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pre
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<title> Introduction to Maxima</title>
<h1 align="center">Introduction to Maxima </h1>
<h3 align="center">Richard H. Rand<br />
Dept. of Theoretical and Applied Mechanics, Cornell University
<a href="#tthFtNtAAB" name="tthFrefAAB"><sup>1</sup></a> </h3>
<h3 align="center"> </h3>
<div class="p"><!----></div>
Copyright (c) 1988-2010 Richard H. Rand.
<div class="p"><!----></div>
This document is free; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free
Software Foundation. See the GNU General Public License for more
details at http://www.gnu.org/copyleft/gpl.html
<div class="p"><!----></div>
<h1>Contents </h1><a href="#tth_sEc1"
>1 Introduction </a><br />
<a href="#tth_sEc2"
>2 Special keys and symbols </a><br />
<a href="#tth_sEc3"
>3 Arithmetic </a><br />
<a href="#tth_sEc4"
>4 Algebra </a><br />
<a href="#tth_sEc5"
>5 Calculus </a><br />
<a href="#tth_sEc6"
>6 Matrix calculations </a><br />
<a href="#tth_sEc7"
>7 Programming in Maxima </a><br />
<a href="#tth_sEc8"
>8 A partial list of Maxima functions</a><br />
<div class="p"><!----></div>
<h2><a name="tth_sEc1">
1</a> Introduction <a name="sec:introduction">
</a></h2>
<div class="p"><!----></div>
To invoke Maxima in a console, type
<pre>
maxima <enter>
</pre>
<div class="p"><!----></div>
The computer will display a greeting of the sort:
<pre>
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)
</pre>
<div class="p"><!----></div>
The <tt>(%i1)</tt> is a "label". Each input or output line is
labelled and can be referred to by its own label for the rest of the
session. <tt>i</tt> labels denote your commands and <tt>o</tt> labels
denote displays of the machine's response. <em>Never use variable
names like <tt>%i1</tt> or <tt>%o5</tt>, as these will be confused with
the lines so labeled</em>.
<div class="p"><!----></div>
Maxima distinguishes lower and upper case. All built-in functions
have names which are lowercase only (<tt>sin</tt>, <tt>cos</tt>, <tt>save</tt>,
<tt>load</tt>, etc). Built-in constants have lowercase names (<tt>%e</tt>,
<tt>%pi</tt>, <tt>inf</tt>, etc). If you type <tt>SIN(x)</tt> or <tt>
Sin(x)</tt>, Maxima assumes you mean something other than the built-in
<tt>sin</tt> function. User-defined functions and variables can have
names which are lower or upper case or both. <tt>foo(XY)</tt>, <tt>
Foo(Xy)</tt>, <tt>FOO(xy)</tt> are all different.
<div class="p"><!----></div>
<h2><a name="tth_sEc2">
2</a> Special keys and symbols <a name="sec:keys">
</a></h2>
<div class="p"><!----></div>
<ol type="1">
<li> To end a Maxima session, type <tt>quit();</tt>.
<div class="p"><!----></div>
</li>
<li> To abort a computation without leaving Maxima, type <tt>^C</tt>.
(Here <tt>^</tt> stands for the control key, so
that <tt>^C</tt> means first press the key marked control and hold it down while pressing the C key.)
It is important for you to
know how to do this in case, for example, you begin a computation which is taking too long.
For example:
<pre>
(%i1) sum (1/x^2, x, 1, 100000)$
^C
Maxima encountered a Lisp error:
Interactive interrupt at #x7FFFF74A43C3.
Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.
(%i2)
</pre>
<div class="p"><!----></div>
</li>
<li> In order to tell Maxima that you have finished your command, use
the semicolon (<tt>;</tt>), followed by a return. Note that the return
key alone does not signal that you are done with your input.
<div class="p"><!----></div>
</li>
<li> An alternative input terminator to the semicolon (<tt>;</tt>) is
the dollar sign (<tt>$</tt>), which, however, supresses the display of
Maxima's computation. This is useful if you are computing some long
intermediate result, and you don't want to waste time having it
displayed on the screen.
<div class="p"><!----></div>
</li>
<li> If you wish to repeat a command which you have already given,
say on line <tt>(%i5)</tt>, you may do so without typing it over again
by preceding its label with two single quotes (<tt>"</tt>), i.e., <tt>
"%i5</tt>. (Note that simply inputing <tt>%i5</tt> will not do the job
- try it.)
<div class="p"><!----></div>
</li>
<li> If you want to refer to the immediately preceding result
computed by Maxima, you can either use its <tt>o</tt> label, or you can
use the special symbol percent (<tt>%</tt>).
