File: maxima_135.html

package info (click to toggle)
maxima-sage 5.45.1-2
links: PTS, VCS
area: main
in suites: bookworm, sid, trixie
size: 113,788 kB
sloc: lisp: 440,833; fortran: 14,665; perl: 14,369; tcl: 10,997; sh: 4,475; makefile: 2,520; ansic: 447; python: 262; xml: 59; awk: 37; sed: 17
file content (855 lines) | stat: -rwxr-xr-x 40,333 bytes
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>Maxima 5.45.1 Manual: Functions and Variables for Series</title>

<meta name="description" content="Maxima 5.45.1 Manual: Functions and Variables for Series">
<meta name="keywords" content="Maxima 5.45.1 Manual: Functions and Variables for Series">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="maxima_0.html#Top" rel="start" title="Top">
<link href="maxima_368.html#Function-and-Variable-Index" rel="index" title="Function and Variable Index">
<link href="maxima_toc.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="maxima_132.html#Sums-Products-and-Series" rel="up" title="Sums Products and Series">
<link href="maxima_136.html#Introduction-to-Fourier-series" rel="next" title="Introduction to Fourier series">
<link href="maxima_134.html#Introduction-to-Series" rel="previous" title="Introduction to Series">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
body {color: black; background: white;  margin-left: 8%; margin-right: 13%;
      font-family: "FreeSans", sans-serif}
h1 {font-size: 150%; font-family: "FreeSans", sans-serif}
h2 {font-size: 125%; font-family: "FreeSans", sans-serif}
h3 {font-size: 100%; font-family: "FreeSans", sans-serif}
a[href] {color: rgb(0,0,255); text-decoration: none;}
a[href]:hover {background: rgb(220,220,220);}
div.textbox {border: solid; border-width: thin; padding-top: 1em;
    padding-bottom: 1em; padding-left: 2em; padding-right: 2em}
div.titlebox {border: none; padding-top: 1em; padding-bottom: 1em;
    padding-left: 2em; padding-right: 2em; background: rgb(200,255,255);
    font-family: sans-serif}
div.synopsisbox {
    border: none; padding-top: 1em; padding-bottom: 1em; padding-left: 2em;
    padding-right: 2em; background: rgb(255,220,255);}
pre.example {border: 1px solid rgb(180,180,180); padding-top: 1em;
    padding-bottom: 1em; padding-left: 1em; padding-right: 1em;
    background-color: rgb(238,238,255)}
div.spacerbox {border: none; padding-top: 2em; padding-bottom: 2em}
div.image {margin: 0; padding: 1em; text-align: center}
div.categorybox {border: 1px solid gray; padding-top: 0px; padding-bottom: 0px;
    padding-left: 1em; padding-right: 1em; background: rgb(247,242,220)}
img {max-width:80%; max-height: 80%; display: block; margin-left: auto; margin-right: auto}

-->
</style>

<link rel="icon" href="figures/favicon.ico">

</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Functions-and-Variables-for-Series"></a>
<div class="header">
<p>
Next: <a href="maxima_136.html#Introduction-to-Fourier-series" accesskey="n" rel="next">Introduction to Fourier series</a>, Previous: <a href="maxima_134.html#Introduction-to-Series" accesskey="p" rel="previous">Introduction to Series</a>, Up: <a href="maxima_132.html#Sums-Products-and-Series" accesskey="u" rel="up">Sums Products and Series</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_368.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-Series-1"></a>
<h3 class="section">28.3 Functions and Variables for Series</h3>

<a name="cauchysum"></a><a name="Item_003a-Series_002fdefvr_002fcauchysum"></a><dl>
<dt><a name="index-cauchysum"></a>Option variable: <strong>cauchysum</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When multiplying together sums with <code>inf</code> as their upper limit,
if <code>sumexpand</code> is <code>true</code> and <code>cauchysum</code> is <code>true</code>
then the Cauchy product will be used rather than the usual
product.
In the Cauchy product the index of the inner summation is a
function of the index of the outer one rather than varying
independently.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) sumexpand: false$
(%i2) cauchysum: false$
</pre><pre class="example">(%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
                      inf         inf
                      ====        ====
                      \           \
(%o3)                ( &gt;    f(i))  &gt;    g(j)
                      /           /
                      ====        ====
                      i = 0       j = 0
</pre><pre class="example">(%i4) sumexpand: true$
(%i5) cauchysum: true$
</pre><pre class="example">(%i6) expand(s,0,0);
                 inf     i1
                 ====   ====
                 \      \
(%o6)             &gt;      &gt;     g(i1 - i2) f(i2)
                 /      /
                 ====   ====
                 i1 = 0 i2 = 0
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Sums-and-products">Sums and products</a>
</div></p></dd></dl>

