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This is maxima.info, produced by makeinfo version 4.1 from maxima.texi. This is a Texinfo Maxima Manual Copyright 1994,2001 William F. Schelter START-INFO-DIR-ENTRY * Maxima: (maxima). A computer algebra system. END-INFO-DIR-ENTRY File: maxima.info, Node: Definitions for Operators, Prev: PREFIX, Up: Operators Definitions for Operators ========================= - operator: "!" The factorial operator, which is the product of all the integers from 1 up to its argument. Thus 5! = 1*2*3*4*5 = 120. The value of /the option FACTLIM (default: [-1]) gives the highest factorial which is automatically expanded. If it is -1 then all integers are expanded. See also the FACTORIAL, MINFACTORIAL, and FACTCOMB commands. - operator: "!!" Stands for double factorial which is defined as the product of all the consecutive odd (or even) integers from 1 (or 2) to the odd (or even) argument. Thus 8!! is 2*4*6*8 = 384. - operator: "#" The logical operator "Not equals". - operator: "." The dot operator, for matrix (non-commutative) multiplication. When "." is used in this way, spaces should be left on both sides of it, e.g. A . B. This distinguishes it plainly from a decimal point in a floating point number. Do APROPOS(DOT); for a list of the switches which affect the dot operator. DESCRIBE(switch-name); will explain them. - operator: ":" The assignment operator. E.g. A:3 sets the variable A to 3. - operator: "::" Assignment operator. :: assigns the value of the expression on its right to the value of the quantity on its left, which must evaluate to an atomic variable or subscripted variable. - operator: "::=" The "::=" is used instead of ":=" to indicate that what follows is a macro definition, rather than an ordinary functional definition. See DESCRIBE(MACROS). - operator: ":=" The function definition operator. E.g. F(X):=SIN(X) defines a function F. - operator: "=" denotes an equation to MACSYMA. To the pattern matcher in MACSYMA it denotes a total relation that holds between two expressions if and only if the expressions are syntactically identical. - Function: ABS (exp) returns the absolute value exp. If exp is complex, returns the complex modulus of exp. - special symbol: ADDITIVE - If DECLARE(F,ADDITIVE) has been executed, then: (1) If F is univariate, whenever the simplifier encounters F applied to a sum, F will be distributed over that sum. I.e. F(Y+X); will simplify to F(Y)+F(X). (2) If F is a function of 2 or more arguments, additivity is defined as additivity in the first argument to F, as in the case of 'SUM or 'INTEGRATE, i.e. F(H(X)+G(X),X); will simplify to F(H(X),X)+F(G(X),X). This simplification does not occur when F is applied to expressions of the form SUM(X[I],I,lower-limit,upper-limit). - keyword: ALLBUT works with the PART commands (i.e. PART, INPART, SUBSTPART, SUBSTINPART, DPART, and LPART). For example, if EXPR is E+D+C+B+A, then PART(EXPR,[2,5]); ==> D+A while PART(EXPR,ALLBUT(2,5))==>E+C+B It also works with the KILL command, KILL(ALLBUT(name1,...,namek)) will do a KILL(ALL) except it will not KILL the names specified. Note: namei means a name such as function name such as U, F, FOO, or G, not an infolist such as FUNCTIONS. - declaration: ANTISYMMETRIC - If DECLARE(H,ANTISYMMETRIC); is done, this tells the simplifier that H is antisymmetric. E.g. H(X,Z,Y) will simplify to - H(X, Y, Z). That is, it will give (-1)^n times the result given by SYMMETRIC or COMMUTATIVE, where n is the number of interchanges of two arguments necessary to convert it to that form. - Function: CABS (exp) returns the complex absolute value (the complex modulus) of exp. - declaration: COMMUTATIVE - If DECLARE(H,COMMUTATIVE); is done, this tells the simplifier that H is a commutative function. E.g. H(X,Z,Y) will simplify to H(X, Y, Z). This is the same as SYMMETRIC. - Function: ENTIER (X) largest integer <= X where X is numeric. FIX (as in FIXnum) is a synonym for this, so FIX(X); is precisely the same. - Function: EQUAL (expr1,expr2) used with an "IS", returns TRUE (or FALSE) if and only if expr1 and expr2 are equal (or not equal) for all possible values of their variables (as determined by RATSIMP). Thus IS(EQUAL((X+1)**2,X**2+2*X+1)) returns TRUE whereas if X is unbound IS((X+1)**2=X**2+2*X+1) returns FALSE. Note also that IS(RAT(0)=0) gives FALSE but IS(EQUAL(RAT(0),0)) gives TRUE. If a determination can't be made with EQUAL then a simplified but equivalent form is returned whereas = always causes either TRUE or FALSE to be returned. All variables occurring in exp are presumed to be real valued. EV(exp,PRED) is equivalent to IS(exp). (C1) IS(X**2 >= 2*X-1); (D1) TRUE (C2) ASSUME(A>1); (D2) DONE (C3) IS(LOG(LOG(A+1)+1)>0 AND A^2+1>2*A); (D3) TRUE - Function: EVAL causes an extra post-evaluation of exp to occur. - Function: EVENP (exp) is TRUE if exp is an even integer. FALSE is returned in all other cases. - Function: FIX (x) a synonym for ENTIER(X) - largest integer <= X where X is numeric. - Function: FULLMAP (fn, exp1, ...) is similar to MAP but it will keep mapping down all subexpressions until the main operators are no longer the same. The user should be aware that