<div class="p"><!----></div>
</li>
<li> The standard quantities e (natural log base), i (square root
of <font face="symbol">-</font
>1) and <font face="symbol">p</font
> (3.14159<font face="symbol">¼</font
>) are respectively referred to as
<tt>%e</tt>, <tt>%i</tt>,
and <tt>%pi</tt>. Note that the use of <tt>%</tt> here as a prefix
is completely unrelated to the use of <tt>%</tt> to refer to the
preceding result computed.
<div class="p"><!----></div>
</li>
<li> In order to assign a value to a variable, Maxima uses the colon
(<tt>:</tt>), not the equal sign. The equal sign is used for
representing equations.
<div class="p"><!----></div>
</li>
</ol>
<div class="p"><!----></div>
<h2><a name="tth_sEc3">
3</a> Arithmetic <a name="sec:arithmetic">
</a></h2>
<div class="p"><!----></div>
The common arithmetic operations are
<dl compact="compact">
<dt><b><tt>+</tt></b></dt>
<dd> addition</dd>
<dt><b><tt>-</tt></b></dt>
<dd> subtraction</dd>
<dt><b><tt>*</tt></b></dt>
<dd> scalar multiplication</dd>
<dt><b><tt>/</tt></b></dt>
<dd> division</dd>
<dt><b><tt>^</tt></b></dt>
<dd> or <tt>**</tt> exponentiation</dd>
<dt><b><tt>.</tt></b></dt>
<dd> matrix multiplication</dd>
<dt><b><tt>sqrt(x)</tt></b></dt>
<dd> square root of <tt>x</tt>.</dd>
</dl>
Maxima's output is characterized by exact (rational) arithmetic. For example,
<pre>
(%i1) 1/100 + 1/101;
201
(%o1) -----
10100
</pre>
If irrational numbers are involved in a computation, they are kept in symbolic form:
<pre>
(%i2) (1 + sqrt(2))^5;
5
(%o2) (sqrt(2) + 1)
(%i3) expand (%);
7/2
(%o3) 3 2 + 5 sqrt(2) + 41
</pre>
However, it is often useful to express a result in decimal notation.
This may be accomplished by following the expression you want expanded
by "<tt>,numer</tt>":
<pre>
(%i4) %, numer;
(%o4) 82.01219330881977
</pre>
Note the use here of <tt>%</tt>
to refer to the previous result. In this version of Maxima, <tt>
numer</tt> gives 16 significant figures, of which the last is often
unreliable. However, Maxima can offer <em>arbitrarily high
precision</em> by using the <tt>bfloat</tt> function:
<pre>
(%i5) bfloat (%o3);
(%o5) 8.201219330881976b1
</pre>
The number of significant figures displayed is controlled by the
Maxima variable <tt>fpprec</tt>, which has the default value of 16:
<pre>
(%i6) fpprec;
(%o6) 16
</pre>
Here we reset <tt>fpprec</tt> to yield 100 digits:
<pre>
(%i7) fpprec: 100;
(%o7) 100
(%i8) ''%i5;
(%o8) 8.20121933088197564152489730020812442785204843859314941221\
2371240173124187540110412666123849550160561b1
</pre>
Note the use of two single quotes (<tt>"</tt>) in <tt>(%i8)</tt> to repeat
command <tt>(%i5)</tt>. Maxima can handle very large numbers without
approximation:
<pre>
(%i9) 100!;
(%o9) 9332621544394415268169923885626670049071596826438162146859\
2963895217599993229915608941463976156518286253697920827223758251\
185210916864000000000000000000000000
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc4">
4</a> Algebra <a name="sec:algebra">
</a></h2>
<div class="p"><!----></div>
Maxima's importance as a computer tool to facilitate analytical
calculations becomes more evident when we see how easily it does
algebra for us. Here's an example in which a polynomial is expanded:
<pre>
(%i1) (x + 3*y + x^2*y)^3;
2 3
(%o1) (x y + 3 y + x)
(%i2) expand (%);
6 3 4 3 2 3 3 5 2 3 2
(%o2) x y + 9 x y + 27 x y + 27 y + 3 x y + 18 x y
2 4 2 3
+ 27 x y + 3 x y + 9 x y + x
</pre>
Now suppose we wanted to substitute <tt>5/z</tt> for <tt>x</tt> in the above
expression:
<div class="p"><!----></div>
<table border="0"><tr><td></td><td><table border="0"><tr><td></td><td width="1000">
<pre>
(%i3) %o2, x=5/z;
2 3 2 3
135 y 675 y 225 y 2250 y 125 5625 y 1875 y
(%o3) ------ + ------ + ----- + ------- + --- + ------- + ------
