (EXPONENTIATE (LIST '(MTIMES) RESULT PARM))
		      (LIST '(MPLUS)
			    1.
			    (LIST '(MTIMES)
				  -1.
				   (LIST '(%ERF)
						 (LIST '(MEXPT)
						       (LIST '(MTIMES)
							     RESULT
							     PARM)
						       '((RAT)
							 1.
							 2.)))
					   ))) 1 NIL)))
	      (T (GO NOLUCK)))
	%E-CASE-LIN
	     (SETQ 
	      RESULT
	      (COND
	       (REST ($RATSIMP ($AT (LAPTIMES REST)
				    (LIST '(MEQUAL SIMP)
					  PARM
					  (LIST '(MPLUS SIMP)
						PARM
						(AFIXSIGN (CADR AB)
							  NIL))))))
	       (T (LIST '(MEXPT)
			(LIST '(MPLUS)
			      PARM
			      (AFIXSIGN (CADR AB) NIL))
			-1.))))
	     (RETURN (SIMPTIMES (LIST '(MTIMES)
				      (EXPONENTIATE (CADDR AB))
				      RESULT)
				1.
				NIL))
	%E-CASE-QUAD
	     (SETQ RESULT (AFIXSIGN (CAR AB) NIL))
	     (SETQ 
	      RESULT
	       (LIST
		'(MTIMES)
		(DIV* (LIST '(MEXPT)
			    (DIV* '$%PI RESULT)
			    '((RAT) 1. 2.))
		      2.)
		(EXPONENTIATE (DIV* (LIST '(MEXPT) PARM 2.)
				    (LIST '(MTIMES)
					  4.
					  RESULT)))
		(LIST '(MPLUS)
		      1.
		      (LIST '(MTIMES)
			    -1.
			     (LIST '(%ERF)
					   (DIV* PARM
						 (LIST '(MTIMES)
						       2.
						       (LIST '(MEXPT)
							     RESULT
							     '((RAT)
							       1.
							       2.)))))
				))))
	     (AND (NULL (EQUAL (CADR AB) 0.))
		  (SETQ RESULT
			(MAXIMA-SUBSTITUTE (LIST '(MPLUS)
					  PARM
					  (LIST '(MTIMES)
						-1.
						(CADR AB)))
				    PARM
				    RESULT)))
	     (RETURN (SIMPTIMES  (LIST '(MTIMES)
			   (EXPONENTIATE (CADDR AB))
			   RESULT) 1 NIL))
	VAR-CASE
	     (COND ((OR (NULL REST) (FREEOF VAR (FIXUPREST REST)))
		    (GO VAR-EASY-CASE)))
	     (COND ((POSINT POWER)
		    (RETURN (AFIXSIGN (APPLY '$DIFF
					     (LIST (LAPTIMES REST)
						   PARM
						   POWER))
				      (EVEN POWER))))
		   ((NEGINT POWER)
		    (RETURN (MYDEFINT (HACKIT POWER REST)
				      (CREATENAME PARM (MINUS POWER))
				      PARM)))
		   (T (GO NOLUCK)))
	VAR-EASY-CASE
	     (SETQ POWER
		   (SIMPLUS (LIST '(MPLUS) 1. POWER) 1. T))
	     (OR (EQ (ASKSIGN POWER) '$POSITIVE) (GO NOLUCK))
	     (SETQ RESULT (LIST (LIST '(%GAMMA) POWER)
				(LIST '(MEXPT)
				      PARM
				      (AFIXSIGN POWER NIL))))
	     (AND REST (SETQ RESULT (NCONC RESULT REST)))
	     (RETURN (SIMPTIMES (CONS '(MTIMES) RESULT)
				1.
				NIL))
	NOLUCK
	     (RETURN
	      (COND
	       ((AND (POSINT POWER)
		     (MEMQ (CAAR BASE-OF-FUN)
			   '(MPLUS %SIN %COS %SINH %COSH)))
		(LAPTIMES (CONS BASE-OF-FUN
				(CONS (COND ((= POWER 2.) BASE-OF-FUN)
					    (T (LIST '(MEXPT SIMP)
						     BASE-OF-FUN
						     (SUB1 POWER))))
				      REST))))
	       (T (LAPSHIFT FUN REST)))))) 