FULLMAP is used by the MACSYMA simplifier for certain matrix manipulations; thus, the user might see an error message concerning FULLMAP even though FULLMAP was not explicitly called by the user. (C1) A+B*C$ (C2) FULLMAP(G,%); (D2) G(B) G(C) + G(A) (C3) MAP(G,D1); (D3) G(B C) + G(A) - Function: FULLMAPL (fn, list1, ...) is similar to FULLMAP but it only maps onto lists and matrices (C1) FULLMAPL("+",[3,[4,5]],[[A,1],[0,-1.5]]); (D1) [[A + 3, 4], [4, 3.5]] - Function: IS (exp) attempts to determine whether exp (which must evaluate to a predicate) is provable from the facts in the current data base. IS returns TRUE if the predicate is true for all values of its variables consistent with the data base and returns FALSE if it is false for all such values. Otherwise, its action depends on the setting of the switch PREDERROR (default: TRUE). IS errs out if the value of PREDERROR is TRUE and returns UNKNOWN if PREDERROR is FALSE. - Function: ISQRT (X) takes one integer argument and returns the "integer SQRT" of its absolute value. - Function: MAX (X1, X2, ...) yields the maximum of its arguments (or returns a simplified form if some of its arguments are non-numeric). - Function: MIN (X1, X2, ...) yields the minimum of its arguments (or returns a simplified form if some of its arguments are non-numeric). - Function: MOD (poly) converts the polynomial poly to a modular representation with respect to the current modulus which is the value of the variable MODULUS. MOD(poly,m) specifies a MODULUS m to be used for converting poly, if it is desired to override the current global value of MODULUS. See DESCRIBE(MODULUS); . - Function: ODDP (exp) is TRUE if exp is an odd integer. FALSE is returned in all other cases. - operator: PRED (EVFLAG) causes predicates (expressions which evaluate to TRUE or FALSE) to be evaluated. - Function: RANDOM (X) returns a random integer between 0 and X-1. If no argument is given then a random integer between -2^(29) and 2^(29) -1 is returned. If X is FALSE then the random sequence is restarted from the beginning. Note that the range of the returned result when no argument is given differs in NIL MACSYMA from that of PDP-10 and Multics MACSYMA, which is -2^(35) to 2^(35) -1. This range is the range of the FIXNUM datatype of the underlying LISP. - Function: SIGN (exp) attempts to determine the sign of its specified expression on the basis of the facts in the current data base. It returns one of the following answers: POS (positive), NEG (negative), ZERO, PZ (positive or zero), NZ (negative or zero), PN (positive or negative), or PNZ (positive, negative, or zero, i.e. nothing known). - Function: SIGNUM (X) if X<0 then -1 else if X>0 then 1 else 0. If X is not numeric then a simplified but equivalent form is returned. For example, SIGNUM(-X) gives -SIGNUM(X). - Function: SORT (list,optional-predicate) sorts the list using a suitable optional-predicate of two arguments (such as "<" or ORDERLESSP). If the optional-predicate is not given, then MACSYMA's built-in ordering predicate is used. - Function: SQRT (X) the square root of X. It is represented internally by X^(1/2). Also see ROOTSCONTRACT. RADEXPAND[TRUE] - if TRUE will cause nth roots of factors of a product which are powers of n to be pulled outside of the radical, e.g. SQRT(16*X^2) will become 4*X only if RADEXPAND is TRUE. - Variable: SQRTDISPFLAG default: [TRUE] - if FALSE causes SQRT to display with exponent 1/2. - Function: SUBLIS (list,expr) allows multiple substitutions into an expression in parallel. Sample syntax: SUBLIS([A=B,B=A],SIN(A)+COS(B)); => SIN(B) + COS(A) The variable SUBLIS_APPLY_LAMBDA[TRUE] controls simplification after SUBLIS. For full documentation, see the file SHARE2;SUBLIS INFO. - Function: SUBLIST (L,F) returns the list of elements of the list L for which the function F returns TRUE. E.g., SUBLIST([1,2,3,4],EVENP); returns [2,4]. - Variable: SUBLIS_APPLY_LAMBDA default:[TRUE] - controls whether LAMBDA's substituted are applied in simplification after SUBLIS is used or whether you have to do an EV to get things to apply. TRUE means do the application. - Function: SUBST (a, b, c) substitutes a for b in c. b must be an atom, or a complete subexpression of c. For example, X+Y+Z is a complete subexpression of 2*(X+Y+Z)/W while X+Y is not. When b does not have these characteristics, one may sometimes use SUBSTPART or RATSUBST (see below). Alternatively, if b is of the form e/f then one could use SUBST(a*f,e,c) while if b is of the form e**(1/f) then one could use SUBST(a**f,e,c). The SUBST command also discerns the X^Y in X^-Y so that SUBST(A,SQRT(X),1/SQRT(X)) yields 1/A. a and b may also be operators of an expression enclosed in "s or they may be function names. If one wishes to substitute for the independent variable in derivative forms then the AT function (see below) should be used. Note: SUBST is an alias for SUBSTITUTE. SUBST(eq1,exp) or SUBST([eq1,...,eqk],exp) are other permissible forms. The eqi are equations indicating substitutions to be made. For each equation, the right side will be substituted for the left in the expression exp. EXPTSUBST[FALSE] if TRUE permits substitutions like Y for %E**X