z 2 2 3 3 4 4
z z z z z z
2 3
9375 y 15625 y 3
+ ------- + -------- + 27 y
5 6
z z
</pre>
</td></tr></table><!--vbox-->
</td><td></td></tr></table><!--hboxt-->The Maxima function <tt>ratsimp</tt> will place this over a common denominator:
<pre>
(%i4) ratsimp (%);
3 6 2 5 3 4
(%o4) (27 y z + 135 y z + (675 y + 225 y) z
2 3 3 2 2
+ (2250 y + 125) z + (5625 y + 1875 y) z + 9375 y z
3 6
+ 15625 y )/z
</pre>
Expressions may also be <tt>factor</tt>ed:
<pre>
(%i5) factor (%);
2 3
(3 y z + 5 z + 25 y)
(%o5) ----------------------
6
z
</pre>
Maxima can obtain exact solutions to systems of nonlinear algebraic
equations. In this example we <tt>solve</tt> three equations in the
three unknowns <tt>a</tt>, <tt>b</tt>, <tt>c</tt>:
<pre>
(%i6) a + b*c = 1;
(%o6) b c + a = 1
(%i7) b - a*c = 0;
(%o7) b - a c = 0
(%i8) a + b = 5;
(%o8) b + a = 5
(%i9) solve ([%o6, %o7, %o8], [a, b, c]);
25 sqrt(79) %i + 25 5 sqrt(79) %i + 5
(%o9) [[a = -------------------, b = -----------------,
6 sqrt(79) %i - 34 sqrt(79) %i + 11
sqrt(79) %i + 1 25 sqrt(79) %i - 25
c = ---------------], [a = -------------------,
10 6 sqrt(79) %i + 34
5 sqrt(79) %i - 5 sqrt(79) %i - 1
b = -----------------, c = - ---------------]]
sqrt(79) %i - 11 10
</pre>
Note that the display consists of a "list", i.e., some expression
contained between two brackets <tt>[ ... ]</tt>, which itself contains
two lists. Each of the latter contain a distinct solution to the
simultaneous equations.
<div class="p"><!----></div>
Trigonometric identities are easy to manipulate in Maxima. The
function <tt>trigexpand</tt> uses the sum-of-angles formulas to make the
argument inside each trig function as simple as possible:
<pre>
(%i10) sin(u + v) * cos(u)^3;
3
(%o10) cos (u) sin(v + u)
(%i11) trigexpand (%);
3
(%o11) cos (u) (cos(u) sin(v) + sin(u) cos(v))
</pre>
The function <tt>trigreduce</tt>, on the other hand, converts an
expression into a form which is a sum of terms, each of which contains
only a single <tt>sin</tt> or <tt>cos</tt>:
<pre>
(%i12) trigreduce (%o10);
sin(v + 4 u) + sin(v - 2 u) 3 sin(v + 2 u) + 3 sin(v)
(%o12) --------------------------- + -------------------------
8 8
</pre>
The functions <tt>realpart</tt> and <tt>imagpart</tt> will return the real
and imaginary parts of a complex expression:
<pre>
(%i13) w: 3 + k*%i;
(%o13) %i k + 3
(%i14) w^2 * %e^w;
2 %i k + 3
(%o14) (%i k + 3) %e
(%i15) realpart (%);
3 2 3
(%o15) %e (9 - k ) cos(k) - 6 %e k sin(k)
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc5">
5</a> Calculus <a name="sec:calculus">
</a></h2>
<div class="p"><!----></div>
Maxima can compute derivatives and integrals, expand in Taylor series,
take limits, and obtain exact solutions to ordinary differential
equations. We begin by defining the symbol <tt>f</tt> to be the
following function of <tt>x</tt>:
<pre>
(%i1) f: x^3 * %e^(k*x) * sin(w*x);
3 k x
(%o1) x %e sin(w x)
</pre>
We compute the derivative of <tt>f</tt> with respect to <tt>x</tt>:
<pre>
(%i2) diff (f, x);
3 k x 2 k x
(%o2) k x %e sin(w x) + 3 x %e sin(w x)
3 k x
+ w x %e cos(w x)
</pre>
Now we find the indefinite integral of <tt>f</tt> with respect to <tt>x</tt>:
<pre>
(%i3) integrate (f, x);
6 3 4 5 2 7 3
(%o3) (((k w + 3 k w + 3 k w + k ) x
6 2 4 4 2 6 2
+ (3 w + 3 k w - 3 k w - 3 k ) x
4 3 2 5 4 2 2 4
+ (- 18 k w - 12 k w + 6 k ) x - 6 w + 36 k w - 6 k )
k x 7 2 5 4 3 6 3
%e sin(w x) + ((- w - 3 k w - 3 k w - k w) x
5 3 3 5 2
+ (6 k w + 12 k w + 6 k w) x
5 2 3 4 3 3 k x
+ (6 w - 12 k w - 18 k w) x - 24 k w + 24 k w) %e
8 2 6 4 4 6 2 8
cos(w x))/(w + 4 k w + 6 k w + 4 k w + k )
</pre>
A slight change in syntax gives definite integrals:
<pre>
(%i4) integrate (1/x^2, x, 1, inf);
(%o4) 1
(%i5) integrate (1/x, x, 0, inf);
defint: integral is divergent.