(DEFUN MYDEFINT (F X A) 
       ;;;INTEGRAL FROM A TO INFINITY OF F(X)
       ((LAMBDA (TRYINT) (COND (TRYINT (CAR TRYINT))
			       (T (LIST '(%INTEGRATE SIMP)
					F
					X
					A
					'$INF))))
	(AND (NOT ($UNKNOWN F))
	     (ERRSET ($DEFINT F X A '$INF))))) 

(DEFUN CREATENAME 
 ;;;CREATES HOPEFULLY UNIQUE NAMES FOR VARIABLE OF INTEGRATION
 (HEAD TAIL) 
 (implode (NCONC (EXPLODEC HEAD) (EXPLODEC TAIL))))

(declare-top (FIXNUM EXPONENT)) 

(DEFUN HACKIT (EXPONENT REST) 
       ;;;REDUCES LAPLACE(F(T)/T**N,T,S) CASE TO LAPLACE(F(T)/T**(N-1),T,S) CASE
       (COND ((EQUAL EXPONENT -1.)
	      ((LAMBDA (PARM) (LAPTIMES REST)) (CREATENAME PARM 1.)))
	     (T (MYDEFINT (HACKIT (f1+ EXPONENT) REST)
			  (CREATENAME PARM (DIFFERENCE -1. EXPONENT))
			  (CREATENAME PARM (MINUS EXPONENT)))))) 

(DECLARE-TOP(NOTYPE EXPONENT)) 

(DEFUN AFIXSIGN (FUNCT SIGNSWITCH) 
       ;;;MULTIPLIES FUNCT BY -1 IF SIGNSWITCH IS NIL
       (COND (SIGNSWITCH FUNCT)
	     (T (SIMPTIMES (LIST '(MTIMES) -1. FUNCT) 1. T)))) 

 

(DEFUN LAPSHIFT (FUN REST) 
       (COND ((ATOM FUN) (merror "INTERNAL ERROR"))
	     ((OR (MEMQ 'LAPLACE (CAR FUN)) (NULL REST))
	      (LAPDEFINT (COND (REST (SIMPTIMES (CONS '(MTIMES)
					   (CONS FUN REST)) 1 T))
			       (T FUN))))
	     (T (LAPTIMES (APPEND REST
				  (NCONS (CONS (APPEND (CAR FUN)
						       '(LAPLACE))
					       (CDR FUN)))))))) 

(DEFUN MOSTPART (F PARM SIGN A B) 
       ;;;COMPUTES %E**(W*B*%I)*F(S-W*A*%I) WHERE W=-1 IF SIGN IS T ELSE W=1
       ((LAMBDA (SUBSTINFUN) 
		(COND ((ZEROP1 B) SUBSTINFUN)
		      (T (LIST '(MTIMES)
			       (EXPONENTIATE (AFIXSIGN (LIST '(MTIMES)
							     B
							     '$%I)
						       (NULL SIGN)))
			       SUBSTINFUN))))
	($AT F
	     (LIST '(MEQUAL SIMP)
		   PARM
		   (LIST '(MPLUS SIMP)
			 PARM
			 (AFIXSIGN (LIST '(MTIMES)
					 A
					 '$%I)
				   SIGN)))))) 

(DEFUN COMPOSE 
 ;;;IF WHICHSIGN IS NIL THEN SIN TRANSFORM ELSE COS TRANSFORM
 (FUN PARM WHICHSIGN A B) 
       ((LAMBDA (RESULT) 
		($RATSIMP (SIMPTIMES (CONS '(MTIMES)
					   (COND (WHICHSIGN RESULT)
						 (T (CONS '$%I
							  RESULT))))
				     1 NIL)))
	(LIST '((RAT) 1. 2.)
	      (LIST '(MPLUS)
		    (MOSTPART FUN PARM T A B)
		    (AFIXSIGN (MOSTPART FUN PARM NIL A B)
			      WHICHSIGN))))) 

