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Version:
4.4.5.dfsg-3ubuntu2 ▾
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<HTML>
<HEAD>
<TITLE>Legendre functions and polynomials</TITLE>
</HEAD>
<BODY bgcolor="#FFFFFF" fgcolor="#000000">
<P>
<font size="+3" color="green"><B>Legendre functions and polynomials</B></font></P>
<P>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%"><CODE>
y = LEGENDRE(n,x)<br />
y = PLMU(n,m,x)<br />
y = PLMN(n,m,x)</code>
</TD></TR>
</table></p>
<p>
The most elementary of the Legendre functions, the Legendre polynomials,
<img align="bottom" src="legendreI01.png">, can be defined by
the generating function:</p>
<p><img align="bottom" src="legendreI02.png"></p>
<p>
More explicit representations are:</p>
<p><img align="bottom" src="legendreI03.png"></p>
<p>
where <img align="top" src="legendreI04.png"> is the hypergeometric
function.</p>
<p>
<font size="+1" color="green"><b>Unnormalized associated Legendre functions of degree <i>n</i></b></font></p>
<p>
<img align="bottom" src="legendreI05.png"></p>
<p><font size="+1" color="green"><b>Normalized associated Legendre functions</b></font></p>
<p>
<img align="bottom" src="legendreI06.png"></p>
<p>
where <img align="bottom" src="legendreI07.png">,
<img align="bottom" src="legendreI08.png"> is a non-negative
integer and <img align="top" src="legendreI09.png"> satisfy the
differential equation:</p>
<p><img align="bottom" src="legendreI10.png"></p>
<p>
Note that <code>PLMU(n,0,x) = LEGENDRE(n,x)</code></p>
</BODY>
</HTML>