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微分、積分、テーラ級数、ラプラス変換、常微分方程式、多項式、そして集合、配列、ベクトル、行列、テンソルなどを含む記号的および数値的な数式を操作するシステムであるMaximaをオンラインで提供する。

Maxima 5.11.0 with MaximaPHP 0.1.3

亡き William Schelterの霊にささげる。
グラフのプロットは Gnuplot 4.2.0 のおかげで可能になった。

Text | MathML | TeX | Maxima | 例題

例題

最後の出力と入力はそれぞれ %lo および %li 変数で参照できる。
そして、 %lx 変数はMaximaPHPが内部的に使用する。

これらの例は Maxima コマンドの example(...) を使って作られた。

evaluationEvaluation in Maxima
diff(x*f(x),x);
f(x):=sin(x);
ev(%th(2),diff)
x;
x:3;
x;
'x
f(x):=x^2;
'f(2);
ev(%,f);
'(f(2));
f(2);
sum(i!,i,1,4);
'sum(i!,i,1,4);
remvalue(x);
'integrate(f(x),x,a,b)
for i thru 5 do s:i^2+s;
s;
ev(%,s:0);
ev(%th(2))
'sum(g(i),i,0,n);
z*%e^z;
ev(%,z:x^2);
subst(x^2,z,%th(3));
a:%;
1+a;
kill(a,y);
a
integrate(y^2,y)
f(y):=diff(y*log(y),y,2)
f(y):=1/y
(y+x)^3;
diff(%,x)
equationsEquations in Maxima
1+x = y^2;
x-1 = 1+2*y;
%+%th(2);
%th(3)/y;
1/%
complexComplex number in Maxima
(sqrt(2.25)+sqrt(-4))^2;
expand(%)
expand(sqrt(2*%i))
arraysArrays in Maxima
a[n]:=n*a[n-1];
a[0]:1;
a[5];
a[n]:=n;
a[6];
a[4]
functionsFunctions in Maxima
f(x):=y+x^2;
f(2);
ev(f(2),y:7)
f(x):=1+sin(x)^2;
f(1+x);
g(y,z):=3*y+f(z);
ev(g(z+2*y,-0.5),y:7);
functions
h(n):=sum(i*x^i,i,0,n)
t[n](x):=ratexpand(2*x*t[n-1](x)-t[n-2](x));
t[0](x):=1;
t[1](x):=x;
t[4](y)
g[n](x):=sum(ev(x),i,n,2+n);
h(n,x):=sum(ev(x),i,n,2+n);
g[2](i^2);
h(2,i^2)
p[n](x):=ratsimp(diff((x^2-1)^n,x,n)/(2^n*n!));
q(n,x):=ratsimp(diff((x^2-1)^n,x,n)/(2^n*n!));
p[2];
p[2](1+y);
q(2,y);
p[2](5)
f[i,j](x,y):=y^j+x^i;
g(fun,a,b):=print(fun,"
applied to ",a,"
and ",b,"
is ",fun(a,b));
g(f[2,1],sin(%pi),2*c)
additiveDeclares additivity of a function.
declare(f,additive);
f(3*b+2*a)
algsysSolves the simultaneous polynomials or polynomial equations for some variables.
f1:2*x*(1-l1)-2*(x-1)*l2;
f2:l2-l1;
f3:l1*(-y-x^2+1);
f4:l2*(y-(x-1)^2);
algsys([f1,f2,f3,f4],[x,y,l1,l2])
f1:x^2-y^2;
f2:-1-y+2*y^2-x+x^2;
algsys([f1,f2],[x,y])
allrootsComputes numerical approximations of the real and complex roots of the polynomial or polynomial equation of one variable.
(1+2*x)^3 = 13.5*(1+x^5);
allroots(%)
antisymmetricDeclares antisymmetricity of a function.
declare(h,antisymmetric);
h(x,z,y)
appendReturns a single list of the elements of a list followed by the elements of another list, also works on general expressions.
append([x+y,0,-3.2],[2.5E+20,x])
arrayinfoReturns information about an array.
b[1,x]:1;
array(f,2,3);
arrayinfo(b);
arrayinfo(f)
atEvaluates the expression with the variables assuming the values as specified for them in the list of equations or a single equation.
atvalue(f(x,y),[x = 0,y = 1],a^2);
atvalue('diff(f(x,y),x),x = 0,1+y);
printprops(all,atvalue);
diff(4*f(x,y)^2-u(x,y)^2,x);
at(%,[x = 0,y = 1])
atvalueAssigns the value of variable to an expression at some point.
atvalue(f(x,y),[x = 0,y = 1],a^2);
atvalue('diff(f(x,y),x),x = 0,1+y);
printprops(all,atvalue);
diff(4*f(x,y)^2-u(x,y)^2,x);
at(%,[x = 0,y = 1])
augcoefmatrixReturns the augmented coefficient matrix for the variables of the system of linear equations.
[2*x-(a-1)*y = 5*b,c+b*y+a*x = 0];
augcoefmatrix(%,[x,y])
bezoutReturns a matrix whose determinant is the resultant of two polynomials eliminating a variable.
bezout(1+x^2+a*y,b+x*y+y^2,x);
expand(determinant(%));
%-expand(resultant(1+x^2+a*y,b+x*y+y^2,x))