<a name="deftaylor"></a><a name="Item_003a-Series_002fdeffn_002fdeftaylor"></a><dl>
<dt><a name="index-deftaylor"></a>Function: <strong>deftaylor</strong> <em>(<var>f_1</var>(<var>x_1</var>), <var>expr_1</var>, &hellip;, <var>f_n</var>(<var>x_n</var>), <var>expr_n</var>)</em></dt>
<dd>
<p>For each function <var>f_i</var> of one variable <var>x_i</var>, 
<code>deftaylor</code> defines <var>expr_i</var> as the Taylor series about zero.
<var>expr_i</var> is typically a polynomial in <var>x_i</var> or a summation;
more general expressions are accepted by <code>deftaylor</code> without complaint.
</p>
<p><code>powerseries (<var>f_i</var>(<var>x_i</var>), <var>x_i</var>, 0)</code>
returns the series defined by <code>deftaylor</code>.
</p>
<p><code>deftaylor</code> returns a list of the functions <var>f_1</var>, &hellip;, <var>f_n</var>.
<code>deftaylor</code> evaluates its arguments.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
(%o1)                          [f]
(%i2) powerseries (f(x), x, 0);
                      inf
                      ====      i1
                      \        x         2
(%o2)                  &gt;     -------- + x
                      /       i1    2
                      ====   2   i1!
                      i1 = 4
(%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
                      2         3          4
                     x    3073 x    12817 x
(%o3)/T/     1 + x + -- + ------- + -------- + . . .
                     2     18432     307200
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="maxtayorder"></a><a name="Item_003a-Series_002fdefvr_002fmaxtayorder"></a><dl>
<dt><a name="index-maxtayorder"></a>Option variable: <strong>maxtayorder</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>maxtayorder</code> is <code>true</code>, then during algebraic
manipulation of (truncated) Taylor series, <code>taylor</code> tries to retain
as many terms as are known to be correct.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="niceindices"></a><a name="Item_003a-Series_002fdeffn_002fniceindices"></a><dl>
<dt><a name="index-niceindices"></a>Function: <strong>niceindices</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Renames the indices of sums and products in <var>expr</var>.  <code>niceindices</code>
attempts to rename each index to the value of <code>niceindicespref[1]</code>, unless
that name appears in the summand or multiplicand, in which case
<code>niceindices</code> tries the succeeding elements of <code>niceindicespref</code> in
turn, until an unused variable is found.  If the entire list is exhausted,
additional indices are constructed by appending integers to the value of
<code>niceindicespref[1]</code>, e.g., <code>i0</code>, <code>i1</code>, <code>i2</code>, &hellip;
</p>
<p><code>niceindices</code> returns an expression.
<code>niceindices</code> evaluates its argument.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) niceindicespref;
(%o1)                  [i, j, k, l, m, n]
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                 inf    inf
                /===\   ====
                 ! !    \
(%o2)            ! !     &gt;      f(bar i j + foo)
                 ! !    /
                bar = 1 ====
                        foo = 1
(%i3) niceindices (%);
</pre><pre class="example">                     inf  inf
                    /===\ ====
                     ! !  \
(%o3)                ! !   &gt;    f(i j l + k)
                     ! !  /
                    l = 1 ====
                          k = 1
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Sums-and-products">Sums and products</a>
</div></p></dd></dl>