(DEFUN LAPSIN 
 ;;;FUN IS OF THE FORM SIN(A*T+B)*REST(T) OR COS
 (FUN REST TRIGSWITCH) 
       ((LAMBDA (AB) 
	 (COND
	  (AB
	    (COND
	     (REST (COMPOSE (LAPTIMES REST)
			    PARM
			    TRIGSWITCH
			    (CAR AB)
			    (CDR AB)))
	     (T (SIMPTIMES
	      (LIST
	       '(MTIMES)
	       (COND
		((ZEROP1 (CDR AB))
		 (COND (TRIGSWITCH PARM) (T (CAR AB))))
		(T (COND (TRIGSWITCH (LIST '(MPLUS)
					   (LIST '(MTIMES)
						 PARM
						 (LIST '(%COS)
						       (CDR AB)))
					   (LIST '(MTIMES)
						 -1.
						 (CAR AB)
						 (LIST '(%SIN)
						       (CDR AB)))))
			 (T (LIST '(MPLUS)
				  (LIST '(MTIMES)
					PARM
					(LIST '(%SIN)
					      (CDR AB)))
				  (LIST '(MTIMES)
					(CAR AB)
					(LIST '(%COS)
					      (CDR AB))))))))
	       (LIST '(MEXPT)
		     (LIST '(MPLUS)
			   (LIST '(MEXPT) PARM 2.)
			   (LIST '(MEXPT) (CAR AB) 2.))
		     -1.))
	    1 NIL))))
	  (T (LAPSHIFT FUN REST))))
	(ISLINEAR (CADR FUN) VAR))) 

(DEFUN LAPSINH 
 ;;;FUN IS OF THE FORM SINH(A*T+B)*REST(T) OR IS COSH
 (FUN REST SWITCH) 
	(COND ((ISLINEAR (CADR FUN) VAR)
       ($RATSIMP
	(LAPLUS
	 (SIMPLUS
	  (LIST '(MPLUS)
		(NCONC (LIST '(MTIMES)
			     (LIST '(MEXPT)
				   '$%E
				   (CADR FUN))
			     '((RAT) 1. 2.))
		       REST)
		(AFIXSIGN (NCONC (LIST '(MTIMES)
				       (LIST '(MEXPT)
					     '$%E
					     (AFIXSIGN (CADR FUN)
						       NIL))
				       '((RAT) 1. 2.))
				 REST)
			  SWITCH))
	  1.
	  NIL)))) 
	(T (LAPSHIFT FUN REST))))

(DEFUN LAPLOG 
 ;;;FUN IS OF THE FORM LOG(A*T)
 (FUN) ((LAMBDA (AB) 
		(COND ((AND AB (ZEROP1 (CDR AB)))
		       (SIMPTIMES (LIST '(MTIMES)
					(LIST '(MPLUS)
					      (subfunmake '$PSI
							  '(0)
							  (NCONS 1.))
					      (LIST '(%LOG)
						    (CAR AB))
					      (LIST '(MTIMES)
						    -1.
						    (LIST '(%LOG)
							  PARM)))
					(LIST '(MEXPT)
					      PARM
					      -1.))
				  1 NIL))
		      (T (LAPDEFINT FUN))))
	(ISLINEAR (CADR FUN) VAR))) 

(DEFUN RAISEUP (FBASE EXPONENT) 
       (COND ((EQUAL EXPONENT 1.) FBASE)
	     (T (LIST '(MEXPT) FBASE EXPONENT)))) 

(DEFUN LAPDELTA (FUN REST) 
       ;;TAKES TRANSFORM OF DELTA(A*T+B)*F(T)
       ((LAMBDA (AB SIGN RECIPA) 
	 (COND
	  (AB
	   (SETQ RECIPA (POWER (CAR AB) -1) AB (DIV (CDR AB) (CAR AB)))
	   (SETQ SIGN (ASKSIGN AB) RECIPA (SIMPLIFYA (LIST '(MABS) RECIPA) NIL))
	   (SIMPLIFYA (COND ((EQ SIGN '$POSITIVE) 0)
			    ((EQ SIGN '$ZERO)
			     (LIST '(MTIMES)
				   (MAXIMA-SUBSTITUTE 0 VAR (FIXUPREST REST))
				   RECIPA))
			    (T (LIST '(MTIMES)
				     (MAXIMA-SUBSTITUTE (NEG AB)
						 VAR
						 (FIXUPREST REST))
				     (LIST '(MEXPT)
					   '$%E
					   (CONS '(MTIMES)
						 (CONS PARM (NCONS AB))))
				     RECIPA)))
		      NIL))
	  (T (LAPSHIFT FUN REST))))
	(ISLINEAR (CADR FUN) VAR) NIL NIL)) 

