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# 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:

    [wfs@chromium]$ maxima
    Maxima 5.9.1 http://maxima.sourceforge.net
    Using Lisp CMU Common Lisp 19a
    Distributed under the GNU Public License. See the file COPYING.
    Dedicated to the memory of William Schelter.
    This is a development version of Maxima. 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();
    [wfs@chromium]$

Maxima can search the info pages. Use the `describe` command to show all the commands and variables containing a string, and optionally their documentation. The question mark `?` is an abbreviation for `describe`:

    (%i1) ? integ

     0: (maxima.info)Introduction to Elliptic Functions and Integrals.
     1: Definitions for Elliptic Integrals.
     2: Integration.
     3: Introduction to Integration.
     4: Definitions for Integration.
     5: askinteger :Definitions for Simplification.
     6: integerp :Definitions for Miscellaneous Options.
     7: integrate :Definitions for Integration.
     8: integrate_use_rootsof :Definitions for Integration.
     9: integration_constant_counter :Definitions for Integration.
    Enter space-separated numbers, `all' or `none': 6 5

    Info from file /usr/local/info/maxima.info:
     - Function: integerp (<expr>)
         Returns `true' if <expr> is an integer, otherwise `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' will
         ask 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)                         false

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) 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
    (%i4) kill(labels);
    (%o0)                         done
    (%i1) plot2d (sin(x)/x, [x, -20, 20]);
    (%o1)
    (%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5]);
    (%o2)
    (%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2), [x, -12, 12], [y, -12, 12]);
    (%o3)

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