<a name="niceindicespref"></a><a name="Item_003a-Series_002fdefvr_002fniceindicespref"></a><dl>
<dt><a name="index-niceindicespref"></a>Option variable: <strong>niceindicespref</strong></dt>
<dd><p>Default value: <code>[i, j, k, l, m, n]</code>
</p>
<p><code>niceindicespref</code> is the list from which <code>niceindices</code>
takes the names of indices for sums and products.
</p>
<p>The elements of <code>niceindicespref</code> are typically names of variables,
although that is not enforced by <code>niceindices</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) niceindicespref: [p, q, r, s, t, u]$
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                 inf    inf
                /===\   ====
                 ! !    \
(%o2)            ! !     &gt;      f(bar i j + foo)
                 ! !    /
                bar = 1 ====
                        foo = 1
(%i3) niceindices (%);
                     inf  inf
                    /===\ ====
                     ! !  \
(%o3)                ! !   &gt;    f(i j q + p)
                     ! !  /
                    q = 1 ====
                          p = 1
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Sums-and-products">Sums and products</a>
</div></p></dd></dl>

<a name="nusum"></a><a name="Item_003a-Series_002fdeffn_002fnusum"></a><dl>
<dt><a name="index-nusum"></a>Function: <strong>nusum</strong> <em>(<var>expr</var>, <var>x</var>, <var>i_0</var>, <var>i_1</var>)</em></dt>
<dd>
<p>Carries out indefinite hypergeometric summation of <var>expr</var> with
respect to <var>x</var> using a decision procedure due to R.W. Gosper.
<var>expr</var> and the result must be expressible as products of integer powers,
factorials, binomials, and rational functions.
</p>
<p>The terms &quot;definite&quot;
and &quot;indefinite summation&quot; are used analogously to &quot;definite&quot; and
&quot;indefinite integration&quot;.
To sum indefinitely means to give a symbolic result
for the sum over intervals of variable length, not just e.g. 0 to
inf.  Thus, since there is no formula for the general partial sum of
the binomial series, <code>nusum</code> can&rsquo;t do it.
</p>
<p><code>nusum</code> and <code>unsum</code> know a little about sums and differences of
finite products.  See also <code><a href="#unsum">unsum</a></code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) nusum (n*n!, n, 0, n);

Dependent equations eliminated:  (1)
(%o1)                     (n + 1)! - 1
(%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
                     4        3       2              n
      2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
(%o2) ------------------------------------------------ - ------
                    693 binomial(2 n, n)                 3 11 7
(%i3) unsum (%, n);
                              4  n
                             n  4
(%o3)                   ----------------
                        binomial(2 n, n)
(%i4) unsum (prod (i^2, i, 1, n), n);
                    n - 1
                    /===\
                     ! !   2
(%o4)              ( ! !  i ) (n - 1) (n + 1)
                     ! !
                    i = 1
(%i5) nusum (%, n, 1, n);

Dependent equations eliminated:  (2 3)
                            n
                          /===\
                           ! !   2
(%o5)                      ! !  i  - 1
                           ! !
                          i = 1
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Sums-and-products">Sums and products</a>
</div></p></dd></dl>


<a name="pade"></a><a name="Item_003a-Series_002fdeffn_002fpade"></a><dl>
<dt><a name="index-pade"></a>Function: <strong>pade</strong> <em>(<var>taylor_series</var>, <var>numer_deg_bound</var>, <var>denom_deg_bound</var>)</em></dt>
<dd>
<p>Returns a list of
all rational functions which have the given Taylor series expansion
where the sum of the degrees of the numerator and the denominator is
less than or equal to the truncation level of the power series, i.e.
are &quot;best&quot; approximants, and which additionally satisfy the specified
degree bounds.
</p>
<p><var>taylor_series</var> is an univariate Taylor series.
<var>numer_deg_bound</var> and <var>denom_deg_bound</var>
are positive integers specifying degree bounds on
the numerator and denominator.
</p>
<p><var>taylor_series</var> can also be a Laurent series, and the degree
bounds can be <code>inf</code> which causes all rational functions whose total
degree is less than or equal to the length of the power series to be
returned.  Total degree is defined as <code><var>numer_deg_bound</var> +
<var>denom_deg_bound</var></code>.
Length of a power series is defined as
<code>&quot;truncation level&quot; + 1 - min(0, &quot;order of series&quot;)</code>.
</p>
<div class="example">
<pre class="example">(%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                              2    3
(%o1)/T/             1 + x + x  + x  + . . .
(%i2) pade (%, 1, 1);
                                 1
(%o2)                       [- -----]
                               x - 1
(%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                   + 387072*x^7 + 86016*x^6 - 1507328*x^5
                   + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                   + 67108864*x - 134217728)
       /134217728, x, 0, 10);
                    2    3       4       5       6        7
             x   3 x    x    15 x    23 x    21 x    189 x
(%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
             2    16    32   1024    2048    32768   65536

