File: maxima_338.html

package info (click to toggle)
maxima 5.47.0-9
links: PTS
area: main
in suites: forky, sid
size: 193,104 kB
sloc: lisp: 434,678; fortran: 14,665; tcl: 10,990; sh: 4,577; makefile: 2,763; ansic: 447; java: 328; python: 262; perl: 201; xml: 60; awk: 28; sed: 15; javascript: 2
file content (1049 lines) | stat: -rw-r--r-- 40,328 bytes
parent folder | download | duplicates (2)
<!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 5.47.0 Manual: Functions and Variables for orthogonal polynomials</title>

<meta name="description" content="Maxima 5.47.0 Manual: Functions and Variables for orthogonal polynomials">
<meta name="keywords" content="Maxima 5.47.0 Manual: Functions and Variables for orthogonal polynomials">
<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_toc.html#Top" rel="start" title="Top">
<link href="maxima_423.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_336.html#orthopoly_002dpkg" rel="up" title="orthopoly-pkg">
<link href="maxima_339.html#pslq_002dpkg" rel="next" title="pslq-pkg">
<link href="maxima_337.html#Introduction-to-orthogonal-polynomials" rel="previous" title="Introduction to orthogonal polynomials">
<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: 1em; padding-bottom: 1em;
    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">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6>"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Functions-and-Variables-for-orthogonal-polynomials"></a>
<div class="header">
<p>
Previous: <a href="maxima_337.html#Introduction-to-orthogonal-polynomials" accesskey="p" rel="previous">Introduction to orthogonal polynomials</a>, Up: <a href="maxima_336.html#orthopoly_002dpkg" accesskey="u" rel="up">orthopoly-pkg</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-orthogonal-polynomials-1"></a>
<h3 class="section">80.2 Functions and Variables for orthogonal polynomials</h3>

<a name="assoc_005flegendre_005fp"></a><a name="Item_003a-orthopoly_002fdeffn_002fassoc_005flegendre_005fp"></a><dl>
<dt><a name="index-assoc_005flegendre_005fp"></a>Function: <strong>assoc_legendre_p</strong> <em>(<var>n</var>, <var>m</var>, <var>x</var>)</em></dt>
<dd><p>The associated Legendre function of the first kind of degree <em>n</em> and
order <em>m</em>, 
\(P_{n}^{m}(z)\), is a solution of the differential equation:
</p>
$$
(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0
$$

<p>This is related to the Legendre polynomial, 
\(P_n(x)\) via
</p>
$$
P_n^m(x) = (-1)^m\left(1-x^2\right)^{m/2} {d^m\over dx^m} P_n(x)
$$

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.37</a>, <a href="https://personal.math.ubc.ca/~cbm/aands/page_334.htm">A&amp;S eqn 8.6.6</a>, and <a href="https://personal.math.ubc.ca/~cbm/aands/page_333.htm">A&amp;S eqn 8.2.5</a>.
</p>
<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) assoc_legendre_p(2,0,x);
                                                 2
                                        3 (1 - x)
(%o1)                   (- 3 (1 - x)) + ---------- + 1
                                            2
(%i2) factor(%);
                                      2
                                   3 x  - 1
(%o2)                              --------
                                      2
(%i3) factor(assoc_legendre_p(2,1,x));
                                              2
(%o3)                         - 3 x sqrt(1 - x )

(%i4) (-1)^1*(1-x^2)^(1/2)*diff(legendre_p(2,x),x);
                                                    2
(%o4)                   - (3 - 3 (1 - x)) sqrt(1 - x )

(%i5) factor(%);
                                              2
(%o5)                         - 3 x sqrt(1 - x )
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="assoc_005flegendre_005fq"></a><a name="Item_003a-orthopoly_002fdeffn_002fassoc_005flegendre_005fq"></a><dl>
<dt><a name="index-assoc_005flegendre_005fq"></a>Function: <strong>assoc_legendre_q</strong> <em>(<var>n</var>, <var>m</var>, <var>x</var>)</em></dt>
<dd><p>The associated Legendre function of the second kind of degree <em>n</em>
and order <em>m</em>, 
\(Q_{n}^{m}(z)\), is a solution of the differential equation:
</p>
$$
(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0
$$