-- an error. To debug this try: debugmode(true);
</pre>
Next we define the simbol <tt>g</tt> in terms of <tt>f</tt> (previously
defined in <tt>%i1</tt>) and the hyperbolic sine function, and find its
Taylor series expansion (up to, say, order 3 terms) about the point
<tt>x = 0</tt>:
<div class="p"><!----></div>
<table border="0"><tr><td></td><td><table border="0"><tr><td></td><td width="1000">
<pre>
(%i6) g: f / sinh(k*x)^4;
3 k x
x %e sin(w x)
(%o6) -----------------
4
sinh (k x)
(%i7) taylor (g, x, 0, 3);
2 3 2 2 3 3
w w x (w k + w ) x (3 w k + w ) x
(%o7)/T/ -- + --- - -------------- - ---------------- + . . .
4 3 4 3
k k 6 k 6 k
</pre>
</td></tr></table><!--vbox-->
</td><td></td></tr></table><!--hboxt-->The limit of <tt>g</tt> as <tt>x</tt> goes to 0 is computed as follows:
<pre>
(%i8) limit (g, x, 0);
w
(%o8) --
4
k
</pre>
Maxima also permits derivatives to be represented in unevaluated form
(note the quote):
<pre>
(%i9) 'diff (y, x);
dy
(%o9) --
dx
</pre>
The quote operator in <tt>(%i9)</tt> means "do not evaluate". Without
it, Maxima would have obtained 0:
<pre>
(%i10) diff (y, x);
(%o10) 0
</pre>
Using the quote operator we can write differential equations:
<pre>
(%i11) 'diff (y, x, 2) + 'diff (y, x) + y;
2
d y dy
(%o11) --- + -- + y
2 dx
dx
</pre>
Maxima's <tt>ode2</tt> function can solve first and second order ODE's:
<pre>
(%i12) ode2 (%o11, y, x);
- x/2 sqrt(3) x sqrt(3) x
(%o12) y = %e (%k1 sin(---------) + %k2 cos(---------))
2 2
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc6">
6</a> Matrix calculations <a name="sec:matrix">
</a></h2>
<div class="p"><!----></div>
Maxima can compute the determinant, inverse and eigenvalues and
eigenvectors of matrices which have symbolic elements (i.e., elements
which involve algebraic variables.) We begin by entering a matrix <tt>
m</tt> element by element:
<pre>
(%i1) m: entermatrix (3, 3);
Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General
Answer 1, 2, 3 or 4 :
4;
Row 1 Column 1:
0;
Row 1 Column 2:
1;
Row 1 Column 3:
a;
Row 2 Column 1:
1;
Row 2 Column 2:
0;
Row 2 Column 3:
1;
Row 3 Column 1:
1;
Row 3 Column 2:
1;
Row 3 Column 3:
0;
Matrix entered.
[ 0 1 a ]
[ ]
(%o1) [ 1 0 1 ]
[ ]
[ 1 1 0 ]
</pre>
Next we find its transpose, determinant and inverse:
<pre>
(%i2) transpose (m);
[ 0 1 1 ]
[ ]
(%o2) [ 1 0 1 ]
[ ]
[ a 1 0 ]
(%i3) determinant (m);
(%o3) a + 1
(%i4) invert (m), detout;
[ - 1 a 1 ]
[ ]
[ 1 - a a ]
[ ]
[ 1 1 - 1 ]
(%o4) -----------------
a + 1
</pre>
In <tt>(%i4)</tt>, the modifier <tt>detout</tt> keeps the determinant
outside the inverse. As a check, we multiply <tt>m</tt> by its inverse
(note the use of the period to represent matrix multiplication):
<pre>
(%i5) m . %o4;
[ - 1 a 1 ]
[ ]
[ 1 - a a ]
[ 0 1 a ] [ ]
[ ] [ 1 1 - 1 ]
(%o5) [ 1 0 1 ] . -----------------
[ ] a + 1
[ 1 1 0 ]
(%i6) expand (%);
[ a 1 ]
[ ----- + ----- 0 0 ]
[ a + 1 a + 1 ]
[ ]
[ a 1 ]
(%o6) [ 0 ----- + ----- 0 ]
[ a + 1 a + 1 ]
[ ]
[ a 1 ]
[ 0 0 ----- + ----- ]
[ a + 1 a + 1 ]
(%i7) factor (%);
[ 1 0 0 ]
[ ]
(%o7) [ 0 1 0 ]
[ ]
[ 0 0 1 ]
</pre>
In order to find the eigenvalues and eigenvectors of <tt>m</tt>, we use the function <tt>
eigenvectors</tt>:
<div class="p"><!----></div>
<table border="0"><tr><td></td><td><table border="0"><tr><td></td><td width="1000">
<pre>
(%i8) eigenvectors (m);
sqrt(4 a + 5) - 1 sqrt(4 a + 5) + 1
(%o8) [[[- -----------------, -----------------, - 1],
2 2
sqrt(4 a + 5) - 1 sqrt(4 a + 5) - 1
[1, 1, 1]], [[[1, - -----------------, - -----------------]],
2 a + 2 2 a + 2
sqrt(4 a + 5) + 1 sqrt(4 a + 5) + 1
[[1, -----------------, -----------------]], [[1, - 1, 0]]]]
2 a + 2 2 a + 2
</pre>
In <tt>%o8</tt>, the first triple gives the eigenvalues of <tt>m</tt> and
the next gives their respective multiplicities (here each is
unrepeated). The next three triples give the corresponding
eigenvectors of <tt>m</tt>. In order to extract from this expression
one of these eigenvectors, we may use the <tt>part</tt> function:
<pre>
(%i9) part (%o23, 2, 1, 1);