(DEFUN LAPERF (FUN  )
       ((LAMBDA (AB) 
	 (COND
	  ((AND AB (EQUAL (CDR AB) 0.))
	   (SIMPTIMES (LIST '(MTIMES)
		 (DIV* (EXPONENTIATE (DIV* (LIST '(MEXPT)
						 PARM
						 2.)
					   (LIST '(MTIMES)
						 4.
						 (LIST '(MEXPT)
						       (CAR AB)
						       2.))))
		       PARM)
		 (LIST '(MPLUS)
		       1.
		       (LIST '(MTIMES)
			     -1.
			      (LIST '(%ERF)
					    (DIV* PARM
						  (LIST '(MTIMES)
							2.
							(CAR AB))))
				      ))) 1 NIL))
	  (T (LAPDEFINT FUN))))
	(ISLINEAR (CADR FUN) VAR)))
(DEFUN LAPDEFINT (FUN)
  (PROG (TRYINT MULT)
    (AND ($UNKNOWN FUN)(GO SKIP))
    (SETQ MULT (SIMPTIMES (LIST '(MTIMES) (EXPONENTIATE
					   (LIST '(MTIMES SIMP) -1 VAR PARM)) FUN) 1 NIL))
    (MEVAL `(($ASSUME) ,@(LIST (LIST '(MGREATERP) PARM 0))))
    (SETQ TRYINT (ERRSET ($DEFINT MULT VAR 0 '$INF)))
    (MEVAL `(($FORGET) ,@(LIST (LIST '(MGREATERP) PARM 0))))
    (AND TRYINT (NOT (EQ (CAAAR TRYINT) '%INTEGRATE))  (RETURN (CAR TRYINT)))
   SKIP (RETURN (LIST '(%LAPLACE SIMP) FUN VAR PARM))))
 

(DECLARE-TOP(FIXNUM ORDER)) 

(DEFUN LAPDIFF 
 ;;;FUN IS OF THE FORM DIFF(F(T),T,N) WHERE N IS A POSITIVE INTEGER
 (FUN) (PROG (DIFFLIST DEGREE FRONTEND RESULTLIST NEWDLIST ORDER
	      ARG2) 
	     (SETQ NEWDLIST (SETQ DIFFLIST (COPY (CDDR FUN))))
	     (SETQ ARG2 (LIST '(MEQUAL SIMP) VAR 0.))
	A    (COND ((NULL DIFFLIST)
		    (RETURN (CONS '(%DERIVATIVE SIMP)
				  (CONS (LIST '(%LAPLACE SIMP)
					      (CADR FUN)
					      VAR
					      PARM)
					NEWDLIST))))
		   ((EQ (CAR DIFFLIST) VAR)
		    (SETQ DEGREE (CADR DIFFLIST) 
			  DIFFLIST (CDDR DIFFLIST))
		    (GO OUT)))
	     (SETQ DIFFLIST (CDR (SETQ FRONTEND (CDR DIFFLIST))))
	     (GO A)
	OUT  (COND ((NULL (POSINT DEGREE))
		    (RETURN (LIST '(%LAPLACE SIMP) FUN VAR PARM))))
	     (COND (FRONTEND (RPLACD FRONTEND DIFFLIST))
		   (T (SETQ NEWDLIST DIFFLIST)))
	     (COND (NEWDLIST (SETQ FUN (CONS '(%DERIVATIVE SIMP)
					     (CONS (CADR FUN)
						   NEWDLIST))))
		   (T (SETQ FUN (CADR FUN))))
	     (SETQ ORDER 0.)
	LOOP (SETQ DEGREE (f1- DEGREE))
	     (SETQ RESULTLIST
		   (CONS (LIST '(MTIMES)
			       (RAISEUP PARM DEGREE)
			       ($AT ($DIFF FUN VAR ORDER) ARG2))
			 RESULTLIST))
	     (SETQ ORDER (f1+ ORDER))
	     (AND (> DEGREE 0.) (GO LOOP))
	     (SETQ RESULTLIST (COND ((CDR RESULTLIST)
				     (CONS '(MPLUS)
					   RESULTLIST))
				    (T (CAR RESULTLIST))))
	     (RETURN (SIMPLUS (LIST '(MPLUS)
				      (LIST '(MTIMES)
					    (RAISEUP PARM ORDER)
					    (LAPLACE FUN))
				      (LIST '(MTIMES)
					    -1.
					    RESULTLIST))
				1 NIL)))) 