in %E**(A*X) to take place. OPSUBST[TRUE] if FALSE, SUBST will not attempt to substitute into the operator of an expression. E.g. (OPSUBST:FALSE, SUBST(X^2,R,R+R[0])); will work. (C1) SUBST(A,X+Y,X+(X+Y)**2+Y); 2 (D1) Y + X + A (C2) SUBST(-%I,%I,A+B*%I); (D2) A - %I B (Note that C2 is one way of obtaining the complex conjugate of an analytic expression.) For further examples, do EXAMPLE(SUBST); - Function: SUBSTINPART (x, exp, n1, ...) is like SUBSTPART but works on the internal representation of exp. (C1) X.'DIFF(F(X),X,2); 2 d (D1) X . (--- F(X)) 2 dX (C2) SUBSTINPART(D**2,%,2); 2 (D2) X . D (C3) SUBSTINPART(F1,F[1](X+1),0); (D3) F1(X + 1) Additional Information If the last argument to a part function is a list of indices then several subexpressions are picked out, each one corresponding to an index of the list. Thus (C1) PART(X+Y+Z,[1,3]); (D1) Z+X PIECE holds the value of the last expression selected when using the part functions. It is set during the execution of the function and thus may be referred to in the function itself as shown below. If PARTSWITCH[FALSE] is set to TRUE then END is returned when a selected part of an expression doesn't exist, otherwise an error message is given. (C1) 27*Y**3+54*X*Y**2+36*X**2*Y+Y+8*X**3+X+1; 3 2 2 3 (D1) 27 Y + 54 X Y + 36 X Y + Y + 8 X + X + 1 (C2) PART(D1,2,[1,3]); 2 (D2) 54 Y (C3) SQRT(PIECE/54); (D3) Y (C4) SUBSTPART(FACTOR(PIECE),D1,[1,2,3,5]); 3 (D4) (3 Y + 2 X) + Y + X + 1 (C5) 1/X+Y/X-1/Z; 1 Y 1 (D5) - - + - + - Z X X (C6) SUBSTPART(XTHRU(PIECE),%,[2,3]); Y + 1 1 (D6) ----- - - X Z Also, setting the option INFLAG to TRUE and calling PART/SUBSTPART is the same as calling INPART/SUBSTINPART. - Function: SUBSTPART (x, exp, n1, ..., nk) substitutes x for the subexpression picked out by the rest of the arguments as in PART. It returns the new value of exp. x may be some operator to be substituted for an operator of exp. In some cases it needs to be enclosed in "s (e.g. SUBSTPART("+",A*B,0); -> B + A ). (C1) 1/(X**2+2); 1 (D1) ------ 2 X + 2 (C2) SUBSTPART(3/2,%,2,1,2); 1 (D2) -------- 3/2 X + 2 (C3) A*X+F(B,Y); (D3) A X + F(B, Y) (C4) SUBSTPART("+",%,1,0); (D4) X + F(B, Y) + A Also, setting the option INFLAG to TRUE and calling PART/SUBSTPART is the same as calling INPART/SUBSTINPART. - Function: SUBVARP (exp) is TRUE if exp is a subscripted variable, for example A[I]. - Function: SYMBOLP (exp) returns TRUE if "exp" is a "symbol" or "name", else FALSE. I.e., in effect, SYMBOLP(X):=ATOM(X) AND NOT NUMBERP(X)$ . - Function: UNORDER () stops the aliasing created by the last use of the ordering commands ORDERGREAT and ORDERLESS. ORDERGREAT and ORDERLESS may not be used more than one time each without calling UNORDER. Do DESCRIBE(ORDERGREAT); and DESCRIBE(ORDERLESS);, and also do EXAMPLE(UNORDER); for specifics. - Function: VECTORPOTENTIAL (givencurl) Returns the vector potential of a given curl vector, in the current coordinate system. POTENTIALZEROLOC has a similar role as for POTENTIAL, but the order of the left-hand sides of the equations must be a cyclic permutation of the coordinate variables. - Function: XTHRU (exp) combines all terms of exp (which should be a sum) over a common denominator without expanding products and exponentiated sums as RATSIMP does. XTHRU cancels common factors in the numerator and denominator of rational expressions but only if the factors are explicit. Sometimes it is better to use XTHRU before RATSIMPing an expression in order to cause explicit factors of the gcd of the numerator and denominator to be canceled thus simplifying the expression to be RATSIMPed. (C1) ((X+2)**20-2*Y)/(X+Y)**20+(X+Y)**-19-X/(X+Y)**20; 20 1 X (X + 2) - 2 Y (D1) --------- - --------- + --------------- 19 20 20 (Y + X) (Y + X) (Y + X) (C2) XTHRU(%); 20 (X + 2) - Y (D2) ------------- 20 (Y + X) - Function: ZEROEQUIV (exp,var) tests whether the expression exp in the variable var is equivalent to zero. It returns either TRUE, FALSE, or DONTKNOW. For example ZEROEQUIV(SIN(2*X) - 2*SIN(X)*COS(X),X) returns TRUE and ZEROEQUIV(%E^X+X,X) returns FALSE. On the other hand ZEROEQUIV(LOG(A*B) - LOG(A) - LOG(B),A) will return DONTKNOW because of the presence of an extra parameter. The restrictions are: (1) Do not use functions that MACSYMA does not know how to differentiate and evaluate. (2) If the expression has poles on the real line, there may be errors in the result (but this is unlikely to occur). (3) If the expression contains functions which are not solutions to first order differential equations (e.g. Bessel functions) there may be incorrect results. (4) The algorithm uses evaluation at randomly chosen points for carefully selected subexpressions. This is always a somewhat hazardous business, although the algorithm tries to minimize the potential for error. File: maxima.info, Node: Expressions, Next: Simplification, Prev: Operators, Up: Top Expressions *********** * Menu: * Introduction to Expressions:: * ASSIGNMENT:: * COMPLEX:: * INEQUALITY:: * SYNTAX:: * Definitions for Expressions:: File: maxima.info, Node: Introduction to Expressions, Next: ASSIGNMENT, Prev: Expressions, Up: Expressions Introduction to Expressions =========================== There