Einfache Berechnungen

> 1/3 + 1/7;

> evalf( 1/3 + 1/7);

> evalf(Pi, 50);

> abs(2+3*I);

> (3+2*I)/(2-I);

Polynome

> factor(x^4+2*x^3-12*x^2-40*x-32);

> expand((x-1)^4);

> simplify( exp(x*log(y)) );

> simplify(sin(x)^2+cos(x)^2);

> expand( cos(4*x)+4*cos(2*x)+3, trig);

> combine(4*cos(x)^3, trig);

Lösung von Gleichungen

> solve( x^2-x=5, x);

> glsys:={ 2*x + 3*y + z = 1,

> x - y - z = 4,

> 3*x + 7*z = 5}:

> solve( glsys );

> solve( {x^2+y^2=10, x^y=2}, {x,y} );

> evalf(%);

> fsolve( {x^2+y^2=10, x^y=2}, {x,y} );

> display({implicitplot(x^2+y^2=10,x=-4..4,y=-4..4,color=black),implicitplot(x^y=2,x=0.01..4,y=-4..4,color=black)},scaling=constrained);

> fsolve( {x^2+y^2=10, x^y=2}, {x=0..1,y=-4..0} );

Lineare Algebra

> A:=<<1,3>|<2,4>>;B:=<<w|x>,<y|z>>;

> A+B,A.B;

> A^(-1),B^(-1);

> glmat:=<<2|3|1>,<1|-1|1>,<3|0|7>>;

>

> glmat.<x,y,z> = <1,4,5>;

> <x,y,z> = glmat^(-1).<1,4,5>;

>

Limites, Summen und Produkte

> Limit( (sqrt(1+x)-1)/x, x=0)=limit( (sqrt(1+x)-1)/x, x=0);

> Limit(x!/x^x, x=infinity)=limit(x!/x^x, x=infinity);

> Sum(1/i^2,i=1..infinity)=sum(1/i^2,i=1..infinity);

> Product( 1+1/x^2, x=1..infinity )=product( 1+1/x^2, x=1..infinity );

Differentiation, Integration

> diff((x-1) / (x^2+1), x);

> simplify(%);

> diff( sin(x*y), x);

> int( 1/(1+x^3), x);

> int( sin(x^2), x=a..b);

Differentialgleichungen

> dgl := diff(y(x),x) * y(x) * (1+x^2) = x;

> dsolve( {dgl, y(0)=0 }, y(x));

> dgl := (y(x)^2 -x) * D(y)(x) + x^2-y(x) =0;

> dsolve( dgl, y(x),implicit);

Potenzreihen

> series( sin(x), x=0, 10);

> Order:=10:

> dgl := diff(y(x), x$2) + diff(y(x),x) + y(x) = x + sin(x);

> lsg := dsolve( {dgl, y(0)=0, D(y)(0)=0}, y(x), series);

Laplace- und Fourier Transformationen

> with(inttrans);

Warning, the name hilbert has been redefined

Laplace

> laplace( cos(t-a), t, s);

> invlaplace(%,s,t);

> combine(%, trig);

Fourier

> alias(sigma=Heaviside):

> f:=sigma(t+1) - sigma(t-1);

> plot(f,t=-2..2);

> g:=fourier(f,t,w);

> simplify(g);

> plot(g, w=-10..10);

> invfourier(g,w,t); simplify(%);

Vereinfachungen (simplify)

> limit(g, w=0);

> g1:=subs(Dirac(w)=0, g);

> simplify(g1);

> plot(g1, w=-10..10);

> invfourier(g1,w,t); simplify(%);

Interpolation, Approximation

Interpolation

> datax:=[seq(i,i=1..10)]:

> datay:=[seq(rand(10)(), i=1..10)]:

> dataxy:=zip( (x,y)->[x,y], datax, datay);

> f:=interp(datax, datay, y);

Approximation

> with(numapprox);

> x0:=solve(x^2=Pi/2)[1];

> f:=pade(tan(x^2), x=x0, [3,3]):

> evalf(normal(f));

> p1:=plot( tan(x^2), x=x0-.1..x0+.1, -10..10,style=point, color=black):

> p2:=plot(f, x=x0-.1..x0+.1, -10..10, color=red):

> with(plots):
display(p1,p2);

Funktionen und Terme

> f:=x->(x+a)^2/sin(x);

> f(b);f(a);f(sin(Pi/2));f(f(3));

>

>

Graphik

> plot(sin(x)*exp(1)^(-x/5), x=0..4*Pi);

> plot3d(sin(x)*exp(1)^y, x=0..2*Pi, y=0..Pi, axes=boxed);

Programmierung

> f:=proc(x::nonnegint)
option remember;
if x=0 then 0
elif x=1 then 1
else f(x-1)+f(x-2) fi
end:

> f(50);