(DECLARE-TOP(NOTYPE ORDER)) 

(DEFUN LAPINT 
 ;;;FUN IS OF THE FORM INTEGRATE(F(X)*G(T)*H(T-X),X,0,T)
 (FUN) (PROG (NEWFUN PARM-LIST F) 
	     (AND DVAR (GO CONVOLUTION))
	     (SETQ DVAR (CADR (SETQ NEWFUN (CDR FUN))))
	     (AND (CDDR NEWFUN)
		  (ZEROP1 (CADDR NEWFUN))
		  (EQ (CADDDR NEWFUN) VAR)
		  (GO CONVOLUTIONTEST))
	NOTCON
	     (SETQ NEWFUN (CDR FUN))
	     (COND ((CDDR NEWFUN)
		    (COND ((AND (FREEOF VAR (CADDR NEWFUN))
				(FREEOF VAR (CADDDR NEWFUN)))
			   (RETURN (LIST '(%INTEGRATE SIMP)
					 (LAPLACE (CAR NEWFUN))
					 DVAR
					 (CADDR NEWFUN)
					 (CADDDR NEWFUN))))
			  (T (GO GIVEUP))))
		   (T (RETURN (LIST '(%INTEGRATE SIMP)
				    (LAPLACE (CAR NEWFUN))
				    DVAR))))
	GIVEUP
	     (RETURN (LIST '(%LAPLACE SIMP) FUN VAR PARM))
	CONVOLUTIONTEST
	     (SETQ NEWFUN ($FACTOR (CAR NEWFUN)))
	     (COND ((EQ (CAAR NEWFUN) 'MTIMES)
		    (SETQ F (CADR NEWFUN) NEWFUN (CDDR NEWFUN)))
		   (T (SETQ F NEWFUN NEWFUN NIL)))
	GOTHRULIST
	     (COND ((FREEOF DVAR F)
		    (SETQ PARM-LIST (CONS F PARM-LIST)))
		   ((FREEOF VAR F) (SETQ VAR-LIST (CONS F VAR-LIST)))
		   ((FREEOF DVAR
			    ($RATSIMP (MAXIMA-SUBSTITUTE (LIST '(MPLUS)
							VAR
							DVAR)
						  VAR
						  F)))
		    (SETQ VAR-PARM-LIST (CONS F VAR-PARM-LIST)))
		   (T (GO NOTCON)))
	     (COND (NEWFUN (SETQ F (CAR NEWFUN) NEWFUN (CDR NEWFUN))
			   (GO GOTHRULIST)))
	     (AND
	      PARM-LIST
	      (RETURN
	       (LAPLACE
		(CONS
		 '(MTIMES)
		 (NCONC PARM-LIST
			(NCONS (LIST '(%INTEGRATE)
				     (CONS '(MTIMES)
					   (APPEND VAR-LIST
						   VAR-PARM-LIST))
				     DVAR
				     0.
				     VAR)))))))
	CONVOLUTION
	     (RETURN
	      (SIMPTIMES
	       (LIST
		'(MTIMES)
		(LAPLACE ($EXPAND (MAXIMA-SUBSTITUTE VAR
					      DVAR
					      (FIXUPREST VAR-LIST))))
		(LAPLACE
		 ($EXPAND (MAXIMA-SUBSTITUTE 0.
				      DVAR
				      (FIXUPREST VAR-PARM-LIST)))))
	       1.
	       T)))) 