                                  8         9          10
                            5853 x    2847 x    83787 x
                          + ------- + ------- - --------- + . . .
                            4194304   8388608   134217728
(%i4) pade (t, 4, 4);
(%o4)                          []
</pre></div>

<p>There is no rational function of degree 4 numerator/denominator, with this
power series expansion.  You must in general have degree of the numerator and
degree of the denominator adding up to at least the degree of the power series,
in order to have enough unknown coefficients to solve.
</p>
<div class="example">
<pre class="example">(%i5) pade (t, 5, 5);
                     5                4                 3
(%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

                  2
 - 1619100813312 x  - 2176885157888 x - 2386516803584)

               5                 4                  3
/(47041365435 x  + 381702613848 x  + 1360678489152 x

                  2
 + 2856700692480 x  + 3370143559680 x + 2386516803584)]
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="powerseries"></a><a name="Item_003a-Series_002fdeffn_002fpowerseries"></a><dl>
<dt><a name="index-powerseries"></a>Function: <strong>powerseries</strong> <em>(<var>expr</var>, <var>x</var>, <var>a</var>)</em></dt>
<dd>
<p>Returns the general form of the power series expansion for <var>expr</var> in the 
variable <var>x</var> about the point <var>a</var> (which may be <code>inf</code> for infinity):
</p><div class="example">
<pre class="example">           inf
           ====
           \               n
            &gt;    b  (x - a)
           /      n
           ====
           n = 0
</pre></div>

<p>If <code>powerseries</code> is unable to expand <var>expr</var>,
<code>taylor</code> may give the first several terms of the series.
</p>
<p>When <code>verbose</code> is <code>true</code>,
<code>powerseries</code> prints progress messages.
</p>
<div class="example">
<pre class="example">(%i1) verbose: true$
(%i2) powerseries (log(sin(x)/x), x, 0);
can't expand 
                                 log(sin(x))
so we'll try again after applying the rule:
                                        d
                                      / -- (sin(x))
                                      [ dx
                        log(sin(x)) = i ----------- dx
                                      ]   sin(x)
                                      /
in the first simplification we have returned:
                             /
                             [
                             i cot(x) dx - log(x)
                             ]
                             /
                    inf
                    ====        i1  2 i1             2 i1
                    \      (- 1)   2     bern(2 i1) x
                     &gt;     ------------------------------
                    /                i1 (2 i1)!
                    ====
                    i1 = 1
(%o2)                -------------------------------------
                                      2
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="psexpand"></a><a name="Item_003a-Series_002fdefvr_002fpsexpand"></a><dl>
<dt><a name="index-psexpand"></a>Option variable: <strong>psexpand</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>psexpand</code> is <code>true</code>,
an extended rational function expression is displayed fully expanded.
The switch <code>ratexpand</code> has the same effect.
</p>
<p>When <code>psexpand</code> is <code>false</code>,
a multivariate expression is displayed just as in the rational function package.
</p>
<p>When <code>psexpand</code> is  <code>multi</code>,
then terms with the same total degree in the variables are grouped together.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Display-flags-and-variables">Display flags and variables</a>
</div></p></dd></dl>