<p>Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.
</p>
<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) assoc_legendre_q(0,0,x);
                                       x + 1
                                 log(- -----)
                                       x - 1
(%o1)                            ------------
                                      2
(%i2) assoc_legendre_q(1,0,x);
                                    x + 1
                              log(- -----) x - 2
                                    x - 1
(%o2)/R/                      ------------------
                                      2
(%i3) assoc_legendre_q(1,1,x);
(%o3)/R/ 
          x + 1            2   2               2            x + 1            2
    log(- -----) sqrt(1 - x ) x  - 2 sqrt(1 - x ) x - log(- -----) sqrt(1 - x )
          x - 1                                             x - 1
  - ---------------------------------------------------------------------------
                                        2
                                     2 x  - 2
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="chebyshev_005ft"></a><a name="Item_003a-orthopoly_002fdeffn_002fchebyshev_005ft"></a><dl>
<dt><a name="index-chebyshev_005ft"></a>Function: <strong>chebyshev_t</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The Chebyshev polynomial of the first kind of degree <em>n</em>, 
\(T_n(x).\)</p>

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.47</a>.
</p>
<p>The polynomials 
\(T_n(x)\) can be written in terms of a hypergeometric function:
</p>
$$
T_n(x) = {_{2}}F_{1}\left(-n, n; {1\over 2}; {1-x\over 2}\right)
$$

<p>The polynomials can also be defined in terms of the sum
</p>
$$
T_n(x) = {n\over 2} \sum_{r=0}^{\lfloor {n/2}\rfloor} {(-1)^r\over n-r} {n-r\choose k}(2x)^{n-2r}
$$

<p>or the Rodrigues formula
</p>
$$
T_n(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= 1/\sqrt{1-x^2} \cr
\kappa_n &= (-2)^n\left(1\over 2\right)_n
}
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) chebyshev_t(2,x);
                                                 2
(%o1)                   (- 4 (1 - x)) + 2 (1 - x)  + 1
(%i2) factor(%);
                                      2
(%o2)                              2 x  - 1
(%i3) factor(chebyshev_t(3,x));
                                       2
(%o3)                            x (4 x  - 3)
(%i4) factor(hgfred([-3,3],[1/2],(1-x)/2));
                                       2
(%o4)                            x (4 x  - 3)
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="chebyshev_005fu"></a><a name="Item_003a-orthopoly_002fdeffn_002fchebyshev_005fu"></a><dl>
<dt><a name="index-chebyshev_005fu"></a>Function: <strong>chebyshev_u</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The Chebyshev polynomial of the second kind of degree <em>n</em>, 
\(U_n(x)\).
</p>
<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.48</a>.
</p>
<p>The polynomials 
\(U_n(x)\) can be written in terms of a hypergeometric function:
</p>
$$
U_n(x) = (n+1)\; {_{2}F_{1}}\left(-n, n+2; {3\over 2}; {1-x\over 2}\right)
$$

<p>The polynomials can also be defined in terms of the sum
</p>
$$
U_n(x) = \sum_{r=0}^{\lfloor n/2 \rfloor} (-1)^r {n-r \choose r} (2x)^{n-2r}
$$

<p>or the Rodrigues formula
</p>
$$
U_n(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= \sqrt{1-x^2} \cr
\kappa_n &= {(-2)^n\left({3\over 2}\right)_n \over n+1}
}
$$<p>.
</p>
<div class="example">
<pre class="example">(%i1) chebyshev_u(2,x);
                                                  2
                            8 (1 - x)    4 (1 - x)
(%o1)                 3 ((- ---------) + ---------- + 1)
                                3            3
(%i2) expand(%);
                                      2
(%o2)                              4 x  - 1
(%i3) expand(chebyshev_u(3,x));
                                     3
(%o3)                             8 x  - 4 x
(%i4) expand(4*hgfred([-3,5],[3/2],(1-x)/2));
                                     3
(%o4)                             8 x  - 4 x
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="gen_005flaguerre"></a><a name="Item_003a-orthopoly_002fdeffn_002fgen_005flaguerre"></a><dl>
<dt><a name="index-gen_005flaguerre"></a>Function: <strong>gen_laguerre</strong> <em>(<var>n</var>, <var>a</var>, <var>x</var>)</em></dt>
<dd><p>The generalized Laguerre polynomial of degree <em>n</em>, 
\(L_n^{(\alpha)}(x)\).
</p>
<p>These can be defined by
</p>
$$
L_n^{(\alpha)}(x) = {n+\alpha \choose n}\; {_1F_1}(-n; \alpha+1; x)
$$

