Laguerre polynomials: Difference between revisions
Carl McBride (talk | contribs) (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...) |
Carl McBride (talk | contribs) No edit summary |
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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 | ||
Latest revision as of 10:48, 31 May 2007
Laguerre polynomials are solutions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_n(x)} to the Laguerre differential equation with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu =0} . The Laguerre polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_n(z)} can be defined by the contour integral
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_n (z) = \frac{1}{2 \pi i} \oint \frac{e^{-zt/(1-t)}}{(1-t)t^{n+1}}{\rm d}t}
The first four Laguerre polynomials are:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. L_0 (x) \right.=1}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. L_1 (x) \right.=-x +1}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2 (x) =\frac{1}{2}(x^2 -4x +2)}
Generalized Laguerre function[edit]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_n^{\alpha}(x)= \frac{(\alpha + 1)_n}{n!} ~_1F_1(-n; \alpha + 1;x)}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a)_n} is the Pochhammer symbol and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~_1F_1(a;b;x)} is a confluent hyper-geometric function.