Carl McBride at 09:48, 31 May 2007

2007-05-31T09:48:44Z

← Older revision		Revision as of 11:48, 31 May 2007
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	The first four Laguerre polynomials are:		The first four Laguerre polynomials are:

	<math>\left. L_0 (x) \right.=1</math>		:<math>\left. L_0 (x) \right.=1</math>


	<math>\left. L_1 (x) \right.=-x +1</math>		:<math>\left. L_1 (x) \right.=-x +1</math>


	<math>L_2 (x) =\frac{1}{2}(x^2 -4x +2)</math>		:<math>L_2 (x) =\frac{1}{2}(x^2 -4x +2)</math>


	<math>L_3 (x) =\frac{1}{6}(-x^3 +9x^2 -18x +6)</math>		:<math>L_3 (x) =\frac{1}{6}(-x^3 +9x^2 -18x +6)</math>


	===Generalized Laguerre function===		===Generalized Laguerre function===

	<math>L_n^{\alpha}(x)= \frac{(\alpha + 1)_n}{n!} ~_1F_1(-n; \alpha + 1;x)</math>		:<math>L_n^{\alpha}(x)= \frac{(\alpha + 1)_n}{n!} ~_1F_1(-n; \alpha + 1;x)</math>

	where <math>(a)_n</math> is the Pochhammer symbol		where <math>(a)_n</math> is the Pochhammer symbol

Carl McBride: New page: Laguerre polynomials are solutions $L_n(x)$ to the Laguerre differential equation with $\nu =0$ . The Laguerre polynomial $H_n(z)$ can be defined by the con...

2007-05-31T09:47:51Z

New page: Laguerre polynomials are solutions <math>L_n(x)</math> to the Laguerre differential equation with <math>\nu =0</math>. The Laguerre polynomial <math>H_n(z)</math> can be defined by the con...

New page

Laguerre polynomials are solutions <math>L_n(x)</math> to the Laguerre differential equation with <math>\nu =0</math>.
The Laguerre polynomial <math>H_n(z)</math> can be defined by the contour integral

:<math>L_n (z) = \frac{1}{2 \pi i} \oint \frac{e^{-zt/(1-t)}}{(1-t)t^{n+1}}{\rm d}t</math>

The first four Laguerre polynomials are:

<math>\left. L_0 (x) \right.=1</math>

<math>\left. L_1 (x) \right.=-x +1</math>

<math>L_2 (x) =\frac{1}{2}(x^2 -4x +2)</math>

<math>L_3 (x) =\frac{1}{6}(-x^3 +9x^2 -18x +6)</math>

===Generalized Laguerre function===

<math>L_n^{\alpha}(x)= \frac{(\alpha + 1)_n}{n!} ~_1F_1(-n; \alpha + 1;x)</math>

where <math>(a)_n</math> is the Pochhammer symbol
and <math> ~_1F_1(a;b;x)</math> is a confluent hyper-geometric function.
==See also==
*[http://mathworld.wolfram.com/LaguerrePolynomial.html Laguerre Polynomial -- from Wolfram MathWorld]
[[category: mathematics]]

Laguerre polynomials - Revision history

Carl McBride at 09:48, 31 May 2007

Carl McBride: New page: Laguerre polynomials are solutions L_n(x) to the Laguerre differential equation with \nu =0. The Laguerre polynomial H_n(z) can be defined by the con...

Carl McBride: New page: Laguerre polynomials are solutions $L_n(x)$ to the Laguerre differential equation with $\nu =0$ . The Laguerre polynomial $H_n(z)$ can be defined by the con...