<p>The polynomials can also be defined by the sum
</p>
$$
L_n^{(\alpha)}(x) = \sum_{k=0}^n {(\alpha + k + 1)_{n-k} \over (n-k)! k!} (-x)^k
$$

<p>or the Rodrigues formula
</p>
$$
L_n^{(\alpha)}(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)x^n\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= e^{-x}x^{\alpha} \cr
\kappa_n &= n!
}
$$

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_780.htm">A&amp;S eqn 22.5.54</a>.
</p>
<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) gen_laguerre(1,k,x);
                                             x
(%o1)                         (k + 1) (1 - -----)
                                           k + 1
(%i2) gen_laguerre(2,k,x);
                                         2
                                        x            2 x
                 (k + 1) (k + 2) (--------------- - ----- + 1)
                                  (k + 1) (k + 2)   k + 1
(%o2)            ---------------------------------------------
                                       2
(%i3) binomial(2+k,2)*hgfred([-2],[1+k],x);
                                         2
                                        x            2 x
                 (k + 1) (k + 2) (--------------- - ----- + 1)
                                  (k + 1) (k + 2)   k + 1
(%o3)            ---------------------------------------------
                                       2

</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="hermite"></a><a name="Item_003a-orthopoly_002fdeffn_002fhermite"></a><dl>
<dt><a name="index-hermite"></a>Function: <strong>hermite</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The Hermite polynomial of degree <em>n</em>, 
\(H_n(x)\).
</p>
<p>These polynomials may be defined by a hypergeometric function
</p>
$$
H_n(x) = (2x)^n\; {_2F_0}\left(-{1\over 2} n, -{1\over 2}n+{1\over 2};;-{1\over x^2}\right)
$$

<p>or by the series
</p>
$$
H_n(x) = n! \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k(2x)^{n-2k} \over k! (n-2k)!}
$$

<p>or the Rodrigues formula
</p>
$$
H_n(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= e^{-{x^2/2}} \cr
\kappa_n &= (-1)^n
}
$$

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_780.htm">A&amp;S eqn 22.5.55</a>.
</p>
<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) hermite(3,x);
                                              2
                                           2 x
(%o1)                          - 12 x (1 - ----)
                                            3
(%i2) expand(%);
                                     3
(%o2)                             8 x  - 12 x
(%i3) expand(hermite(4,x));
                                  4       2
(%o3)                         16 x  - 48 x  + 12
(%i4) expand((2*x)^4*hgfred([-2,-2+1/2],[],-1/x^2));
                                  4       2
(%o4)                         16 x  - 48 x  + 12
(%i5) expand(4!*sum((-1)^k*(2*x)^(4-2*k)/(k!*(4-2*k)!),k,0,floor(4/2)));
                                  4       2
(%o5)                         16 x  - 48 x  + 12
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="intervalp"></a><a name="Item_003a-orthopoly_002fdeffn_002fintervalp"></a><dl>
<dt><a name="index-intervalp"></a>Function: <strong>intervalp</strong> <em>(<var>e</var>)</em></dt>
<dd><p>Return <code>true</code> if the input is an interval and return false if it isn&rsquo;t. 
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Predicate-functions">Predicate functions</a>
&middot;</div>
</dd></dl>