<a name="revert"></a><a name="Item_003a-Series_002fdeffn_002frevert"></a><dl>
<dt><a name="index-revert"></a>Function: <strong>revert</strong> <em>(<var>expr</var>, <var>x</var>)</em></dt>
<dd><a name="Item_003a-Series_002fdeffn_002frevert2"></a></dd><dt><a name="index-revert2"></a>Function: <strong>revert2</strong> <em>(<var>expr</var>, <var>x</var>, <var>n</var>)</em></dt>
<dd>
<p>These functions return the reversion of <var>expr</var>, a Taylor series about zero
in the variable <var>x</var>.  <code>revert</code> returns a polynomial of degree equal to
the highest power in <var>expr</var>.  <code>revert2</code> returns a polynomial of degree
<var>n</var>, which may be greater than, equal to, or less than the degree of
<var>expr</var>.
</p>
<p><code>load (&quot;revert&quot;)</code> loads these functions.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) load (&quot;revert&quot;)$
(%i2) t: taylor (exp(x) - 1, x, 0, 6);
                   2    3    4    5     6
                  x    x    x    x     x
(%o2)/T/      x + -- + -- + -- + --- + --- + . . .
                  2    6    24   120   720
(%i3) revert (t, x);
               6       5       4       3       2
           10 x  - 12 x  + 15 x  - 20 x  + 30 x  - 60 x
(%o3)/R/ - --------------------------------------------
                                60
(%i4) ratexpand (%);
                     6    5    4    3    2
                    x    x    x    x    x
(%o4)             - -- + -- - -- + -- - -- + x
                    6    5    4    3    2
(%i5) taylor (log(x+1), x, 0, 6);
                    2    3    4    5    6
                   x    x    x    x    x
(%o5)/T/       x - -- + -- - -- + -- - -- + . . .
                   2    3    4    5    6
(%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
(%o6)                           0
(%i7) revert2 (t, x, 4);
                          4    3    2
                         x    x    x
(%o7)                  - -- + -- - -- + x
                         4    3    2
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taylor"></a><a name="Item_003a-Series_002fdeffn_002ftaylor"></a><dl>
<dt><a name="index-taylor"></a>Function: <strong>taylor</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>taylor</tt> (<var>expr</var>, <var>x</var>, <var>a</var>, <var>n</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>taylor</tt> (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, &hellip;], <var>a</var>, <var>n</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>taylor</tt> (<var>expr</var>, [<var>x</var>, <var>a</var>, <var>n</var>, 'asymp]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>taylor</tt> (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, &hellip;], [<var>a_1</var>, <var>a_2</var>, &hellip;], [<var>n_1</var>, <var>n_2</var>, &hellip;]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>taylor</tt> (<var>expr</var>, [<var>x_1</var>, <var>a_1</var>, <var>n_1</var>], [<var>x_2</var>, <var>a_2</var>, <var>n_2</var>], &hellip;)</em></dt>
<dd>
<p><code>taylor (<var>expr</var>, <var>x</var>, <var>a</var>, <var>n</var>)</code> expands the expression
<var>expr</var> in a truncated Taylor or Laurent series in the variable <var>x</var>
around the point <var>a</var>,
containing terms through <code>(<var>x</var> - <var>a</var>)^<var>n</var></code>.
</p>
<p>If <var>expr</var> is of the form <code><var>f</var>(<var>x</var>)/<var>g</var>(<var>x</var>)</code> and
<code><var>g</var>(<var>x</var>)</code> has no terms up to degree <var>n</var> then <code>taylor</code>
attempts to expand <code><var>g</var>(<var>x</var>)</code> up to degree <code>2 <var>n</var></code>.
If there are still no nonzero terms, <code>taylor</code> doubles the degree of the
expansion of <code><var>g</var>(<var>x</var>)</code> so long as the degree of the expansion is
less than or equal to <code><var>n</var> 2^taylordepth</code>.
</p>
<p><code>taylor (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, ...], <var>a</var>, <var>n</var>)</code>
returns a truncated power series 
of degree <var>n</var> in all variables <var>x_1</var>, <var>x_2</var>, &hellip;
about the point <code>(<var>a</var>, <var>a</var>, ...)</code>.
</p>
<p><code>taylor (<var>expr</var>, [<var>x_1</var>, <var>a_1</var>, <var>n_1</var>], [<var>x_2</var>,
<var>a_2</var>, <var>n_2</var>], ...)</code> returns a truncated power series in the variables
<var>x_1</var>, <var>x_2</var>, &hellip; about the point
<code>(<var>a_1</var>, <var>a_2</var>, ...)</code>, truncated at <var>n_1</var>, <var>n_2</var>, &hellip;
</p>
<p><code>taylor (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, ...], [<var>a_1</var>,
<var>a_2</var>, ...], [<var>n_1</var>, <var>n_2</var>, ...])</code> returns a truncated power series
in the variables <var>x_1</var>, <var>x_2</var>, &hellip; about the point
<code>(<var>a_1</var>, <var>a_2</var>, ...)</code>, truncated at <var>n_1</var>, <var>n_2</var>, &hellip;
</p>
<p><code>taylor (<var>expr</var>, [<var>x</var>, <var>a</var>, <var>n</var>, 'asymp])</code> returns an
expansion of <var>expr</var> in negative powers of <code><var>x</var> - <var>a</var></code>.
The highest order term is <code>(<var>x</var> - <var>a</var>)^<var>-n</var></code>.
</p>
<p>When <code>maxtayorder</code> is <code>true</code>, then during algebraic
manipulation of (truncated) Taylor series, <code>taylor</code> tries to retain
as many terms as are known to be correct.
</p>
<p>When <code>psexpand</code> is <code>true</code>,
an extended rational function expression is displayed fully expanded.
The switch <code>ratexpand</code> has the same effect.
When <code>psexpand</code> is <code>false</code>,
a multivariate expression is displayed just as in the rational function package.
When <code>psexpand</code> is  <code>multi</code>,
then terms with the same total degree in the variables are grouped together.
</p>
<p>See also the <code><a href="#taylor_005flogexpand">taylor_logexpand</a></code> switch for controlling expansion.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                           2             2
             (a + 1) x   (a  + 2 a + 1) x
(%o1)/T/ 1 + --------- - -----------------
                 2               8