catchEvaluates expressions one by one; if any leads to the evaluation of an expression of some form, then the value of the catch is the value of that form, and no further expressions are evaluated.
g(l):=catch(map(lambda([x],if x <
0 then throw(x) else f(x)),l));
g([1,2,3,7]);
g([1,2,-3,7])
cfConverts expression into a continued fraction.
cf([1,-2,1]+[1,2,-3]);
cfdisrep(%)
cflength:4;
cf(sqrt(3));
cfexpand(%);
ev(%[1,2]/%[2,2],numer)
cfdisrepConstructs and returns an ordinary arithmetic expression from the list representation of a continued fraction.
cf([1,-2,1]+[1,2,-3]);
cfdisrep(%)
cfexpandReturns a matrix of the numerators and denominators of the last and next-to-last convergents of the continued fraction.
cflength:4;
cf(sqrt(3));
cfexpand(%);
ev(%[1,2]/%[2,2],numer)
charpolyReturns the characteristic polynomial for a matrix with respect to a variable.
a:matrix([3,1],[2,4]);
expand(charpoly(a,lambda));
(programmode:true,solve(%));
matrix([x1],[x2]);
ev(a . %-lambda*%,%th(2)[1]);
%[1,1] = 0;
x2^2+x1^2 = 1;
solve([%th(2),%],[x1,x2])
coeffReturns the coefficient of x^n in expression.
coeff(b+tan(x)+2*a*tan(x) = 3+5*tan(x),tan(x))
coeff(1+x*%e^x+y,x,0)
combineSimplifies the sum expression by combining terms with the same denominator into a single term.
combine(b/y+a/y+b/x+a/x)
commutativeDeclares commutativity of a function.
declare(h,commutative);
h(x,z,y)
contentReturns a list whose first element is the greatest common divisor of the coefficients of the terms of a polynomial in a variable (this is the content) and whose second element is another polynomial divided by the content.
content(4*x^2*y^2+2*x*y,y)
defmatchCreates a function which tests an expression to see if it matches a pattern.
nonzeroandfreeof(x,e):=is(e # 0 and freeof(x,e));
matchdeclare(a,nonzeroandfreeof(x),b,freeof(x));
defmatch(linear,b+a*x,x);
linear(y^2+(1+y)*z+3*z,z);
matchdeclare([a,f],true);
constinterval(l,h):=constantp(h-l);
matchdeclare(b,constinterval(a));
matchdeclare(x,atom);
block(remove(integrate,outative), defmatch(checklimits,'integrate(f,x,a,b)), declare(integrate,outative));
'integrate(sin(t),t,%pi+x,2*%pi+x);
checklimits(%);
'integrate(sin(t),t,0,x);
checklimits(%);
remvalue(a,b,f,x)
deftaylorFor each function of one variable, deftaylor defines an expression as the Taylor series about zero.
deftaylor(f(x),sum(x^i/(2^i*i!^2),i,4,inf)+x^2);
taylor(%e^sqrt(f(x)),x,0,4)
deleteRemoves all occurrences of an expression from another expression
delete(sin(x),y+sin(x)+x)
dependsDeclares functional dependencies among variables for the purpose of computing derivatives.
depends(a,x);
diff(a . a,x)
depends(f,[x,y],[x,y],t);
diff(f,t)
derivdegreeReturns the highest degree of the derivative of a dependent variable with respect to an independent variable occuring in expression.
'diff(y,x)*x^2+'diff(y,z,3)*2+'diff(y,x,2);
derivdegree(%,y,x)
desolveSolves systems of linear ordinary differential equations using Laplace transform.
eqn1:'diff(f(x),x) = sin(x)+'diff(g(x),x);
eqn2:'diff(g(x),x,2) = 'diff(f(x),x)-cos(x);
atvalue('diff(g(x),x),x = 0,a);
atvalue(f(x),x = 0,1);
desolve([eqn1,eqn2],[f(x),g(x)]);
ev([eqn1,eqn2],%,diff)
diffReturns the derivative or differential of expression with respect to some or all variables.
diff(2*x^2+x^3+sin(x),x)
diff(sin(x)*cos(x),x)