<a name="jacobi_005fp"></a><a name="Item_003a-orthopoly_002fdeffn_002fjacobi_005fp"></a><dl>
<dt><a name="index-jacobi_005fp"></a>Function: <strong>jacobi_p</strong> <em>(<var>n</var>, <var>a</var>, <var>b</var>, <var>x</var>)</em></dt>
<dd><p>The Jacobi polynomial, 
\(P_n^{(a,b)}(x)\).
</p>
<p>The Jacobi polynomials are actually defined for all
<em>a</em> and <em>b</em>; however, the Jacobi polynomial
weight <em>(1 - x)^a (1 + x)^b</em> isn&rsquo;t integrable
for 
\(a \le -1\) or 
\(b \le -1\). 
</p>
<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.42</a>.
</p>
<p>The polynomial may be defined in terms of hypergeometric functions:
</p>
$$
P_n^{(a,b)}(x) = {n+a\choose n} {_1F_2}\left(-n, n + a + b + 1; a+1; {1-x\over 2}\right)
$$

<p>or the Rodrigues formula
</p>
$$
P_n^{(a, b)}(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= (1-x)^a(1-x)^b \cr
\kappa_n &= (-2)^n n!
}
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) jacobi_p(0,a,b,x);
(%o1)                                  1
(%i2) jacobi_p(1,a,b,x);
                                    (b + a + 2) (1 - x)
(%o2)                  (a + 1) (1 - -------------------)
                                         2 (a + 1)
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="laguerre"></a><a name="Item_003a-orthopoly_002fdeffn_002flaguerre"></a><dl>
<dt><a name="index-laguerre"></a>Function: <strong>laguerre</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The Laguerre polynomial, 
\(L_n(x)\) of degree <em>n</em>.
</p>
<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_778.htm">A&amp;S eqn 22.5.16</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_780.htm">A&amp;S eqn 22.5.54</a>.
</p>
<p>These are related to the generalized Laguerre polynomial by
</p>
$$
L_n(x) = L_n^{(0)}(x)
$$

<p>The polynomials are given by the sum
</p>
$$
L_n(x) = \sum_{k=0}^{n} {(-1)^k\over k!}{n \choose k} x^k
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) laguerre(1,x);
(%o1)                                1 - x
(%i2) laguerre(2,x);
                                  2
                                 x
(%o2)                            -- - 2 x + 1
                                 2
(%i3) gen_laguerre(2,0,x);
                                  2
                                 x
(%o3)                            -- - 2 x + 1
                                 2
(%i4) sum((-1)^k/k!*binomial(2,k)*x^k,k,0,2);
                                  2
                                 x
(%o4)                            -- - 2 x + 1
                                 2
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="legendre_005fp"></a><a name="Item_003a-orthopoly_002fdeffn_002flegendre_005fp"></a><dl>
<dt><a name="index-legendre_005fp"></a>Function: <strong>legendre_p</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The Legendre polynomial of the first kind, 
\(P_n(x)\), of degree <em>n</em>.
</p>
<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.50</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.51</a>.
</p>
<p>The Legendre polynomial is related to the Jacobi polynomials by
</p>
$$
P_n(x) = P_n^{(0,0)}(x)
$$