                                   3      2             3
                               (3 a  + 9 a  + 9 a - 1) x
                             + -------------------------- + . . .
                                           48
(%i2) %^2;
                                    3
                                   x
(%o2)/T/           1 + (a + 1) x - -- + . . .
                                   6
(%i3) taylor (sqrt (x + 1), x, 0, 5);
                       2    3      4      5
                  x   x    x    5 x    7 x
(%o3)/T/      1 + - - -- + -- - ---- + ---- + . . .
                  2   8    16   128    256
(%i4) %^2;
(%o4)/T/                  1 + x + . . .
(%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
</pre><pre class="example">                         inf
                        /===\
                         ! !    i     2.5
                         ! !  (x  + 1)
                         ! !
                        i = 1
(%o5)                   -----------------
                              2
                             x  + 1
</pre><pre class="example">(%i6) ev (taylor(%, x,  0, 3), keepfloat);
                               2           3
(%o6)/T/    1 + 2.5 x + 3.375 x  + 6.5625 x  + . . .
(%i7) taylor (1/log (x + 1), x, 0, 3);
                               2       3
                 1   1   x    x    19 x
(%o7)/T/         - + - - -- + -- - ----- + . . .
                 x   2   12   24    720
(%i8) taylor (cos(x) - sec(x), x, 0, 5);
                                4
                           2   x
(%o8)/T/                - x  - -- + . . .
                               6
(%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
(%o9)/T/                    0 + . . .
(%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
                                               2          4
            1     1       11      347    6767 x    15377 x
(%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
             6      4        2   15120   604800    7983360
            x    2 x    120 x

                                                          + . . .
(%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
               2  2       4      2   4
              k  x    (3 k  - 4 k ) x
(%o11)/T/ 1 - ----- - ----------------
                2            24

                                    6       4       2   6
                               (45 k  - 60 k  + 16 k ) x
                             - -------------------------- + . . .
                                          720
(%i12) taylor ((x + 1)^n, x, 0, 4);
</pre><pre class="example">                      2       2     3      2         3
                    (n  - n) x    (n  - 3 n  + 2 n) x
(%o12)/T/ 1 + n x + ----------- + --------------------
                         2                 6