diff(sin(x)*cos(x),x,2)
derivabbrev:true;
diff(exp(f(x)),x,2)
derivabbrev:true;
'integrate(f(x,y),y,g(x),h(x));
diff(%,x)
displayDisplays equations whose left side is unevaluated, and whose right side is the value of the expression centered on the line.
display(b[1,2])
divideComputes the quotient and remainder of a polynomial divided by another polynomial, in a main polynomial variable.
divide(y+x,x-y,x)
divide(y+x,x-y)
dotscrulesGives simplification rule for expression A.SC or SC.A and A.(SC*B).
declare(l,scalar,[m1,m2,m3],nonscalar);
expand((1-l*m1) . (1-l*m2) . (1-l*m3));
ev(%,dotscrules);
rat(%,l)
dpartSelects parts of the displayed form of expression and returns the whole expression with the selected subexpression displayed inside a box.
dpart(y/z^2+x,1,2,1)
expand((a+b)^4);
(a+b)^2*(x+y)^2;
expand(%);
%th(3)/%;
factor(%);
dpart(%th(2),2,4);
part(%th(3),2,4)
echelonReturns the echelon form of a matrix as produced by Gaussian elimination.
matrix([2,1-a,-5*b],[a,b,c]);
echelon(%)
M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
echelon (M)
eliminateEliminates variables from equations (or expressions assumed equal to zero) by taking successive resultants.
exp1:z+y*x+2*x^2;
exp2:-1-z+5*y+3*x;
exp3:5-y^2+x+z^2;
eliminate([exp3,exp2,exp1],[y,z])
evEvaluates an expression in the environment specified by the arguments.
'diff(sin(w),w)+(1+w)^2+cos(y)+sin(x);
ev(%,sin,expand,diff,x = 2,y = 1)
ev(y+x,x:y+a,y:2)
'diff(x^2+x*y+y^2,x,2,y,1);
ev(%,diff)
2*x-3*y = 3;
2*y-3*x = -4;
solve([%th(2),%]);
ev(%th(3),%)
1/x+x >
gamma(1/2);
ev(%,numer,x = 1/2);
ev(%,pred)
evenfunDeclares a function as even function.
declare(g,evenfun);
g(-x)
expandExpand an expression.
(1/(y+x)^4-3/(z+y)^3)^2;
expand(%,2,0)
expand(a . (f+c . (e+d)+b))
expand((1+x)^3)
(1+x)^7;
expand(%);
expand(%th(2),7,7)
ev(a*(c+b)^2+a*(c+b),expop:1)
exponentializeConverts circular and hyperbolic functions in an expression to exponentials.
ev(%e^x*sin(x)^2,exponentialize)
ev(%e^x*sin(x)^2,exponentialize);
integrate(%,x);
ev(%,demoivre);
ans:ev(%,ratexpand);
ev(%,x:1,numer)-ev(%,x:0,numer);
integrate(%e^x*sin(x)^2,x);
trigreduce(%);
%-ans;
ev(sin(x),%emode)
factcombTries to combine the coefficients of factorials in an expression with the factorials themselves by converting, for example, (n + 1)*n! into (n + 1)!.
(1+n)^2*n!^2;
factcomb(%)
factorFactors an expression containing any number of variables or functions into factors irreducible over the integers.
factor(2^63-1)
factor(-8*y-4*x+z^2*(2*y+x))
-1-2*x-x^2+y^2+2*x*y^2+x^2*y^2;
block([dontfactor:[x]],factor(%/36/(1+2*y+y^2)))
factor(1+%e^(3*x))
factor(1+x^4,a^2-2)
factor(-y^2*z^2-x*z^2+x^2*y^2+x^3)
(2+x)/(3+x)/(b+x)/(c+x)^2;
ratsimp(%);
partfrac(%,x);
map('factor,%)
ratsimp((x^5-1)/(x-1));
subst(a,x,%);
factor(%th(2),%)
factor(1+x^12)
factor(1+x^99)
factorsumTries to group terms in factors of an expression which are sums into groups of terms such that their sum is factorable.
ev((1+x)*(a*(z+w)^2+(v+u)^2),expand);
factorsum(%)
freeofReturns true if no subexpression of an expression is equal to some expression or if it occurs only as a dummy variable, and returns false otherwise.
freeof(y,sin(2*y+x))
freeof(cos(y),"*",cos(x)+sin(y))
featurepAttempts to determine whether an object has some feature on the basis of the facts in the current database.