(DECLARE-TOP(SPECIAL VARLIST RATFORM ILS ILT)) 

(DEFMFUN $ILT (EXP ILS ILT) 
 ;;;EXP IS F(S)/G(S) WHERE F AND G ARE POLYNOMIALS IN S AND DEGR(F) < DEGR(G)
	 (LET (VARLIST ($SAVEFACTORS T) CHECKFACTORS $RATFAC $KEEPFLOAT) 
		;;; MAKES ILS THE MAIN VARIABLE
		(SETQ VARLIST (LIST ILS))
		(NEWVAR EXP)
		(ORDERPOINTER VARLIST)
		(SETQ VAR (CAADR (RATREP* ILS)))
		(COND ((AND (NULL (ATOM EXP))
			    (EQ (CAAR EXP) 'MEQUAL))
		       (LIST '(MEQUAL)
			     ($ILT (CADR EXP) ILS ILT)
			     ($ILT (CADDR EXP) ILS ILT)))
		      ((ZEROP1 EXP) 0.)
		      ((FREEOF ILS EXP)
		       (LIST '(%ILT SIMP) EXP ILS ILT))
		      (T (ILT0 EXP))))) 

(DEFUN MAXIMA-RATIONALP (LE V) 
       (COND ((NULL LE))
	     ((AND (NULL (ATOM (CAR LE))) (NULL (FREEOF V (CAR LE))))
	      NIL)
	     (T (MAXIMA-RATIONALP (CDR LE) V)))) 

(DEFUN ILT0 
 ;;;THIS FUNCTION DOES THE PARTIAL FRACTION DECOMPOSITION
 (EXP) (PROG (WHOLEPART FRPART NUM DENOM Y CONTENT REAL FACTOR
	      APART BPART PARNUMER RATARG RATFORM) 
	     (AND (MPLUSP EXP)
		  (RETURN (SIMPLUS  (CONS '(MPLUS)
				(MAPCAR (FUNCTION (LAMBDA(F)($ILT F ILS ILT))) (CDR EXP))) 1 T)))
	     (AND (NULL (ATOM EXP))
		  (EQ (CAAR EXP) '%LAPLACE)
		  (EQ (CADDDR EXP) ILS)
		  (RETURN (COND ((EQ (CADDR EXP) ILT) (CADR EXP))
				(T (SUBST ILT
					       (CADDR EXP)
					       (CADR EXP))))))
	     (SETQ RATARG (RATREP* EXP))
	     (OR (MAXIMA-RATIONALP VARLIST ILS)
		 (RETURN (LIST '(%ILT SIMP) EXP ILS ILT)))
	     (SETQ RATFORM (CAR RATARG))
	     (SETQ DENOM (RATDENOMINATOR (CDR RATARG)))
	     (SETQ FRPART (PDIVIDE (RATNUMERATOR (CDR RATARG)) DENOM))
	     (SETQ WHOLEPART (CAR FRPART))
	     (SETQ FRPART (RATQU (CADR FRPART) DENOM))
	     (COND ((NOT (ZEROP1 (CAR WHOLEPART)))
		    (RETURN (LIST '(%ILT SIMP) EXP ILS ILT)))
		   ((ZEROP1 (CAR FRPART)) (RETURN 0)))
	     (SETQ NUM (CAR FRPART) DENOM (CDR FRPART))
	     (SETQ Y (OLDCONTENT DENOM))
	     (SETQ CONTENT (CAR Y))
	     (SETQ REAL (CADR Y))
	     (SETQ FACTOR (PFACTOR REAL))
	LOOP (COND ((NULL (CDDR FACTOR))
		    (SETQ APART REAL 
			  BPART 1 
			  Y '((0 . 1) 1 . 1))
		    (GO SKIP)))
	     (SETQ APART (PEXPT (CAR FACTOR) (CADR FACTOR)))
	     (SETQ BPART (CAR (RATQU REAL APART)))
	     (SETQ Y (BPROG APART BPART))
	SKIP (SETQ FRPART
		   (CDR (RATDIVIDE (RATTI (RATNUMERATOR NUM)
					  (CDR Y)
					  T)
				   (RATTI (RATDENOMINATOR NUM)
					  (RATTI CONTENT APART T)
					  T))))
	     (SETQ 
	      PARNUMER
	      (CONS (ILT1 (RATQU (RATNUMERATOR FRPART)
				 (RATTI (RATDENOMINATOR FRPART)
					(RATTI (RATDENOMINATOR NUM)
					       CONTENT
					       T)
					T))
			  (CAR FACTOR)
			  (CADR FACTOR))
		    PARNUMER))
	     (SETQ FACTOR (CDDR FACTOR))
	     (COND ((NULL FACTOR)
		    (RETURN (SIMPLUS (CONS '(MPLUS) PARNUMER)
				     1.
				     T))))
	     (SETQ NUM (CDR (RATDIVIDE (RATTI NUM (CAR Y) T)
				       (RATTI CONTENT BPART T))))
	     (SETQ REAL BPART)
	     (GO LOOP))) 