are a number of reserved words which cannot be used as variable names. Their use would cause a possibly cryptic syntax error. INTEGRATE NEXT FROM DIFF IN AT LIMIT SUM FOR AND ELSEIF THEN ELSE DO OR IF UNLESS PRODUCT WHILE THRU STEP Most things in MAXIMA are expressions. A sequence of expressions can be made into an expression by separating them by commas and putting parentheses around them. This is similar to the C comma expression. (C29) x:3$ (C30) joe:(x:x+1,x:x*x); (D30) 16 (C31) joe:(if (x >17) then 2 else 4); (D31) 4 (C32) joe:(if (x >17) then x:2 else joe:4,joe+x); (D32) 20 Even loops in maxima are expressions, although the value they return is the not too useful `DONE' (C33) joe:(x:1,for i from 1 thru 10 do (x:x*i)); (D33) DONE whereas what you really want is probably to include a third term in the comma expression which actually gives back the value. (C34) joe:(x:1,for i from 1 thru 10 do (x:x*i),x); (D34) 3628800 File: maxima.info, Node: ASSIGNMENT, Next: COMPLEX, Prev: Introduction to Expressions, Up: Expressions ASSIGNMENT ========== - There are two assignment operators in MACSYMA, : and :: . E.g. A:3 sets the variable A to 3. :: assigns the value of the expression on its right to the value of the quantity on its left, which must evaluate to an atomic variable or subscripted variable. File: maxima.info, Node: COMPLEX, Next: INEQUALITY, Prev: ASSIGNMENT, Up: Expressions COMPLEX ======= - A complex expression is specified in MACSYMA by adding the real part of the expression to %I times the imaginary part. Thus the roots of the equation X^2-4*X+13=0 are 2+3*%I and 2-3*%I. Note that simplification of products of complex expressions can be effected by expanding the product. Simplification of quotients, roots, and other functions of complex expressions can usually be accomplished by using the REALPART, IMAGPART, RECTFORM, POLARFORM, ABS, CARG functions. File: maxima.info, Node: INEQUALITY, Next: SYNTAX, Prev: COMPLEX, Up: Expressions INEQUALITY ========== - MACSYMA has the usual inequality operators: less than: < greater than: > greater than or equal to: >= less than or equal to: <= File: maxima.info, Node: SYNTAX, Next: Definitions for Expressions, Prev: INEQUALITY, Up: Expressions SYNTAX ====== - It is possible to add new operators to MACSYMA (infix, prefix, postfix, unary, or matchfix with given precedences), to remove existing operators, or to redefine the precedence of existing operators. While MACSYMA's syntax should be adequate for most ordinary applications, it is possible to define new operators or eliminate predefined ones that get in the user's way. The extension mechanism is rather straightforward and should be evident from the examples below. (C1) PREFIX("DDX")$ (C2) DDX Y$ /* means "DDX"(Y) */ (C3) INFIX("<-")$ (C4) A<-DDX Y$ /* means "<-"(A,"DDX"(Y)) */ For each of the types of operator except SPECIAL, there is a corresponding creation function that will give the lexeme specified the corresponding parsing properties. Thus "PREFIX("DDX")" will make "DDX" a prefix operator just like "-" or "NOT". Of course, certain extension functions require additional information such as the matching keyword for a matchfix operator. In addition, binding powers and parts of speech must be specified for all keywords defined. This is done by passing additional arguments to the extension functions. If a user does not specify these additional parameters, MACSYMA will assign default values. The six extension functions with binding powers and parts of speech defaults (enclosed in brackets) are summarized below. PREFIX(operator, rbp[180], rpos[ANY], pos[ANY]) POSTFIX(operator, lbp[180], lpos[ANY], pos[ANY]) INFIX(operator, lbp[180], rbp[180], lpos[ANY], rpos[ANY],pos[ANY]) NARY(operator, bp[180], argpos[ANY], pos[ANY]) NOFIX(operator, pos[ANY]) MATCHFIX(operator, match, argpos[ANY], pos[ANY]) The defaults have been provided so that a user who does not wish to concern himself with parts of speech or binding powers may simply omit those arguments to the extension functions. Thus the following are all equivalent. PREFIX("DDX",180,ANY,ANY)$ PREFIX("DDX",180)$ PREFIX("DDX")$ It is also possible to remove the syntax properties of an operator by using the functions REMOVE or KILL. Specifically, "REMOVE("DDX",OP)" or "KILL("DDX")" will return "DDX" to operand status; but in the second case all the other properties of "DDX" will also be removed. (C20) PREFIX("DDX",180,ANY,ANY)$ (C21) DDXYZ; (D21) DDX YZ (C26) "ddx"(u):=u+4; (D26) DDX u := u + 4 (C27) ddx 8; (D27) 12 File: maxima.info, Node: Definitions for Expressions, Prev: SYNTAX, Up: Expressions Definitions for Expressions =========================== - Function: AT (exp, list) will evaluate exp (which may be any expression) with the variables assuming the values as specified for them in the list of equations or the single equation similar to that given to the ATVALUE function. If a subexpression depends on any of the variables in list but it hasn't had an atvalue specified and it can't be evaluated then a noun form of the AT will be returned which will display in a two-dimensional form. Do EXAMPLE(AT); for an example. - Function: BOX (expr) returns expr