4.8.1 Tutorial

We follow the tutorial at http://maxima.sourceforge.net/docs/intromax/.

sage: maxima('1/100 + 1/101')
201/10100

sage: a = maxima('(1 + sqrt(2))^5'); a
(sqrt(2) + 1)^5
sage: a.expand()
29*sqrt(2) + 41

sage: a = maxima('(1 + sqrt(2))^5')
sage: float(a)                
82.012193308819747
sage: a.numer()
82.01219330881975

sage: maxima.eval('fpprec : 100')
'100'
sage: a.bfloat()
8.2012193308819756415248973002081244278520484385931494122123712401731241875
4011041266612384955016056B1

sage: maxima('100!')
933262154439441526816992388562667004907159682643816214685929638952175999932
299156089414639761565182862536979208272237582511852109168640000000000000000
00000000

sage: f = maxima('(x + 3*y + x^2*y)^3')
sage: f.expand()
x^6*y^3 + 9*x^4*y^3 + 27*x^2*y^3 + 27*y^3 + 3*x^5*y^2 + 18*x^3*y^2 +
27*x*y^2 + 3*x^4*y + 9*x^2*y + x^3
sage: f.subst('x=5/z')
(5/z + 25*y/z^2 + 3*y)^3
sage: g = f.subst('x=5/z')
sage: h = g.ratsimp(); h
(27*y^3*z^6 + 135*y^2*z^5 + (675*y^3 + 225*y)*z^4 + (2250*y^2 + 125)*z^3 +
(5625*y^3 + 1875*y)*z^2 + 9375*y^2*z + 15625*y^3)/z^6
sage: h.factor()
(3*y*z^2 + 5*z + 25*y)^3/z^6

sage: eqn = maxima(['a+b*c=1', 'b-a*c=0', 'a+b=5'])
sage: s = eqn.solve('[a,b,c]'); s
[[a = (25*sqrt(79)*%i + 25)/(6*sqrt(79)*%i - 34),b = (5*sqrt(79)*%i +
5)/(sqrt(79)*%i + 11),c = (sqrt(79)*%i + 1)/10],[a = (25*sqrt(79)*%i -
25)/(6*sqrt(79)*%i + 34),b = (5*sqrt(79)*%i - 5)/(sqrt(79)*%i - 11),c =  -
(sqrt(79)*%i - 1)/10]]

Here is an example of solving an algebraic equation:

sage: maxima('x^2+y^2=1').solve('y')
[y =  - sqrt(1 - x^2),y = sqrt(1 - x^2)]
sage: maxima('x^2 + y^2 = (x^2 - y^2)/sqrt(x^2 + y^2)').solve('y')
[y =  - sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2),y = sqrt(( - y^2 -
x^2)*sqrt(y^2 + x^2) + x^2)]

You can even nicely typeset the solution in latex:

sage: print latex(s)
\left[\left[a = \frac{25\sqrt{79}i + 25}{6\sqrt{79}i - 34},b =
\frac{5\sqrt{79}i + 5}{\sqrt{79}i + 11},c = \frac{\sqrt{79}i +
1}{10}\right],\left[a = \frac{25\sqrt{79}i - 25}{6\sqrt{79}i + 34},b =
\frac{5\sqrt{79}i - 5}{\sqrt{79}i - 11},c =  - \frac{\sqrt{79}i -
1}{10}\right]\right]

To have the above appear onscreen via xdvi, type view(s). (TODO: For OS X should create pdf output and use preview instead?)

sage: e = maxima('sin(u + v) * cos(u)^3'); e
cos(u)^3*sin(v + u)
sage: f = e.trigexpand(); f
cos(u)^3*(cos(u)*sin(v) + sin(u)*cos(v))
sage: f.trigreduce()
(sin(v + 4*u) + sin(v - 2*u))/8 + (3*sin(v + 2*u) + 3*sin(v))/8
sage: w = maxima('3 + k*%i')
sage: f = w^2 + maxima('%e')^w
sage: f.realpart()
%e^3*cos(k) - k^2 + 9

sage: f = maxima('x^3 * %e^(k*x) * sin(w*x)'); f
x^3*%e^(k*x)*sin(w*x)
sage: f.diff('x')
k*x^3*%e^(k*x)*sin(w*x) + 3*x^2*%e^(k*x)*sin(w*x) + w*x^3*%e^(k*x)*cos(w*x)
sage: f.integrate('x')
(((k*w^6 + 3*k^3*w^4 + 3*k^5*w^2 + k^7)*x^3 + (3*w^6 + 3*k^2*w^4 -
3*k^4*w^2 - 3*k^6)*x^2 + ( - 18*k*w^4 - 12*k^3*w^2 + 6*k^5)*x - 6*w^4 +
36*k^2*w^2 - 6*k^4)*%e^(k*x)*sin(w*x) + (( - w^7 - 3*k^2*w^5 - 3*k^4*w^3 -
k^6*w)*x^3 + (6*k*w^5 + 12*k^3*w^3 + 6*k^5*w)*x^2 + (6*w^5 - 12*k^2*w^3 -
18*k^4*w)*x - 24*k*w^3 + 24*k^3*w)*%e^(k*x)*cos(w*x))/(w^8 + 4*k^2*w^6 +
6*k^4*w^4 + 4*k^6*w^2 + k^8)

sage: f = maxima('1/x^2')
sage: f.integrate('x', 1, 'inf')
1
sage: g = maxima('f/sinh(k*x)^4')
sage: g.taylor('x', 0, 3)
f/(k^4*x^4) - 2*f/(3*k^2*x^2) + 11*f/45 - 62*k^2*f*x^2/945

sage: maxima.taylor('asin(x)','x',0, 10)
x + x^3/6 + 3*x^5/40 + 5*x^7/112 + 35*x^9/1152

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