                               4      3       2         4
                             (n  - 6 n  + 11 n  - 6 n) x
                           + ---------------------------- + . . .
                                          24
</pre><pre class="example">(%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
               3                 2
              y                 y
(%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
              6                 2

                    3                       2
               y   y            2      1   y            3
          + (- - + -- + . . .) x  + (- - + -- + . . .) x  + . . .
               2   12                  6   12
(%i14) taylor (sin (y + x), [x, y], 0, 3);
                     3        2      2      3
                    x  + 3 y x  + 3 y  x + y
(%o14)/T/   y + x - ------------------------- + . . .
                                6
(%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
          1   y              1    1               1            2
(%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
          y   6               2   6                3
                             y                    y

                                           1            3
                                      + (- -- + . . .) x  + . . .
                                            4
                                           y
(%i16) taylor (1/sin (y + x), [x, y], 0, 3);
                             3         2       2        3
            1     x + y   7 x  + 21 y x  + 21 y  x + 7 y
(%o16)/T/ ----- + ----- + ------------------------------- + . . .
          x + y     6                   360
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taylordepth"></a><a name="Item_003a-Series_002fdefvr_002ftaylordepth"></a><dl>
<dt><a name="index-taylordepth"></a>Option variable: <strong>taylordepth</strong></dt>
<dd><p>Default value: 3
</p>
<p>If there are still no nonzero terms, <code>taylor</code> doubles the degree of the
expansion of <code><var>g</var>(<var>x</var>)</code> so long as the degree of the expansion is
less than or equal to <code><var>n</var> 2^taylordepth</code>.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taylorinfo"></a><a name="Item_003a-Series_002fdeffn_002ftaylorinfo"></a><dl>
<dt><a name="index-taylorinfo"></a>Function: <strong>taylorinfo</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Returns information about the Taylor series <var>expr</var>.
The return value is a list of lists.
Each list comprises the name of a variable,
the point of expansion, and the degree of the expansion.
</p>
<p><code>taylorinfo</code> returns <code>false</code> if <var>expr</var> is not a Taylor series.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                  2                       2
(%o1)/T/ - (y - a)  - 2 a (y - a) + (1 - a )

         2                        2
 + (1 - a  - 2 a (y - a) - (y - a) ) x

         2                        2   2
 + (1 - a  - 2 a (y - a) - (y - a) ) x

         2                        2   3
 + (1 - a  - 2 a (y - a) - (y - a) ) x  + . . .
(%i2) taylorinfo(%);
(%o2)               [[y, a, inf], [x, 0, 3]]
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taylorp"></a><a name="Item_003a-Series_002fdeffn_002ftaylorp"></a><dl>
<dt><a name="index-taylorp"></a>Function: <strong>taylorp</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Returns <code>true</code> if <var>expr</var> is a Taylor series,
and <code>false</code> otherwise.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Predicate-functions">Predicate functions</a>
&middot;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>


<a name="taylor_005flogexpand"></a><a name="Item_003a-Series_002fdefvr_002ftaylor_005flogexpand"></a><dl>
<dt><a name="index-taylor_005flogexpand"></a>Option variable: <strong>taylor_logexpand</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>taylor_logexpand</code> controls expansions of logarithms in
<code>taylor</code> series.
</p>
<p>When <code>taylor_logexpand</code> is <code>true</code>, all logarithms are expanded fully
so that zero-recognition problems involving logarithmic identities do not
disturb the expansion process.  However, this scheme is not always
mathematically correct since it ignores branch information.
</p>
<p>When <code>taylor_logexpand</code> is set to <code>false</code>, then the only expansion of
logarithms that occur is that necessary to obtain a formal power series.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
&middot;<a href="maxima_369.html#Category_003a-Exponential-and-logarithm-functions">Exponential and logarithm functions</a>
</div></p></dd></dl>