enclosed in a box. The box is actually part of the expression. BOX(expr,label) encloses expr in a labelled box. label is a name which will be truncated in display if it is too long. BOXCHAR["] - is the character used to draw the box in this and in the DPART and LPART functions. - Variable: BOXCHAR default: ["] is the character used to draw the box in the BOX and in the DPART and LPART functions. - special operator: CONSTANT - makes ai a constant as is %PI. - Function: CONSTANTP (exp) is TRUE if exp is a constant (i.e. composed of numbers and %PI, %E, %I or any variables bound to a constant or DECLAREd constant) else FALSE. Any function whose arguments are constant is also considered to be a constant. - Function: CONTRACT (exp) carries out all possible contractions in exp, which may be any well-formed combination of sums and products. This function uses the information given to the DEFCON function. Since all tensors are considered to be symmetric in all indices, the indices are sorted into alphabetical order. Also all dummy indices are renamed using the symbols !1,!2,... to permit the expression to be simplified as much as possible by reducing equivalent terms to a canonical form. For best results exp should be fully expanded. RATEXPAND is the fastest way to expand products and powers of sums if there are no variables in the denominators of the terms. The GCD switch should be FALSE if gcd cancellations are unnecessary. - Function: DECLARE (a1, f1, a2, f2, ...) gives the atom ai the flag fi. The ai's and fi's may also be lists of atoms and flags respectively in which case each of the atoms gets all of the properties. The possible flags and their meanings are: CONSTANT - makes ai a constant as is %PI. MAINVAR - makes ai a MAINVAR. The ordering scale for atoms: numbers < constants (e.g. %E,%PI) < scalars < other variables < mainvars. SCALAR - makes ai a scalar. NONSCALAR - makes ai behave as does a list or matrix with respect to the dot operator. NOUN - makes the function ai a noun so that it won't be evaluated automatically. EVFUN - makes ai known to the EV function so that it will get applied if its name is mentioned. Initial evfuns are FACTOR, TRIGEXPAND, TRIGREDUCE, BFLOAT, RATSIMP, RATEXPAND, and RADCAN EVFLAG - makes ai known to the EV function so that it will be bound to TRUE during the execution of EV if it is mentioned. Initial evflags are FLOAT, PRED, SIMP, NUMER, DETOUT, EXPONENTIALIZE, DEMOIVRE, KEEPFLOAT, LISTARITH, TRIGEXPAND, SIMPSUM, ALGEBRAIC, RATALGDENOM, FACTORFLAG, %EMODE, LOGARC, LOGNUMER, RADEXPAND, RATSIMPEXPONS, RATMX, RATFAC, INFEVAL, %ENUMER, PROGRAMMODE, LOGNEGINT, LOGABS, LETRAT, HALFANGLES, EXPTISOLATE, ISOLATE_WRT_TIMES, SUMEXPAND, CAUCHYSUM, NUMER_PBRANCH, M1PBRANCH, DOTSCRULES, and LOGEXPAND BINDTEST - causes ai to signal an error if it ever is used in a computation unbound. DECLARE([var1, var2, ...], BINDTEST) causes MACSYMA to give an error message whenever any of the vari occur unbound in a computation. MACSYMA currently recognizes and uses the following features of objects: EVEN, ODD, INTEGER, RATIONAL, IRRATIONAL, REAL, IMAGINARY, and COMPLEX he useful features of functions include: INCREASING, DECREASING, ODDFUN (odd function), EVENFUN (even function), COMMUTATIVE (or SYMMETRIC), ANTISYMMETRIC, LASSOCIATIVE and RASSOCIATIVE DECLARE(F,INCREASING) is in all respects equivalent to ASSUME(KIND(F,INCREASING)) The ai and fi may also be lists of objects or features. The command FEATUREP(object,feature) may be used to determine if an object has been DECLAREd to have "feature". See DESCRIBE(FEATURES); . - Function: DISOLATE (exp, var1, var2, ..., varN) is similar to ISOLATE(exp, var) (Do DESCRIBE(ISOLATE);) except that it enables the user to isolate more than one variable simultaneously. This might be useful, for example, if one were attempting to change variables in a multiple integration, and that variable change involved two or more of the integration variables. This function is autoloaded from `simplification/disol.mac'. A demo is available by `demo("disol")$'. - Function: DISPFORM (exp) returns the external representation of exp (wrt its main operator). This should be useful in conjunction with PART which also deals with the external representation. Suppose EXP is -A . Then the internal representation of EXP is "*"(-1,A), while the external representation is "-"(A). DISPFORM(exp,ALL) converts the entire expression (not just the top-level) to external format. For example, if EXP:SIN(SQRT(X)), then FREEOF(SQRT,EXP) and FREEOF(SQRT,DISPFORM(EXP)) give TRUE, while FREEOF(SQRT,DISPFORM(EXP,ALL)) gives FALSE. - Function: DISTRIB (exp) distributes sums over products. It differs from EXPAND in that it works at only the top level of an expression, i.e. it doesn't recurse and it is faster than EXPAND. It differs from MULTTHRU in that it expands all sums at that level. For example, DISTRIB((A+B)*(C+D)) -> A C + A D + B C + B D MULTTHRU ((A+B)*(C+D)) -> (A + B) C + (A + B) D DISTRIB (1/((A+B)*(C+D))) -> 1/ ((A+B) *(C+D)) EXPAND(1/((A+B)*(C+D)),1,0) -> 1/(A C + A D + B C + B D) - Function: DPART (exp, n1, ..., nk) selects the same subexpression as PART, but instead of just returning