sqrt(4 a + 5) - 1 sqrt(4 a + 5) - 1
(%o9) [1, - -----------------, - -----------------]
2 a + 2 2 a + 2
</pre>
</td></tr></table><!--vbox-->
</td><td></td></tr></table><!--hboxt--> <h2><a name="tth_sEc7">
7</a> Programming in Maxima <a name="sec:programming">
</a></h2>
<div class="p"><!----></div>
So far, we have used Maxima in the interactive mode, rather like a
calculator. However, for computations which involve a repetitive
sequence of commands, it is better to execute a program. Here we
present a short sample program to calculate the critical points of a
function <tt>f</tt> of two variables <tt>x</tt> and <tt>y</tt>. The program
cues the user to enter the function <tt>f</tt>, then it computes the
partial derivatives <tt>f</tt><sub><tt>x</tt></sub> and <tt>f</tt><sub><tt>y</tt></sub>, and then it
uses the Maxima command <tt>solve</tt> to obtain solutions to
<tt>f</tt><sub><tt>x</tt></sub><tt> = </tt><tt>f</tt><sub><tt>y</tt></sub><tt> = </tt><tt>0</tt>. The program is written outside of Maxima
with a text editor, and then loaded into Maxima with the <tt>batch</tt>
command. Here is the program listing:
<pre>
/* --------------------------------------------------------------------------
this is file critpts.max:
as you can see, comments in maxima are like comments in C
Nelson Luis Dias, nldias@simepar.br
created 20000707
updated 20000707
--------------------------------------------------------------------------- */
critpts():=(
print("program to find critical points"),
/* ---------------------------------------------------------------------------
asks for a function
--------------------------------------------------------------------------- */
f:read("enter f(x,y)"),
/* ---------------------------------------------------------------------------
echoes it, to make sure
--------------------------------------------------------------------------- */
print("f = ",f),
/* ---------------------------------------------------------------------------
produces a list with the two partial derivatives of f
--------------------------------------------------------------------------- */
eqs:[diff(f,x),diff(f,y)],
/* ---------------------------------------------------------------------------
produces a list of unknowns
--------------------------------------------------------------------------- */
unk:[x,y],
/* ---------------------------------------------------------------------------
solves the system
--------------------------------------------------------------------------- */
solve(eqs,unk)
)$
</pre>
The program (which is actually a function with no argument) is called
<tt>critpts</tt>. Each line is a valid Maxima command which could be
executed from the keyboard, and which is separated by the next command
by a comma. The partial derivatives are stored in a variable named
<tt>eqs</tt>, and the unknowns are stored in <tt>unk</tt>. Here is a sample
run:
<pre>
(%i1) batch ("critpts.max");
batching #p/home/robert/tmp/maxima-clean/maxima/critpts.max
(%i2) critpts() := (print("program to find critical points"),
f : read("enter f(x,y)"), print("f = ", f),
eqs : [diff(f, x), diff(f, y)], unk : [x, y], solve(eqs, unk))
(%i3) critpts ();
program to find critical points
enter f(x,y)
%e^(x^3 + y^2)*(x + y);
2 3
y + x
f = (y + x) %e
(%o3) [[x = .4588955685487001 %i + .3589790871086935,
y = .4942017368275118 %i - .1225787367783657],
[x = .3589790871086935 - .4588955685487001 %i,
y = - .4942017368275118 %i - .1225787367783657],
[x = .4187542327234816 %i - .6923124204420268,
y = 0.455912070111699 - .8697262692814121 %i],
[x = - .4187542327234816 %i - .6923124204420268,
y = .8697262692814121 %i + 0.455912070111699]]
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc8">
8</a> A partial list of Maxima functions</h2>
<div class="p"><!----></div>
See the Maxima reference manual <tt>doc/html/maxima_toc.html</tt> (under
the main Maxima installation directory). From Maxima itself, you can
use <tt>describe(<i>function name</i>)</tt>.
