1、factor - Factor a Multivariate PolynomialCalling Sequencesfactor(a)factor(a,K)Parametersa - an expressionK - a field extension over which to factorDescription The function factor computes the factorization of a multivariate polynomial with integer, rational, (complex) numeric, or algebraic number co
2、efficients. The factor function does NOT factor integers. Nor does it factor integer coefficients in a polynomial. Use the ifactor function to factor integers. If the second argument K is not given, the polynomial is factored over the field implied by the coefficients. For example, if the coefficien
3、ts are all integers then factor computes all irreducible factors with integer coefficients. Thus factor does not necessarily factor into linear factors. Note that any integer content (see first example below) is not factored. If the input, a, is a rational expression, then it is first normalized (se
4、e normal) and the numerator and denominator of the resulting expression are then factored. This provides a fully-factored form which can be used to simplify an expression in the same way the normal function is used. However, it is more expensive to compute. If the input, a, is a list, set, equation,
5、 range, series, relation, or function, then factor is applied recursively to the components of a. If the second argument K is the keyword real or complex, a floating point factorization is performed over the reals and complexes respectively. At present this is only implemented for univariate polynom
6、ials. If the second argument K is a single RootOf, a list or set of RootOfs, a single radical, or a list or set of radicals, then the expression is factored over the algebraic number field defined by K. Examples factor(6*x2+18*x-24); 6()x41 factor(6); ifactor(6); () 23 factor(x3-y3)/(x4-y4);xy facto
7、r(x3+5); 35 factor(x3+5, 5(1/3);()x2()/1(/2)x(/13 factor(x3+5, 5(1/3),(-3)(1/2);14(25)/13-)/3(/-(/)x5(/13 factor(x3+5.0);x.7094(x21.709542.9074 factor(x3+5,complex);()x1.70954( )x.89731.48026I)831026I factor(y4-2,sqrt(2); y2 alias(alpha = RootOf(x2-2):factor(y4-2,alpha); ()2 factor(x3+y3); xy)y fact
8、or(x3+y3,(-3)(1/2);14()2-3(-3()xcombine - combine terms into a single termCalling Sequencecombine(f)combine(f, n)combine(f, n, opt1, opt2, .)Parametersf - any expression, or set or list of expressions n - a name or a list or set of names opt1, opt2, . - (optional) names of optionsDescription The com
9、bine function applies transformations which combine terms in sums, products, and powers into a single term. This function is applied recursively to the components of lists, sets, and relations; that is, f and n may be lists (or sets) of expressions and names, respectively. For many functions, the tr
10、ansformations applied by combine are the inverse of the transformations that are applied by expand. For example, consider the well-known identity sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b) The expand function applies the identity from left to right whereas the combine function does the reverse. Subexp
11、ressions involving Int, Sum, and Limit are combined into one expression where possible using linearity; that is, c1*f(a,range) + c2*f(b,range) = f(c1*a+c2*b,range). A specific set of transformations is obtained by specifying a second (optional) argument n (a name) which is one or a list of the follo
12、wing: abs arctan conjugate exp icombine ln piecewise polylog power product Psi radical range signum trig For additional information and examples about the transformations applied by each of these, see combinen. The combine function maps over functions. Therefore combine(f(a) is the same as map(combi
13、ne,f(a). Examples combine(Int(x,x=ab)-Int(x2,x=ab);dabx2 combine(Limit(x,x=a)*Limit(x2,x=a)+c);limx3c combine(4*sin(x)3,trig);()sn()six combine(exp(x)2*exp(y),exp);e2y combine(exp(sin(a)*cos(b)*exp(cos(a)*sin(b),trig,exp);()sinab assume(y0,z0);combine(2*ln(y)-ln(z),ln); ly2z combine(xa)2,power); x()
14、a combine(Psi(-x)+Psi(x),Psi);2cot1x combine(2*sin(x)*cos(x),2*cos(x)2-1,trig);,()sinxs2 combine(exp(sin(a)*cos(b)*exp(cos(a)*sin(b),trig,exp);eiabsimplify - Apply simplification rules to an expressionCalling Sequencesimplify(expr)simplify(expr, n1, n2, .)simplify(expr, assume=prop)simplify(expr, sy
15、mbolic)Parametersexpr - any expressionn1, n2,. - (optional) names or sets or listsprop - any property (optional)Description The simplify function is used to apply simplification rules to an expression. If only one argument is present, then simplify will search the expression for function calls, squa
16、re roots, radicals, and powers. Next it will invoke the appropriate simplification procedures, which include:BesselI, BesselJ, BesselK, BesselY, D, Ei, GAMMA, RootOf, LambertW, dilog, exp, ln, sqrt, polylog, pochhammer, trig (for trig functions), hypergeom (for hypergeometrics), radical (occurrence