(DECLARE-TOP(FIXNUM K) (SPECIAL Q Z)) 

(DEFUN ILT1 (P Q K)
  ((LAMBDA (Z)
  (COND (( ONEP1 K)(ILT3 P ))
	(T (SETQ Z (BPROG Q (PDERIVATIVE Q VAR)))(ILT2 P K)))) NIL))


(DEFUN ILT2 
 ;;;INVERTS P(S)/Q(S)**K WHERE Q(S)  IS IRREDUCIBLE
 ;;;DOESN'T CALL ILT3 IF Q(S) IS LINEAR
  (P K)
       (PROG (Y A B) 
		(AND (ONEP1 K)(RETURN (ILT3 P)))
		(SETQ K (f1- K))
	     (SETQ A (RATTI P (CAR Z) T))
	     (SETQ B (RATTI P (CDR Z) T))
	     (SETQ Y (PEXPT Q K))
	     (COND
	      ((OR (NULL (EQUAL (PDEGREE Q VAR) 1.))
		   (> (PDEGREE (CAR P) VAR) 0.))
	       (RETURN
		(SIMPLUS
		 (LIST
		  '(MPLUS)
		  (ILT2
		   (CDR (RATDIVIDE (RATPLUS A
					    (RATQU (RATDERIVATIVE B
								  VAR)
						   K))
				   Y))
		   K)
		  ($MULTTHRU (SIMPTIMES (LIST '(MTIMES)
				   ILT
				   (POWER K -1)
				   (ILT2 (CDR (RATDIVIDE B Y)) K))
			     1.
			     T)))
		 1.
		 T))))
	     (SETQ A (DISREP (POLCOEF Q 1.)) 
		   B (DISREP (POLCOEF Q 0.)))
	     (RETURN
	      (SIMPTIMES (LIST '(MTIMES)
			       (DISREP P)
			       (RAISEUP ILT K)
			       (SIMPEXPT (LIST '(MEXPT)
					       '$%E
					       (LIST '(MTIMES)
						     -1.
						     ILT
						     B
						     (LIST '(MEXPT)
							   A
							   -1.)))
					 1.
					 NIL)
			       (LIST '(MEXPT)
				     A
				     (DIFFERENCE -1. K))
			       (LIST '(MEXPT)
				     (FACTORIAL K)
				     -1.))
			 1.
			 NIL)))) 

(DECLARE-TOP(NOTYPE K)) 

;(DEFUN COEF MACRO (POL) (SUBST (CADR POL) (QUOTE DEG)
;  '(DISREP (RATQU (POLCOEF (CAR P) DEG) (CDR P)))))

(defmacro coef (pol)
  `(DISREP (RATQU (POLCOEF (CAR P) ,pol) (CDR P))))