<div class="p"><!----></div>
<dl compact="compact">
<dt><b><tt>allroots(a)</tt></b></dt>
<dd> Finds all the (generally complex) roots of
the polynomial equation <tt>A</tt>, and lists them in <tt>numer</tt>ical
format (i.e. to 16 significant figures).</dd>
<dt><b><tt>append(a,b)</tt></b></dt>
<dd> Appends the list <tt>b</tt> to the list <tt>a</tt>,
resulting in a single list.</dd>
<dt><b><tt>batch(a)</tt></b></dt>
<dd> Loads and runs a program with filename <tt>a</tt>.</dd>
<dt><b><tt>coeff(a,b,c)</tt></b></dt>
<dd> Gives the coefficient of <tt>b</tt> raised to
the power <tt>c</tt> in expression <tt>a</tt>.</dd>
<dt><b><tt>concat(a,b)</tt></b></dt>
<dd> Creates the symbol <tt>ab</tt>.</dd>
<dt><b><tt>cons(a,b)</tt></b></dt>
<dd> Adds <tt>a</tt> to the list <tt>b</tt> as its first element.</dd>
<dt><b><tt>demoivre(a)</tt></b></dt>
<dd> Transforms all complex exponentials in <tt>
a</tt> to their trigonometric equivalents.</dd>
<dt><b><tt>denom(a)</tt></b></dt>
<dd> Gives the denominator of <tt>a</tt>.</dd>
<dt><b><tt>depends(a,b)</tt></b></dt>
<dd> Declares <tt>a</tt> to be a function of <tt>
b</tt>. This is useful for writing unevaluated derivatives, as in
specifying differential equations.</dd>
<dt><b><tt>desolve(a,b)</tt></b></dt>
<dd> Attempts to solve a linear system <tt>a</tt> of
ODE's for unknowns <tt>b</tt> using Laplace transforms.</dd>
<dt><b><tt>determinant(a)</tt></b></dt>
<dd> Returns the determinant of the square
matrix <tt>a</tt>.</dd>
<dt><b><tt>diff(a,b1,c1,b2,c2,...,bn,cn)</tt></b></dt>
<dd> Gives the mixed partial
derivative of <tt>a</tt> with respect to each <tt>bi</tt>, <tt>ci</tt> times.
For brevity, <tt>diff(a,b,1)</tt> may be represented by <tt>
diff(a,b)</tt>. <tt>'diff(...)</tt> represents the unevaluated
derivative, useful in specifying a differential equation.</dd>
<dt><b><tt>eigenvalues(a)</tt></b></dt>
<dd> Returns two lists, the first being the
eigenvalues of the square matrix <tt>a</tt>, and the second being their
respective multiplicities.</dd>
<dt><b><tt>eigenvectors(a)</tt></b></dt>
<dd> Does everything that <tt>eigenvalues</tt>
does, and adds a list of the eigenvectors of <tt>a</tt>.</dd>
<dt><b><tt>entermatrix(a,b)</tt></b></dt>
<dd> Cues the user to enter an <tt>a</tt> × <tt>b</tt> matrix, element by element.</dd>
<dt><b><tt>ev(a,b1,b2,...,bn)</tt></b></dt>
<dd> Evaluates <tt>a</tt> subject to the
conditions <tt>bi</tt>. In particular the <tt>bi</tt> may be equations,
lists of equations (such as that returned by <tt>solve</tt>), or
assignments, in which cases <tt>ev</tt> "plugs" the <tt>bi</tt> into
<tt>a</tt>. The <tt>Bi</tt> may also be words such as <tt>numer</tt> (in
which case the result is returned in numerical format), <tt>detout</tt>
(in which case any matrix inverses in <tt>a</tt> are performed with the
determinant factored out), or <tt>diff</tt> (in which case all
differentiations in <tt>a</tt> are evaluated, i.e., <tt>'diff</tt> in <tt>
a</tt> is replaced by <tt>diff</tt>). For brevity in a manual command
(i.e., not inside a user-defined function), the <tt>ev</tt> may be
dropped, shortening the syntax to <tt>a,b1,b2,...,bn</tt>.</dd>
<dt><b><tt>expand(a)</tt></b></dt>
<dd> Algebraically expands <tt>a</tt>. In particular
multiplication is distributed over addition.</dd>
<dt><b><tt>exponentialize(a)</tt></b></dt>
<dd> Transforms all trigonometric functions
in <tt>a</tt> to their complex exponential equivalents.</dd>
<dt><b><tt>factor(a)</tt></b></dt>
<dd> Factors <tt>a</tt>.</dd>
<dt><b><tt>freeof(a,b)</tt></b></dt>