<p>or the Rodrigues formula
</p>
$$
P_n(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= 1 \cr
\kappa_n &= (-2)^n n!
}
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) legendre_p(1,x);
(%o1)                                  x
(%i2) legendre_p(2,x);
                                                 2
                                        3 (1 - x)
(%o2)                   (- 3 (1 - x)) + ---------- + 1
                                            2
(%i3) expand(%);
                                      2
                                   3 x    1
(%o3)                              ---- - -
                                    2     2
(%i4) expand(legendre_p(3,x));
                                     3
                                  5 x    3 x
(%o4)                             ---- - ---
                                   2      2
(%i5) expand(jacobi_p(3,0,0,x));
                                     3
                                  5 x    3 x
(%o5)                             ---- - ---
                                   2      2
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="legendre_005fq"></a><a name="Item_003a-orthopoly_002fdeffn_002flegendre_005fq"></a><dl>
<dt><a name="index-legendre_005fq"></a>Function: <strong>legendre_q</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The Legendre function of the second kind, 
\(Q_n(x)\) of degree <em>n</em>.
</p>
<p>Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.
</p>
<p>These are related to 
\(Q_n^m(x)\) by
</p>
$$
Q_n(x) = Q_n^0(x)
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) legendre_q(0,x);
                                       x + 1
                                 log(- -----)
                                       x - 1
(%o1)                            ------------
                                      2
(%i2) legendre_q(1,x);
                                    x + 1
                              log(- -----) x - 2
                                    x - 1
(%o2)/R/                      ------------------
                                      2
(%i3) assoc_legendre_q(1,0,x);
                                    x + 1
                              log(- -----) x - 2
                                    x - 1
(%o3)/R/                      ------------------
                                      2
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="orthopoly_005frecur"></a><a name="Item_003a-orthopoly_002fdeffn_002forthopoly_005frecur"></a><dl>
<dt><a name="index-orthopoly_005frecur"></a>Function: <strong>orthopoly_recur</strong> <em>(<var>f</var>, <var>args</var>)</em></dt>
<dd><p>Returns a recursion relation for the orthogonal function family
<var>f</var> with arguments <var>args</var>. The recursion is with 
respect to the polynomial degree.
</p>
<div class="example">
<pre class="example">(%i1) orthopoly_recur (legendre_p, [n, x]);
                    (2 n + 1) P (x) x - n P     (x)
                               n           n - 1
(%o1)   P     (x) = -------------------------------
         n + 1                   n + 1
</pre></div>

<p>The second argument to <code>orthopoly_recur</code> must be a list with the 
correct number of arguments for the function <var>f</var>; if it isn&rsquo;t, 
Maxima signals an error.
</p>
<div class="example">
<pre class="example">(%i1) orthopoly_recur (jacobi_p, [n, x]);

Function jacobi_p needs 4 arguments, instead it received 2
 -- an error.  Quitting.  To debug this try debugmode(true);
</pre></div>

<p>Additionally, when <var>f</var> isn&rsquo;t the name of one of the 
families of orthogonal polynomials, an error is signalled.
</p>
<div class="example">
<pre class="example">(%i1) orthopoly_recur (foo, [n, x]);

A recursion relation for foo isn't known to Maxima
 -- an error.  Quitting.  To debug this try debugmode(true);
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="Item_003a-orthopoly_002fdefvr_002forthopoly_005freturns_005fintervals"></a><dl>
<dt><a name="index-orthopoly_005freturns_005fintervals"></a>Variable: <strong>orthopoly_returns_intervals</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>orthopoly_returns_intervals</code> is <code>true</code>, floating point results are returned in
the form <code>interval (<var>c</var>, <var>r</var>)</code>, where <var>c</var> is the center of an interval
and <var>r</var> is its radius. The center can be a complex number; in that
case, the interval is a disk in the complex plane.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="orthopoly_005fweight"></a><a name="Item_003a-orthopoly_002fdeffn_002forthopoly_005fweight"></a><dl>
<dt><a name="index-orthopoly_005fweight"></a>Function: <strong>orthopoly_weight</strong> <em>(<var>f</var>, <var>args</var>)</em></dt>
<dd>
<p>Returns a three element list; the first element is 
the formula of the weight for the orthogonal polynomial family
<var>f</var> with arguments given by the list <var>args</var>; the 
second and third elements give the lower and upper endpoints
of the interval of orthogonality. For example,
</p>
<div class="example">
<pre class="example">(%i1) w : orthopoly_weight (hermite, [n, x]);
                            2
                         - x
(%o1)                 [%e    , - inf, inf]
(%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
(%o2)                           0
</pre></div>