17、of fractional powers), power (occurrence of powers), exp, ln), and (for operators). In the case of two or more arguments where the additional arguments are names, simplify will only invoke the simplification procedures specified by the additional arguments. Further information on the simplification
18、procedures supported is available in the help pages simplifyname where name is one of: Ei, GAMMA, RootOf, , hypergeom, ln, polar, power, radical, sqrt, trig The symbolic option indicates that formal symbolic manipulation of expressions is allowed without regard to the analytical issue of branches fo
19、r multi-valued functions. For example, the expression sqrt(x2) will simplify to x under the symbolic option whereas without this option the simplified result must take into account the different possible values of the (complex) sign of x. Note that when the symbolic option is specified, any branch o
20、f a multi-valued function can be chosen during the simplification process. The result of such an operation is in general not valid over the whole complex plane, and in fact it can lead to incorrect results in the common situation where the expressions are understood to represent analytical functions
21、. A user can make his or her own simplifications known to the simplify function by defining a Maple procedure. If the procedure simplify/f is defined then the function call simplify(a,f) will invoke simplify/f(a). The case of two or more arguments where the additional arguments are sets or lists is
22、used for simplification with respect to side relations. See the subtopic simplifysiderels for details. Whenever the last argument is assume=prop, all the indeterminates in expr are assumed to have the property prop to compute the simplified expression. Examples simplify(4(1/2)+3); 5 simplify(xa)b+4(
23、1/2), power);()xab2 simplify(exp(a+ln(b*exp(c);ec simplify(sin(x)2+cos(x)2, trig);1 e := cos(x)5 + sin(x)4 + 2*cos(x)2 - 2*sin(x)2 - cos(2*x):simplify(e); ()cosx5()4 f := -1/3*x5*y + x4*y2 + 1/3*x*y3 + 1:simplify(f, x3 = x*y, y2 = x+1);1y423y g:=sqrt(x2); :gx simplify(g); ()csn simplify(g,assume=rea
24、l); simplify(g,assume=positive); x simplify(g,symbolic);series - generalized series expansionCalling Sequenceseries(expr, eqn)series(expr, eqn, n)Parametersexpr - an expression eqn - an equation (such as x = a) or name (such as x) n - (optional) a non-negative integer Description The series function
25、 computes a truncated series expansion of expr, with respect to the variable x, about the point a, up to order n. If a is infinity then an asymptotic expansion is given. If eqn evaluates to a name x then the equation x = 0 is assumed. If the third argument n is present then it specifies the truncati
26、on order of the series calculations. This does not mean the truncation order of the actual series. See Order for more information about this. If n is not present, the truncation order is determined by the global variable Order. The user may assign any non-negative integer to Order. The default value
27、 of Order is 6. See Order for more information. If the series is not exact then an order term (for example O(x6) ) is the last term in the series. It is possible to invoke series on user-defined functions. For example, if the procedure series/f is defined then the function call series(f(x,y),x) will
28、 invoke series/f(x,y,x) to compute the series. Note that this user-defined function series/f must return a series data structure, not just a polynomial (see typeseries). If series is applied to an unevaluated integral then the series expansion of the integral will be computed (if possible). The resu
29、lt of the series function is a generalized series expansion. This could be a Taylor series or a Laurent series or a more general series. Formally, the coefficients in a generalized series are such that k1*(x-a)eps 0 as x approaches a. In other words, the coefficients may depend on x but their growth
30、 must be less than the polynomial in x. O(1) represents such a coefficient, rather than an arbitrary constant. Usually, the result of the series function is represented in the form of Maples series data structure. For an explanation of the data structure, see the help page for typeseries. However, t
31、he result of the series function will be represented in ordinary sum-of-products form, rather than in Maples series data structure, if it is a generalized series requiring fractional exponents. Examples series(x/(1-x-x2), x=0);x23x45()Ox6 convert(%,polynom); series(x+1/x, x=1, 3 );()12()3 series(exp
32、(x), x=0, 8 );1x264x057x61047()x8 series(exp(x)/x, x=0, 8 );x-126x14320x417504x6()O7 series(GAMMA(x), x=0, 2 );-()2 series(x3/(x4+4*x-5),x=infinity);1x45O1x7 int(exp(x3), x );13()-/2 ()-/13x()/-()/3,x3/1 series(%, x=0); 4()O7 series(xx, x=0, 3);1()lnx2lx23 s := series(sqrt(sin(x), x=0, 4);:=s()/5)(/9 type(s, series); false whattype(s); +