that subexpression as its value, it returns the whole expression with the selected subexpression displayed inside a box. The box is actually part of the expression. (C1) DPART(X+Y/Z**2,1,2,1); Y (D1) ---- + X 2 ***** * Z * ***** - Function: EXP (X) the exponential function. It is represented internally as %E^X. DEMOIVRE[FALSE] - if TRUE will cause %E^(A+B*%I) to become %E^A*(COS(B)+%I*SIN(B)) if B is free of %I. A and B are not expanded. %EMODE[TRUE] - when TRUE %E^(%PI*%I*X) will be simplified as follows: it will become COS(%PI*X)+%I*SIN(%PI*X) if X is an integer or a multiple of 1/2, 1/3, 1/4, or 1/6 and thus will simplify further. For other numerical X it will become %E^(%PI*%I*Y) where Y is X-2*k for some integer k such that ABS(Y)<1. If %EMODE is FALSE no simplification of %E^(%PI*%I*X) will take place. %ENUMER[FALSE] - when TRUE will cause %E to be converted into 2.718... whenever NUMER is TRUE. The default is that this conversion will take place only if the exponent in %E^X evaluates to a number. - Variable: EXPTISOLATE default: [FALSE] if TRUE will cause ISOLATE(expr,var); to examine exponents of atoms (like %E) which contain var. - Variable: EXPTSUBST default: [FALSE] if TRUE permits substitutions such as Y for %E**X in %E**(A*X) to take place. - Function: FREEOF (x1, x2, ..., exp) yields TRUE if the xi do not occur in exp and FALSE otherwise. The xi are atoms or they may be subscripted names, functions (e.g. SIN(X) ), or operators enclosed in "s. If 'var' is a "dummy variable" of 'exp', then FREEOF(var,exp); will return TRUE. "Dummy variables" are mathematical things like the index of a sum or product, the limit variable, and the definite integration variable. Example: FREEOF(I,'SUM(F(I),I,0,N)); returns TRUE. Do EXAMPLE(FREEOF); for more examples. - Function: GENFACT (X, Y, Z) is the generalized factorial of X which is: X*(X-Z)*(X-2*Z)*...*(X-(Y-1)*Z). Thus, for integral X, GENFACT(X,X,1)=X! and GENFACT(X,X/2,2)=X!! - Function: IMAGPART (exp) returns the imaginary part of the expression exp. - Function: INDICES (exp) returns a list of two elements. The first is a list of the free indices in exp (those that occur only once); the second is the list of dummy indices in exp (those that occur exactly twice). - Function: INFIX (op) - INFIX operators are used to denote functions of two arguments, one given before the operator and one after, e.g. A^2 . The INFIX("x") function is a syntax extention function to declare x to be an INFIX operator. Do DESCRIBE(SYNTAX); for more details. - Variable: INFLAG default: [FALSE] if set to TRUE, the functions for part extraction will look at the internal form of exp. Note that the simplifier re-orders expressions. Thus FIRST(X+Y) will be X if INFLAG is TRUE and Y if INFLAG is FALSE. (FIRST(Y+X) gives the same results). Also, setting INFLAG to TRUE and calling PART/SUBSTPART is the same as calling INPART/SUBSTINPART. Functions affected by the setting of INFLAG are: PART, SUBSTPART, FIRST, REST, LAST, LENGTH, the FOR ... IN construct, MAP, FULLMAP, MAPLIST, REVEAL and PICKAPART. - Function: INPART (exp, n1, ..., nk) is similar to PART but works on the internal representation of the expression rather than the displayed form and thus may be faster since no formatting is done. Care should be taken with respect to the order of subexpressions in sums and products (since the order of variables in the internal form is often different from that in the displayed form) and in dealing with unary minus, subtraction, and division (since these operators are removed from the expression). PART(X+Y,0) or INPART(X+Y,0) yield +, though in order to refer to the operator it must be enclosed in "s. For example ...IF INPART(D9,0)="+" THEN ... (C1) X+Y+W*Z; (D1) W Z + Y + X (C2) INPART(D1,3,2); (D2) Z (C3) PART(D1,1,2); (D3) Z (C4) 'LIMIT(F(X)**G(X+1),X,0,MINUS); G(X + 1) (D4) LIMIT F(X) X ->0- (C5) INPART(%,1,2); (D5) G(X + 1) - Function: ISOLATE (exp, var) returns exp with subexpressions which are sums and which do not contain var replaced by intermediate expression labels (these being atomic symbols like E1, E2, ...). This is often useful to avoid unnecessary expansion of subexpressions which don't contain the variable of interest. Since the intermediate labels are bound to the subexpressions they can all be substituted back by evaluating the expression in which they occur. EXPTISOLATE[FALSE] if TRUE will cause ISOLATE to examine exponents of atoms (like %E) which contain var. ISOLATE_WRT_TIMES[FALSE] if TRUE, then ISOLATE will also isolate wrt products. E.g. compare both settings of the switch on ISOLATE(EXPAND((A+B+C)^2),C); . Do EXAMPLE(ISOLATE); for examples. - Variable: ISOLATE_WRT_TIMES default: [FALSE] - if set to TRUE, then ISOLATE will also isolate wrt products. E.g. compare both settings of the switch on ISOLATE(EXPAND((A+B+C)^2),C); . - Variable: LISTCONSTVARS default: [FALSE] - if TRUE will cause LISTOFVARS to include %E, %PI, %I, and any variables declared constant in the list it returns if they appear in the expression LISTOFVARS is called on. The default is to omit these. - Variable: LISTDUMMYVARS default: [TRUE] - if FALSE, "dummy variables" in the expression will