(DEFmfUN LAPSUM N (CONS '(MPLUS)(LISTIFY N)))
(DEFmfUN LAPPROD N (CONS '(MTIMES)(LISTIFY N)))
(DEFmfUN EXPO N (CONS '(MEXPT)(LISTIFY N)))
(DEFUN ILT3 
 ;;;INVERTS P(S)/Q(S) WHERE Q(S) IS IRREDUCIBLE
 (P ) (PROG (DISCRIM SIGN A C D E B1 B0 R TERM1 TERM2 DEGR) 
	     (SETQ E (DISREP (POLCOEF Q 0.)) 
		   D (DISREP (POLCOEF Q 1.)) 
		   DEGR (PDEGREE Q VAR))
	      (AND (EQUAL DEGR 1.)
	       (RETURN
		(SIMPTIMES (LAPPROD
				 (DISREP P)
				 (EXPO D -1.)
				 (EXPO
				       '$%E
				       (LAPPROD
					     -1.
					     ILT
					     E
					     (EXPO
						   D
						   -1.))))
			   1.
			   NIL)))
		(SETQ C (DISREP (POLCOEF Q 2)))
	      (AND (EQUAL DEGR 2.) (GO QUADRATIC))
	      (AND (EQUAL DEGR 3.) (ZEROP1 C) (ZEROP1 D)
	       (GO CUBIC))
	(RETURN (LIST '(%ILT SIMP) (DIV* (DISREP P)(DISREP Q)) ILS ILT))
	CUBIC (SETQ  A (DISREP (POLCOEF Q 3))
		R (SIMPNRT (DIV* E A) 3))
		(SETQ D (DIV* (DISREP P)(LAPPROD A (LAPSUM
			(EXPO ILS 3)(EXPO '%R 3)))))
		(RETURN (ILT0 (MAXIMA-SUBSTITUTE R '%R ($PARTFRAC D ILS))))
	QUADRATIC (SETQ B0 (COEF 0) B1 (COEF 1))

	     (SETQ DISCRIM
		   (SIMPLUS (LAPSUM
				  (LAPPROD
					4.
					E
					C)
				  (LAPPROD -1. D D))
			    1.
			    NIL))
	     (SETQ SIGN (COND ((FREE DISCRIM '$%I) (ASKSIGN DISCRIM)) (T '$POSITIVE)) 
		   TERM1 '(%COS) 
		   TERM2 '(%SIN))
	(SETQ DEGR (EXPO '$%E (LAPPROD ILT D (POWER C -1) '((RAT SIMP) -1 2))))
	     (COND ((EQ SIGN '$ZERO)
		    (RETURN (SIMPTIMES (LAPPROD DEGR (LAPSUM (DIV* B1 C)(LAPPROD
		(DIV* (LAPSUM (LAPPROD 2 B0 C)(LAPPROD -1 B1 D))
		(LAPPROD 2 C C)) ILT))) 1 NIL))
)		   ((EQ SIGN '$NEGATIVE)
		    (SETQ TERM1 '(%COSH) 
			  TERM2 '(%SINH) 
			  DISCRIM (SIMPTIMES (LAPPROD
						   -1.
						   DISCRIM)
					     1.
					     T))))
	     (SETQ DISCRIM (SIMPNRT DISCRIM 2))
	     (SETQ 
	      SIGN
	      (SIMPTIMES
	       (LAPPROD
		     (LAPSUM
			   (LAPPROD
				 2.
				 B0
				 C)
			   (LAPPROD
				 -1.
				 B1
				 D))
		     (EXPO DISCRIM -1.))
	       1.
	       NIL))
	     (SETQ C (POWER C -1))
	     (SETQ DISCRIM (SIMPTIMES (LAPPROD
					    DISCRIM
					    ILT
					    '((RAT SIMP) 1. 2.)
					    C)
				      1.
				      T))
	     (RETURN
	      (SIMPTIMES
	       (LAPPROD
		     C
		     DEGR
		     (LAPSUM
			   (LAPPROD
				 B1
				 (LIST TERM1 DISCRIM))
			   (LAPPROD
				 SIGN
				 (LIST TERM2 DISCRIM))))
	       1.
	       NIL)))) 

#-NIL
(DECLARE-TOP(UNSPECIAL DVAR ILS ILT NOUNL PARM Q RATFORM VAR VARLIST
		    VAR-LIST VAR-PARM-LIST Z))