<p>The main variable of <var>f</var> must be a symbol; if it isn&rsquo;t, Maxima
signals an error. 
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="pochhammer"></a><a name="Item_003a-orthopoly_002fdeffn_002fpochhammer"></a><dl>
<dt><a name="index-pochhammer"></a>Function: <strong>pochhammer</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>The Pochhammer symbol, 
\((x)_n\). (See <a href="https://personal.math.ubc.ca/~cbm/aands/page_256.htm">A&amp;S eqn 6.1.22</a> and <a href="https://dlmf.nist.gov/5.2.iii">DLMF 5.2.iii</a>).
</p>
<p>For nonnegative
integers <var>n</var> with <code><var>n</var> &lt;= pochhammer_max_index</code>, the
expression 
\((x)_n\) evaluates to the
product 
\(x(x+1)(x+2)\cdots(x+n-1)\) when 
\(n > 0\) and
to 1 when <em>n = 0</em>.
For negative <em>n</em>, 
\((x)_n\) is
defined as 
\((-1)^n/(1-x)_{-n}.\)</p>
<p>Thus
</p>
<div class="example">
<pre class="example">(%i1) pochhammer (x, 3);
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, -3);
                                 1
(%o2)               - -----------------------
                      (1 - x) (2 - x) (3 - x)
</pre></div>

<p>To convert a Pochhammer symbol into a quotient of gamma functions,
(see <a href="https://personal.math.ubc.ca/~cbm/aands/page_256.htm">A&amp;S eqn 6.1.22</a>) use <code>makegamma</code>; for example 
</p>
<div class="example">
<pre class="example">(%i1) makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)
</pre></div>

<p>When <var>n</var> exceeds <code>pochhammer_max_index</code> or when <var>n</var> 
is symbolic, <code>pochhammer</code> returns a noun form.
</p>
<div class="example">
<pre class="example">(%i1) pochhammer (x, n);
(%o1)                         (x)
                                 n
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;</div>
</dd></dl>

<a name="Item_003a-orthopoly_002fdefvr_002fpochhammer_005fmax_005findex"></a><dl>
<dt><a name="index-pochhammer_005fmax_005findex"></a>Variable: <strong>pochhammer_max_index</strong></dt>
<dd><p>Default value: 100
</p>
<p><code>pochhammer (<var>n</var>, <var>x</var>)</code> expands to a product if and only if
<code><var>n</var> &lt;= pochhammer_max_index</code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) pochhammer (x, 3), pochhammer_max_index : 3;
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, 4), pochhammer_max_index : 3;
(%o2)                         (x)
                                 4
</pre></div>

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_256.htm">A&amp;S eqn 6.1.16</a>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;</div>
</dd></dl>

<a name="spherical_005fbessel_005fj"></a><a name="Item_003a-orthopoly_002fdeffn_002fspherical_005fbessel_005fj"></a><dl>
<dt><a name="index-spherical_005fbessel_005fj"></a>Function: <strong>spherical_bessel_j</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The spherical Bessel function of the first kind, 
\(j_n(x).\)</p>

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_437.htm">A&amp;S eqn 10.1.8</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">A&amp;S eqn 10.1.15</a>.
</p>
<p>It is related to the Bessel function by
</p>
$$
j_n(x) = \sqrt{\pi\over 2x} J_{n+1/2}(x)
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) spherical_bessel_j(1,x);
                                sin(x)
                                ------ - cos(x)
                                  x
(%o1)                           ---------------
                                       x
(%i2) spherical_bessel_j(2,x);
                                3             3 cos(x)
                        (- (1 - --) sin(x)) - --------
                                 2               x
                                x
(%o2)                   ------------------------------
                                      x
(%i3) expand(%);
                          sin(x)    3 sin(x)   3 cos(x)
(%o3)                  (- ------) + -------- - --------
                            x           3          2
                                       x          x
(%i4) expand(sqrt(%pi/(2*x))*bessel_j(2+1/2,x)),besselexpand:true;
                          sin(x)    3 sin(x)   3 cos(x)
(%o4)                  (- ------) + -------- - --------
                            x           3          2
                                       x          x
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
&middot;</div>
</dd></dl>

<a name="spherical_005fbessel_005fy"></a><a name="Item_003a-orthopoly_002fdeffn_002fspherical_005fbessel_005fy"></a><dl>
<dt><a name="index-spherical_005fbessel_005fy"></a>Function: <strong>spherical_bessel_y</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The spherical Bessel function of the second kind, 
\(y_n(x).\)</p>

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_437.htm">A&amp;S eqn 10.1.9</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">A&amp;S eqn 10.1.15</a>.
</p>
<p>It is related to the Bessel function by
</p>
$$
y_n(x) = \sqrt{\pi\over 2x} Y_{n+1/2}(x)
$$

