## 1 Introduction to Maxima ¶

Start Maxima with the command "maxima". Maxima will display version information and a prompt. End each Maxima command with a semicolon. End the session with the command "`quit();`". Here’s a sample session:

    $ maxima
    Maxima 5.45.1 https://maxima.sourceforge.io
    using Lisp SBCL 2.0.1.debian
    Distributed under the GNU Public License. See the file COPYING.
    Dedicated to the memory of William Schelter.
    The function bug_report() provides bug reporting information.
    (%i1) factor(10!);
                                       8  4  2
    (%o1)                             2  3  5  7
    (%i2) expand ((x + y)^6);
               6        5       2  4       3  3       4  2      5      6
    (%o2)     y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
    (%i3) factor (x^6 - 1);
                                         2            2
    (%o3)              (x - 1) (x + 1) (x  - x + 1) (x  + x + 1)
    (%i4) quit();
    $

Maxima can search the info pages. Use the `describe` command to show information about the command or all the commands and variables containing a string. The question mark `?` (exact search) and double question mark `??` (inexact search) are abbreviations for `describe`:

    (%i1) ?? integ
     0: Functions and Variables for Elliptic Integrals
     1: Functions and Variables for Integration
     2: Introduction to Elliptic Functions and Integrals
     3: Introduction to Integration
     4: askinteger  (Functions and Variables for Simplification)
     5: integerp  (Functions and Variables for Miscellaneous Options)
     6: integer_partitions  (Functions and Variables for Sets)
     7: integrate  (Functions and Variables for Integration)
     8: integrate_use_rootsof  (Functions and Variables for Integration)
     9: integration_constant_counter  (Functions and Variables for
        Integration)
     10: nonnegintegerp  (Functions and Variables for linearalgebra)
    Enter space-separated numbers, `all' or `none': 5 4

     -- Function: integerp (<expr>)
         Returns `true' if <expr> is a literal numeric integer, otherwise
         `false'.

         `integerp' returns false if its argument is a symbol, even if the
         argument is declared integer.

         Examples:

              (%i1) integerp (0);
              (%o1)                         true
              (%i2) integerp (1);
              (%o2)                         true
              (%i3) integerp (-17);
              (%o3)                         true
              (%i4) integerp (0.0);
              (%o4)                         false
              (%i5) integerp (1.0);
              (%o5)                         false
              (%i6) integerp (%pi);
              (%o6)                         false
              (%i7) integerp (n);
              (%o7)                         false
              (%i8) declare (n, integer);
              (%o8)                         done
              (%i9) integerp (n);
              (%o9)                         false

     -- Function: askinteger (<expr>, integer)
     -- Function: askinteger (<expr>)
     -- Function: askinteger (<expr>, even)
     -- Function: askinteger (<expr>, odd)
         `askinteger (<expr>, integer)' attempts to determine from the
         `assume' database whether <expr> is an integer.  `askinteger'
         prompts the user if it cannot tell otherwise, and attempt to
         install the information in the database if possible.  `askinteger
         (<expr>)' is equivalent to `askinteger (<expr>, integer)'.

         `askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
         likewise attempt to determine if <expr> is an even integer or odd
         integer, respectively.

    (%o1)                                true

To use a result in later calculations, you can assign it to a variable or refer to it by its automatically supplied label. In addition, `%` refers to the most recent calculated result:

    (%i1) u: expand ((x + y)^6);
           6        5       2  4       3  3       4  2      5      6
    (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x

    (%i2) diff (u, x);
             5         4       2  3       3  2       4        5
    (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x

    (%i3) factor (%o2);
                                        5
    (%o3)                      6 (y + x)

Maxima knows about complex numbers and numerical constants:

    (%i1) cos(%pi);
    (%o1)                          - 1

    (%i2) exp(%i*%pi);
    (%o2)                          - 1

Maxima can do differential and integral calculus:

    (%i1) u: expand ((x + y)^6);
           6        5       2  4       3  3       4  2      5      6
    (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x

    (%i2) diff (%, x);
             5         4       2  3       3  2       4        5
    (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x

    (%i3) integrate (1/(1 + x^3), x);
                                      2 x - 1
                    2            atan(-------)
               log(x  - x + 1)        sqrt(3)    log(x + 1)
    (%o3)    - --------------- + ------------- + ----------
                      6             sqrt(3)          3

Maxima can solve linear systems and cubic equations:

    (%i1) linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
                            7 a - 52        25
    (%o1)              [x = --------, y = -------]
                            3 a - 8       3 a - 8

    (%i2) solve (x^3 - 3*x^2 + 5*x = 15, x);
    (%o2)       [x = - sqrt(5) %i, x = sqrt(5) %i, x = 3]

Maxima can solve nonlinear sets of equations. Note that if you don’t want a result printed, you can finish your command with `$` instead of `;`.

    (%i1) eq_1: x^2 + 3*x*y + y^2 = 0$
    (%i2) eq_2: 3*x + y = 1$

    (%i3) solve ([eq_1, eq_2]);
                  3 sqrt(5) + 7      sqrt(5) + 3
    (%o3) [[y = - -------------, x = -----------],
                        2                 2
                                   3 sqrt(5) - 7        sqrt(5) - 3
                              [y = -------------, x = - -----------]]
                                         2                   2

Maxima can generate plots of one or more functions:

    (%i1) plot2d (sin(x)/x, [x, -20, 20])$

    (%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$

    (%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2),
             [x, -12, 12], [y, -12, 12])$

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