not be included in the list returned by LISTOFVARS. (The meaning of "dummy variables" is as given in DESCRIBE(FREEOF): "Dummy variables" are mathematical things like the index of a sum or product, the limit variable, and the definite integration variable.) Example: LISTOFVARS('SUM(F(I),I,0,N)); gives [I,N] if LISTDUMMYVARS is TRUE, and [N] if LISTDUMMYVARS is FALSE. - Function: LISTOFVARS (exp) yields a list of the variables in exp. LISTCONSTVARS[FALSE] if TRUE will cause LISTOFVARS to include %E, %PI, %I, and any variables declared constant in the list it returns if they appear in exp. The default is to omit these. (C1) LISTOFVARS(F(X[1]+Y)/G**(2+A)); (D1) [X[1], Y, A, G] - Function: LFREEOF (list, exp) For each member m of list, calls FREEOF(m,exp). It returns false if any call to FREEOF does and true otherwise. - Function: LOPOW (exp, v) the lowest exponent of v which explicitly appears in exp. Thus LOPOW((X+Y)**2+(X+Y)**A,X+Y) ==> MIN(A,2) . - Function: LPART (label, expr, n1, ..., nk) is similar to DPART but uses a labelled box. A labelled box is similar to the one produced by DPART but it has a name in the top line. - Function: MULTTHRU (exp) multiplies a factor (which should be a sum) of exp by the other factors of exp. That is exp is f1*f2*...*fn where at least one factor, say fi, is a sum of terms. Each term in that sum is multiplied by the other factors in the product. (Namely all the factors except fi). MULTTHRU does not expand exponentiated sums. This function is the fastest way to distribute products (commutative or noncommutative) over sums. Since quotients are represented as products MULTTHRU can be used to divide sums by products as well. MULTTHRU(exp1, exp2) multiplies each term in exp2 (which should be a sum or an equation) by exp1. If exp1 is not itself a sum then this form is equivalent to MULTTHRU(exp1*exp2). (C1) X/(X-Y)**2-1/(X-Y)-F(X)/(X-Y)**3; 1 X F(X) (D1) - ----- + -------- - -------- X - Y 2 3 (X - Y) (X - Y) (C2) MULTTHRU((X-Y)**3,%); 2 (D2) - (X - Y) + X (X - Y) - F(X) (C3) RATEXPAND(D2); 2 (D3) - Y + X Y - F(X) (C4) ((A+B)**10*S**2+2*A*B*S+(A*B)**2)/(A*B*S**2); 10 2 2 2 (B + A ) S + 2 A B S + A B (D4) -------------------------------- 2 A B S (C5) MULTTHRU(%); 10 2 A B (B + A) (D5) - + --- + ------- S 2 A B S (notice that (B+A)**10 is not expanded) (C6) MULTTHRU(A.(B+C.(D+E)+F)); (D6) A . F + A . (C . (E + D)) + A . B (compare with similar example under EXPAND) - Function: NOUNIFY (f) returns the noun form of the function name f. This is needed if one wishes to refer to the name of a verb function as if it were a noun. Note that some verb functions will return their noun forms if they can't be evaluated for certain arguments. This is also the form returned if a function call is preceded by a quote. - Function: NTERMS (exp) gives the number of terms that exp would have if it were fully expanded out and no cancellations or combination of terms occurred. Note that expressions like SIN(E), SQRT(E), EXP(E), etc. count as just one term regardless of how many terms E has (if it is a sum). - Function: OP (exp) Returns the operator of the expression, and functions the same way as PART(exp,0). It observes the setting of the INPART flag. It may not return the same value as the commercial Macsyma. - Function: OPERATORP (exp, ool) Uses OP to get the operator of the expression and either compares it to ool, if it is a operator, or checks if it is a member of ool if it is a list. - Function: OPTIMIZE (exp) returns an expression that produces the same value and side effects as exp but does so more efficiently by avoiding the recomputation of common subexpressions. OPTIMIZE also has the side effect of "collapsing" its argument so that all common subexpressions are shared. Do EXAMPLE(OPTIMIZE); for examples. - Variable: OPTIMPREFIX default: [%] - The prefix used for generated symbols by the OPTIMIZE command. - Function: ORDERGREAT (V1, ..., Vn) sets up aliases for the variables V1, ..., Vn such that V1 > V2 > ... > Vn > any other variable not mentioned as an argument. See also ORDERLESS. Caveat: do EXAMPLE(ORDERGREAT); for some specifics. - Function: ORDERGREATP (exp1,exp2) returns TRUE if exp2 precedes exp1 in the ordering set up with the ORDERGREAT function (see DESCRIBE(ORDERGREAT);). - Function: ORDERLESS (V1, ..., Vn) sets up aliases for the variables V1, ..., Vn such that V1 < V2 < ... < Vn < any other variable not mentioned as an argument. Thus the complete ordering scale is: numerical constants < declared constants < declared scalars < first argument to ORDERLESS < ... < last argument to ORDERLESS < variables which begin with A < ... < variables which begin with Z < last argument to ORDERGREAT < ... < first argument to ORDERGREAT < declared MAINVARs. Caveat: do EXAMPLE(ORDERLESS); for some specifics. For another ordering scheme, see DESCRIBE(MAINVAR);. - Function: ORDERLESSP (exp1,exp2) returns TRUE if exp1 precedes exp2 in the ordering set up by the ORDERLESS command (see DESCRIBE(ORDERLESS);). - Function: PART (exp, n1, ..., nk) deals with the displayed form of exp. It obtains the part of exp as specified by the indices n1,...,nk. First part n1 of exp is obtained, then part