<a name="taylor_005forder_005fcoefficients"></a><a name="Item_003a-Series_002fdefvr_002ftaylor_005forder_005fcoefficients"></a><dl>
<dt><a name="index-taylor_005forder_005fcoefficients"></a>Option variable: <strong>taylor_order_coefficients</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>taylor_order_coefficients</code> controls the ordering of
coefficients in a Taylor series.
</p>
<p>When <code>taylor_order_coefficients</code> is <code>true</code>,
coefficients of taylor series are ordered canonically.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taylor_005fsimplifier"></a><a name="Item_003a-Series_002fdeffn_002ftaylor_005fsimplifier"></a><dl>
<dt><a name="index-taylor_005fsimplifier"></a>Function: <strong>taylor_simplifier</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Simplifies coefficients of the power series <var>expr</var>.
<code>taylor</code> calls this function.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taylor_005ftruncate_005fpolynomials"></a><a name="Item_003a-Series_002fdefvr_002ftaylor_005ftruncate_005fpolynomials"></a><dl>
<dt><a name="index-taylor_005ftruncate_005fpolynomials"></a>Option variable: <strong>taylor_truncate_polynomials</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>taylor_truncate_polynomials</code> is <code>true</code>,
polynomials are truncated based upon the input truncation levels.
</p>
<p>Otherwise,
polynomials input to <code>taylor</code> are considered to have infinite precision.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="taytorat"></a><a name="Item_003a-Series_002fdeffn_002ftaytorat"></a><dl>
<dt><a name="index-taytorat"></a>Function: <strong>taytorat</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Converts <var>expr</var> from <code>taylor</code> form to canonical rational expression
(CRE) form.  The effect is the same as <code>rat (ratdisrep (<var>expr</var>))</code>, but
faster.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
&middot;<a href="maxima_369.html#Category_003a-Rational-expressions">Rational expressions</a>
</div></p></dd></dl>

<a name="trunc"></a><a name="Item_003a-Series_002fdeffn_002ftrunc"></a><dl>
<dt><a name="index-trunc"></a>Function: <strong>trunc</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Annotates the internal representation of the general expression <var>expr</var>
so that it is displayed as if its sums were truncated Taylor series.
<var>expr</var> is not otherwise modified.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) expr: x^2 + x + 1;
                            2
(%o1)                      x  + x + 1
(%i2) trunc (expr);
                                2
(%o2)                  1 + x + x  + . . .
(%i3) is (expr = trunc (expr));
(%o3)                         true
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="unsum"></a><a name="Item_003a-Series_002fdeffn_002funsum"></a><dl>
<dt><a name="index-unsum"></a>Function: <strong>unsum</strong> <em>(<var>f</var>, <var>n</var>)</em></dt>
<dd>
<p>Returns the first backward difference
<code><var>f</var>(<var>n</var>) - <var>f</var>(<var>n</var> - 1)</code>.
Thus <code>unsum</code> in a sense is the inverse of <code>sum</code>.
</p>
<p>See also <code><a href="#nusum">nusum</a></code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) g(p) := p*4^n/binomial(2*n,n);
                                     n
                                  p 4
(%o1)               g(p) := ----------------
                            binomial(2 n, n)
(%i2) g(n^4);
                              4  n
                             n  4
(%o2)                   ----------------
                        binomial(2 n, n)
(%i3) nusum (%, n, 0, n);
                     4        3       2              n
      2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
(%o3) ------------------------------------------------ - ------
                    693 binomial(2 n, n)                 3 11 7
(%i4) unsum (%, n);
                              4  n
                             n  4
(%o4)                   ----------------
                        binomial(2 n, n)
</pre></div>

<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Sums-and-products">Sums and products</a>
</div></p></dd></dl>

<a name="verbose"></a><a name="Item_003a-Series_002fdefvr_002fverbose"></a><dl>
<dt><a name="index-verbose"></a>Option variable: <strong>verbose</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>verbose</code> is <code>true</code>,
<code>powerseries</code> prints progress messages.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Power-series">Power series</a>
</div></p></dd></dl>

<a name="Item_003a-Series_002fnode_002fIntroduction-to-Fourier-series"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_136.html#Introduction-to-Fourier-series" accesskey="n" rel="next">Introduction to Fourier series</a>, Previous: <a href="maxima_134.html#Introduction-to-Series" accesskey="p" rel="previous">Introduction to Series</a>, Up: <a href="maxima_132.html#Sums-Products-and-Series" accesskey="u" rel="up">Sums Products and Series</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_368.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>