Definitions for Expressions

Maxima Manual. Node: Definitions for Expressions

6.6: 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. To access this function, do LOAD(DISOL)$ . 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: 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: 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.

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).

SYNTAX

Expressions