<dd> Is true if the variable <tt>a</tt> is not part
of the expression <tt>b</tt>.</dd>
<dt><b><tt>grind(a)</tt></b></dt>
<dd> Displays a variable or function <tt>a</tt> in a
compact format. When used with <tt>writefile</tt> and an editor
outside of Maxima, it offers a scheme for producing <tt>batch</tt>
files which include Maxima-generated expressions.</dd>
<dt><b><tt>ident(a)</tt></b></dt>
<dd> Returns an <tt>a</tt> × <tt>a</tt>
identity matrix.</dd>
<dt><b><tt>imagpart(a)</tt></b></dt>
<dd> Returns the imaginary part of <tt>a</tt>.</dd>
<dt><b><tt>integrate(a,b)</tt></b></dt>
<dd> Attempts to find the indefinite integral
of <tt>a</tt> with respect to <tt>b</tt>.</dd>
<dt><b><tt>integrate(a,b,c,d)</tt></b></dt>
<dd> Attempts to find the indefinite
integral of <tt>a</tt> with respect to <tt>b</tt>. taken from
<tt>b</tt><tt>=</tt><tt>c</tt> to <tt>b</tt><tt>=</tt><tt>d</tt>. The limits of integration <tt>c</tt>
and <tt>d</tt> may be taken is <tt>inf</tt> (positive infinity) of <tt>
minf</tt> (negative infinity).</dd>
<dt><b><tt>invert(a)</tt></b></dt>
<dd> Computes the inverse of the square matrix <tt>
a</tt>.</dd>
<dt><b><tt>kill(a)</tt></b></dt>
<dd> Removes the variable <tt>a</tt> with all its
assignments and properties from the current Maxima environment.</dd>
<dt><b><tt>limit(a,b,c)</tt></b></dt>
<dd> Gives the limit of expression <tt>a</tt> as
variable <tt>b</tt> approaches the value <tt>c</tt>. The latter may be
taken as <tt>inf</tt> of <tt>minf</tt> as in <tt>integrate</tt>.</dd>
<dt><b><tt>lhs(a)</tt></b></dt>
<dd> Gives the left-hand side of the equation <tt>a</tt>.</dd>
<dt><b><tt>loadfile(a)</tt></b></dt>
<dd> Loads a disk file with filename <tt>a</tt> from
the current default directory. The disk file must be in the proper
format (i.e. created by a <tt>save</tt> command).</dd>
<dt><b><tt>makelist(a,b,c,d)</tt></b></dt>
<dd> Creates a list of <tt>a</tt>'s (each of
which presumably depends on <tt>b</tt>), concatenated from
<tt>b</tt><tt>=</tt><tt>c</tt> to <tt>b</tt><tt>=</tt><tt>d</tt></dd>
<dt><b><tt>map(a,b)</tt></b></dt>
<dd> Maps the function <tt>a</tt> onto the
subexpressions of <tt>b</tt>.</dd>
<dt><b><tt>matrix(a1,a2,...,an)</tt></b></dt>
<dd> Creates a matrix consisting of the rows <tt>ai</tt>, where each
row <tt>ai</tt> is a list of <tt>m</tt> elements, <tt>[b1, b2, ..., bm]</tt>.</dd>
<dt><b><tt>num(a)</tt></b></dt>
<dd> Gives the numerator of <tt>a</tt>.</dd>
<dt><b><tt>ode2(a,b,c)</tt></b></dt>
<dd> Attempts to solve the first- or second-order
ordinary differential equation <tt>a</tt> for <tt>b</tt> as a function of
<tt>c</tt>.</dd>
<dt><b><tt>part(a,b1,b2,...,bn)</tt></b></dt>
<dd> First takes the <tt>b1</tt>th part
of <tt>a</tt>, then the <tt>b2</tt>th part of that, and so on.</dd>
<dt><b><tt>playback(a)</tt></b></dt>
<dd> Displays the last <tt>a</tt> (an integer)
labels and their associated expressions. If <tt>a</tt> is omitted,
all lines are played back. See the Manual for other options.</dd>
<dt><b><tt>ratsimp(a)</tt></b></dt>
<dd> Simplifies <tt>a</tt> and returns a quotient of
two polynomials.</dd>
<dt><b><tt>realpart(a)</tt></b></dt>
<dd> Returns the real part of <tt>a</tt>.</dd>
<dt><b><tt>rhs(a)</tt></b></dt>
<dd> Gives the right-hand side of the equation <tt>a</tt>.</dd>
<dt><b><tt>save(a,b1,b2,..., bn)</tt></b></dt>
<dd> Creates a disk file with
filename <tt>a</tt> in the current default directory, of variables,
functions, or arrays <tt>bi</tt>. The format of the file permits it to
be reloaded into Maxima using the <tt>loadfile</tt> command.