<div class="example">
<pre class="example">(%i1) spherical_bessel_y(1,x);
                                           cos(x)
                              (- sin(x)) - ------
                                             x
(%o1)                         -------------------
                                       x
(%i2) spherical_bessel_y(2,x);
                           3 sin(x)        3
                           -------- - (1 - --) cos(x)
                              x             2
                                           x
(%o2)                    - --------------------------
                                       x
(%i3) expand(%);
                          3 sin(x)    cos(x)   3 cos(x)
(%o3)                  (- --------) + ------ - --------
                              2         x          3
                             x                    x
(%i4) expand(sqrt(%pi/(2*x))*bessel_y(2+1/2,x)),besselexpand:true;
                          3 sin(x)    cos(x)   3 cos(x)
(%o4)                  (- --------) + ------ - --------
                              2         x          3
                             x                    x
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
&middot;</div>
</dd></dl>

<a name="spherical_005fhankel1"></a><a name="Item_003a-orthopoly_002fdeffn_002fspherical_005fhankel1"></a><dl>
<dt><a name="index-spherical_005fhankel1"></a>Function: <strong>spherical_hankel1</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The spherical Hankel function of the
first kind, 
\(h_n^{(1)}(x).\)</p>

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">A&amp;S eqn 10.1.36</a>.
</p>
<p>This is defined by
</p>
$$
h_n^{(1)}(x) = j_n(x) + iy_n(x)
$$

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
&middot;</div>
</dd></dl>

<a name="spherical_005fhankel2"></a><a name="Item_003a-orthopoly_002fdeffn_002fspherical_005fhankel2"></a><dl>
<dt><a name="index-spherical_005fhankel2"></a>Function: <strong>spherical_hankel2</strong> <em>(<var>n</var>, <var>x</var>)</em></dt>
<dd><p>The spherical Hankel function of the
second kind, 
\(h_n^{(2)}(x).\)</p>

<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">A&amp;S eqn 10.1.17</a>.
</p>
<p>This is defined by
</p>
$$
h_n^{(2)}(x) = j_n(x) + iy_n(x)
$$

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
&middot;</div>
</dd></dl>

<a name="spherical_005fharmonic"></a><a name="Item_003a-orthopoly_002fdeffn_002fspherical_005fharmonic"></a><dl>
<dt><a name="index-spherical_005fharmonic"></a>Function: <strong>spherical_harmonic</strong> <em>(<var>n</var>, <var>m</var>, <var>theta</var>, <var>phi</var>)</em></dt>
<dd><p>The spherical harmonic function, 
\(Y_n^m(\theta, \phi)\).
</p>
<p>Spherical harmonics satisfy the angular part of Laplace&rsquo;s equation in spherical coordinates.
</p>
<p>For integers <em>n</em> and <em>m</em> such
that 
\(n \geq |m|\) and
for 
\(\theta \in [0, \pi]\), Maximaâs
spherical harmonic function can be defined by
</p>
$$
Y_n^m(\theta, \phi) = (-1)^m \sqrt{{2n+1\over 4\pi} {(n-m)!\over (n+m)!}} P_n^m(\cos\theta) e^{im\phi}
$$
 