declare(j,even);
featurep(j,integer)
fullmapSimilar to map, but fullmap keeps mapping down all subexpressions until the main operators are no longer the same.
fullmap(g,b*c+a);
map(g,b*c+a)
fullmaplSimilar to fullmap, but fullmapl only maps onto lists and matrices.
fullmapl("+",[3,[4,5]],[[a,1],[0,-1.5]])
funcsolveReturns a function g(t) or nothing, depending on whether or not there exists a rational function g(t)satisfying the equation, which must be a first order, linear polynomial in (for this case) g(t) and g(t+1).
funcsolve((1+n)*f(n)-(3+n)*f(1+n)/(1+n) = (n-1)/(2+n),f(n))
genmatrixReturns a matrix generated from an array a, taking element a[i_1,j_1] as the upper-left element and a[i_2,j_2] as the lower-right element of the matrix.
h[i,j]:=1/(-1+j+i);
genmatrix(h,3,3)
getRetrieves the user property associated with an atom or returns false if it doesn't have the property.
put(%e,transcendental,type);
put(%i,transcendental,type);
put(%i,algebraic,type);
typeof(x):=block([q],if numberp(x) then return(algebraic), if not atom(x) then return(maplist(typeof,x)),q:get(x,type), if q = false then error("not numeric") else q);
errcatch(typeof(x*%pi+2*%e));
typeof(%pi+2*%e)
gfactorFactors a polynomial over the Gaussian integers (that is, the integers with the imaginary unit %i adjoined).
gfactor(x^4-1)
gradefDefines the partial derivatives (i.e., the components of the gradient) of a function or a variable.
depends(y,x);
kill(f,g,j);
gradef(f(x,y),x^2,g(x,y));
diff(f(x,y),x)
gradef(j(n,z),'diff(j(n,z),n),j(n-1,z)-n*j(n,z)/z);
ratsimp(diff(j(2,x),x,2))
hornerReturns a rearranged representation of an expression as in Horner's rule, using a variable as the main variable if it is specified.
poly:5.2E+20-5.5*x+9.9999999999999995E-21*x^2;
errcatch(ev(%,x = 1.0E+20));
ev(horner(poly,x),keepfloat);
ev(%,x = 1.0E+20)
inpartIs 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.
w*z+y+x;
inpart(%,3,2);
'limit(f(x)^g(1+x),x,0,minus);
inpart(%,1,2)
integrateAttempts to symbolically compute the integral of an expression with respect to a variable.
test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)));
test(sin(x));
test(1/(x+1));
test(1/(x^2+1))
integrate(sin(x)^3,x)
integrate(%e^x/(2+%e^x),x)
integrate(1/(x*log(x)),x)
integrate(sin(3+2*x),x)
integrate(%e^x*erf(x),x)
integrate(x/(1+x^3),x);
diff(%,x);
ratsimp(%)
integrate(x^(5/4)/(1+x)^(5/2),x,0,inf)
gradef(q(x),sin(x^2));
diff(log(q(r(x))),x);
integrate(%,x)
isAttempts to determine whether the predicate expression is provable from the facts in the assume database.
is(x^2 >= 2*x-1)
assume(a >
1);
is(log(1+log(1+a)) >
0 and 1+a^2 >
2*a)
isolateReturns expression with subexpressions which are sums and which do not contain a variable replaced by intermediate expression labels (these being atomic symbols like %t1, %t2, ...).
(b+a)^4*(x*((d+c)^2+2*x)+1);
isolate(%,x);
ratexpand(%);
ev(%)
(b+a)*(b+a+x)^2*%e^(b+a*x+x^2);
ev(isolate(%,x),exptisolate:true)
laplaceAttempts to compute the Laplace transform of an expression with respect to a variable and a transform parameter.
laplace(%e^(a+2*t)*sin(t)*t,t,s)
lassociativeDeclares left-associativity of an operator.
declare(g,lassociative);
g(g(a,b),g(c,d));
g(g(a,b),g(c,d))-g(a,g(b,g(c,d)))

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