Everything (including labels) may be <tt>save</tt>d by taking <tt>b1</tt>
equal to <tt>all</tt>.</dd>
<dt><b><tt>solve(a,b)</tt></b></dt>
<dd> Attempts to solve the algebraic equation <tt>
a</tt> for the unknown <tt>b</tt>. A list of solution equations is
returned. For brevity, if <tt>a</tt> is an equation of the form
<tt>c</tt><tt> = </tt><tt>0</tt>, it may be abbreviated simply by the expression
<tt>c</tt>.</dd>
<dt><b><tt>string(a)</tt></b></dt>
<dd> Converts <tt>a</tt> to Maxima's linear notation
(similar to Fortran's) just as if it had been typed in and puts <tt>
a</tt> into the buffer for possible editing. The <tt>string</tt>'ed
expression should not be used in a computation.</dd>
<dt><b><tt>stringout(a,b1,b2,...,bn)</tt></b></dt>
<dd> Creates a disk file with
filename <tt>a</tt> in the current default directory, of variables
(e.g. labels) <tt>bi</tt>. The file is in a text format and is not
reloadable into Maxima. However the strungout expressions can be
incorporated into a Fortran, Basic or C program with a minimum of
editing.</dd>
<dt><b><tt>subst(a,b,c)</tt></b></dt>
<dd> Substitutes <tt>a</tt> for <tt>b</tt> in <tt>c</tt>.</dd>
<dt><b><tt>taylor(a,b,c,d)</tt></b></dt>
<dd> Expands <tt>a</tt> in a Taylor series in
<tt>b</tt> about <tt>b</tt><tt>=</tt><tt>c</tt>, up to and including the term
<tt>(</tt><tt>b</tt><font face="symbol">-</font
><tt>c</tt><tt>)</tt><sup><tt>d</tt></sup>. Maxima also supports Taylor expansions in more
than one independent variable; see the Manual for details.</dd>
<dt><b><tt>transpose(a)</tt></b></dt>
<dd> Gives the transpose of the matrix <tt>a</tt>.</dd>
<dt><b><tt>trigexpand(a)</tt></b></dt>
<dd> Is a trig simplification function which
uses the sum-of-angles formulas to simplify the arguments of
individual <tt>sin</tt>'s or <tt>cos</tt>'s. For example, <tt>
trigexpand(sin(x+y))</tt> gives <tt>cos(x) sin(y) + sin(x) cos(y)</tt>.</dd>
<dt><b><tt>trigreduce(a)</tt></b></dt>
<dd> Is a trig simplification function which
uses trig identities to convert products and powers of <tt>sin</tt> and
<tt>cos</tt> into a sum of terms, each of which contains only a single
<tt>sin</tt> or <tt>cos</tt>. For example, <tt>trigreduce(sin(x)^2)</tt>
gives <tt>(1 - cos(2x))/2</tt>.</dd>
<dt><b><tt>trigsimp(a)</tt></b></dt>
<dd> Is a trig simplification function which
replaces <tt>tan</tt>, <tt>sec</tt>, etc., by their <tt>sin</tt> and <tt>
cos</tt> equivalents. It also uses the identity <tt>sin</tt><tt>(</tt><tt>)</tt><sup><tt>2</tt></sup> <tt>+</tt> <tt>cos</tt><tt>(</tt><tt>)</tt><sup><tt>2</tt></sup><tt> = </tt><tt>1</tt>.</dd>
</dl>
<div class="p"><!----></div>
<hr /><h3>Footnotes:</h3>
<div class="p"><!----></div>
<a name="tthFtNtAAB"></a><a href="#tthFrefAAB"><sup>1</sup></a>Adapted from "Perturbation Methods, Bifurcation Theory and Computer Algebra"
by Rand and Armbruster, Springer, 1987.
Adapted to <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> and HTML by Nelson L. Dias (nldias@simepar.br),
SIMEPAR Technological Institute and Federal University of Paraná, Brazil.
Updated by Robert Dodier, August 2005.
<br /><br /><hr /><small>File translated from
T<sub><font size="-1">E</font></sub>X
by <a href="http://hutchinson.belmont.ma.us/tth/">
T<sub><font size="-1">T</font></sub>H</a>,
version 3.85.<br />On 26 Apr 2010, 00:45.</small>
</html>
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