<p>Further, when 
\(n < |m|\), the
spherical harmonic function vanishes.
</p>
<p>The factor <em>(-1)^m</em>, frequently used in Quantum mechanics, is
called the <a href="https://en.wikipedia.org/wiki/Spherical_harmonics#Condon%E2%80%93Shortley_phase">Condon-Shortely phase</a>.
Some references, including <em>NIST Digital Library of Mathematical Functions</em> omit
this factor; see <a href="http://dlmf.nist.gov/14.30.E1">http://dlmf.nist.gov/14.30.E1</a>.
</p>
<p>Reference: Merzbacher 9.64.
</p>
<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) spherical_harmonic(1,0,theta,phi);
                              sqrt(3) cos(theta)
(%o1)                         ------------------
                                 2 sqrt(%pi)
(%i2) spherical_harmonic(1,1,theta,phi);
                                    %i phi
                          sqrt(3) %e       sin(theta)
(%o2)                     ---------------------------
                                 3/2
                                2    sqrt(%pi)
(%i3) spherical_harmonic(1,-1,theta,phi);
                                    - %i phi
                          sqrt(3) %e         sin(theta)
(%o3)                   - -----------------------------
                                  3/2
                                 2    sqrt(%pi)
(%i4) spherical_harmonic(2,0,theta,phi);
                                                              2
                                            3 (1 - cos(theta))
          sqrt(5) ((- 3 (1 - cos(theta))) + ------------------- + 1)
                                                     2
(%o4)     ----------------------------------------------------------
                                 2 sqrt(%pi)
(%i5) factor(%);
                                        2
                          sqrt(5) (3 cos (theta) - 1)
(%o5)                     ---------------------------
                                  4 sqrt(%pi)
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<a name="unit_005fstep"></a><a name="Item_003a-orthopoly_002fdeffn_002funit_005fstep"></a><dl>
<dt><a name="index-unit_005fstep"></a>Function: <strong>unit_step</strong> <em>(<var>x</var>)</em></dt>
<dd><p>The left-continuous unit step function; thus
<code>unit_step (<var>x</var>)</code> vanishes for <code><var>x</var> &lt;= 0</code> and equals
1 for <code><var>x</var> &gt; 0</code>.
</p>
<p>If you want a unit step function that takes on the value 1/2 at zero,
use <code><a href="maxima_48.html#hstep">hstep</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
&middot;</div>
</dd></dl>

<a name="ultraspherical"></a><a name="Item_003a-orthopoly_002fdeffn_002fultraspherical"></a><dl>
<dt><a name="index-ultraspherical"></a>Function: <strong>ultraspherical</strong> <em>(<var>n</var>, <var>a</var>, <var>x</var>)</em></dt>
<dd><p>The ultraspherical polynomial, 
\(C_n^{(a)}(x)\) (also known as the Gegenbauer polynomial).
</p>
<p>Reference: <a href="https://personal.math.ubc.ca/~cbm/aands/page_779.htm">A&amp;S eqn 22.5.46</a>.
</p>
<p>These polynomials can be given in terms of Jacobi polynomials:
</p>
$$
C_n^{(\alpha)}(x) = {\Gamma\left(\alpha + {1\over 2}\right) \over \Gamma(2\alpha)}
   {\Gamma(n+2\alpha) \over \Gamma\left(n+\alpha + {1\over 2}\right)}
   P_n^{(\alpha-1/2, \alpha-1/2)}(x)
$$

<p>or the series
</p>
$$
C_n^{(\alpha)}(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k (\alpha)_{n-k} \over k! (n-2k)!}(2x)^{n-2k}
$$

<p>or the Rodrigues formula
</p>
$$
C_n^{(\alpha)}(x) = {1\over \kappa_n  w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)
$$

<p>where
</p>
$$
\eqalign{
w(x) &= \left(1-x^2\right)^{\alpha-{1\over 2}} \cr
\kappa_n &= {(-2)^n\left(\alpha + {1\over 2}\right)_n n!\over (2\alpha)_n} \cr
}
$$

<p>Some examples:
</p><div class="example">
<pre class="example">(%i1) ultraspherical(1,a,x);
                                   (2 a + 1) (1 - x)
(%o1)                     2 a (1 - -----------------)
                                              1
                                       2 (a + -)
                                              2
(%i2) factor(%);
(%o2)                                2 a x
(%i3) factor(ultraspherical(2,a,x));
                                     2      2
(%o3)                        a (2 a x  + 2 x  - 1)
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-orthopoly">Package orthopoly</a>
&middot;</div>
</dd></dl>

<hr>
<div class="header">
<p>
Previous: <a href="maxima_337.html#Introduction-to-orthogonal-polynomials" accesskey="p" rel="previous">Introduction to orthogonal polynomials</a>, Up: <a href="maxima_336.html#orthopoly_002dpkg" accesskey="u" rel="up">orthopoly-pkg</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>