n2 of that, etc. The result is part nk of ... part n2 of part n1 of exp. Thus PART(Z+2*Y,2,1) yields 2. PART can be used to obtain an element of a list, a row of a matrix, etc. If the last argument to a Part function is a list of indices then several subexpressions are picked out, each one corresponding to an index of the list. Thus PART(X+Y+Z,[1,3]) is Z+X. PIECE holds the last expression selected when using the Part functions. It is set during the execution of the function and thus may be referred to in the function itself as shown below. If PARTSWITCH[FALSE] is set to TRUE then END is returned when a selected part of an expression doesn't exist, otherwise an error message is given. For examples, do EXAMPLE(PART); - Function: PARTITION (exp, var) returns a list of two expressions. They are (1) the factors of exp (if it is a product), the terms of exp (if it is a sum), or the list (if it is a list) which don't contain var and, (2) the factors, terms, or list which do. (C1) PARTITION(2*A*X*F(X),X); (D1) [ 2 A , X F(X) ] (C2) PARTITION(A+B,X); (D2) [ A + B , 0 ] (C3) PARTITION([A,B,F(A),C],A); (D3) [[B,C],[A,F(A)]] - Variable: PARTSWITCH default: [FALSE] - if set to TRUE then END is returned when a selected part of an expression doesn't exist, otherwise an error message is given. - Function: PICKAPART (exp,depth) will assign E labels to all subexpressions of exp down to the specified integer depth. This is useful for dealing with large expressions and for automatically assigning parts of an expression to a variable without having to use the part functions. (C1) EXP:(A+B)/2+SIN(X^2)/3-LOG(1+SQRT(X+1)); 2 SIN(X ) B + A (D1) - LOG(SQRT(X + 1) + 1) + ------- + ----- 3 2 (C2) PICKAPART(%,1); (E2) - LOG(SQRT(X + 1) + 1) 2 SIN(X ) (E3) ------- 3 B + A (E4) ----- 2 (D4) E4 + E3 + E2 - Variable: PIECE - holds the last expression selected when using the Part functions. It is set during the execution of the function and thus may be referred to in the function itself. - Function: POWERS (expr, var) gives the powers of var occuring in expr. To use it, do LOAD(POWERS);. For details on usage, do PRINTFILE("powers.usg");. - Function: PRODUCT (exp, ind, lo, hi) gives the product of the values of exp as the index ind varies from lo to hi. The evaluation is similar to that of SUM. No simplification of products is available at this time. If hi is one less than lo, we have an "empty product" and PRODUCT returns 1 rather than erring out. Also see DESCRIBE(PRODHACK). (C1) PRODUCT(X+I*(I+1)/2,I,1,4); (D1) (X + 1) (X + 3) (X + 6) (X + 10) - Function: REALPART (exp) gives the real part of exp. REALPART and IMAGPART will work on expressions involving trigonometic and hyperbolic functions, as well as SQRT, LOG, and exponentiation. - Function: RECTFORM (exp) returns an expression of the form A + B*%I, where A and B are purely real. - Function: REMBOX (expr, arg) removes boxes from expr according to arg. If arg is UNLABELED then all unlabelled boxes are removed. If arg is the name of some label then only boxes with that label are removed. If arg is omitted then all boxes labelled and unlabelled are removed. - Function: SUM (exp, ind, lo, hi) performs a summation of the values of exp as the index ind varies from lo to hi. If the upper and lower limits differ by an integer then each term in the sum is evaluated and added together. Otherwise, if the SIMPSUM [FALSE] is TRUE the result is simplified. This simplification may sometimes be able to produce a closed form. If SIMPSUM is FALSE or if 'SUM is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics. If hi is one less than lo, we have an "empty sum" and SUM returns 0 rather than erring out. Sums may be differentiated, added, subtracted, or multiplied with some automatic simplification being performed. Also see DESCRIBE(SUMHACK). CAUCHYSUM[FALSE] when TRUE causes the Cauchy product to be used when multiplying sums together rather than the usual product. In the Cauchy product the index of the inner summation is a function of the index of the outer one rather than varying independently. GENINDEX[I] is the alphabetic prefix used to generate the next variable of summation. GENSUMNUM[0] is the numeric suffix used to generate the next variable of summation. If it is set to FALSE then the index will consist only of GENINDEX with no numeric suffix. Do EXAMPLE(SUM); for examples. See also SUMCONTRACT, INTOSUM, BASHINDICES, and NICEINDICES. - Function: LSUM (exp, ind, list) performs the sum of EXP for each element IND of the LIST. (C10) lsum(x^i,i,[1,2,7]); 7 2 (D10) x + x + x If the last element LIST argument does not evaluate, or does not evaluate to a Maxima list then the answer is left in noun form (C13) lsum(i^2,i,rootsof(x^3-1)); ==== \ 2 (D13) > i / ==== 3 i in ROOTSOF(x - 1) - special symbol: VERB - the opposite of "noun", i.e. a function form which "does something" ("action" - for most functions the usual case). E.g. INTEGRATE integrates a function, unless it is DECLAREd to be a "noun", in which case it represents the INTEGRAL of the function. See NOUN, NOUNIFY, and VERBIFY. - Function: VERBIFY (f) returns the function name f in its verb form (See also VERB, NOUN, and NOUNIFY).