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laguerrepolynomials

laguerre polynomials

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LAGUERRE POLYNOMIALS - In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: x ...

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Laguerre polynomials might refer to
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:* * * * x * * y * ″ * * + * ( * 1 * − * x * ) * * y * ′ * * + * n * y * = * 0 * * * {\displaystyle xy''+(1-x)y'+ny=0} * which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. * Sometimes the name Laguerre polynomials is used for solutions of * * * * * x * * y * ″ * * + * ( * α * + * 1 * − * x * ) * * y * ′ * * + * n * y * = * 0 * * . * * * {\displaystyle xy''+(\alpha +1-x)y'+ny=0~.} * where n is still a non-negative integer. * Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). * More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. * The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form * * * * * * ∫ * * 0 * * * ∞ * * * f * ( * x * ) * * e * * − * x * * * * d * x * . * * * {\displaystyle \int _{0}^{\infty }f(x)e^{-x}\,dx.} * These polynomials, usually denoted L0, L1, ..., are a polynomial sequence which may be defined by the Rodrigues formula, * * * * * * L * * n * * * ( * x * ) * = * * * * e * * x * * * * n * ! * * * * * * * d * * n * * * * d * * x * * n * * * * * * * ( * * * e * * − * x * * * * x * * n * * * * ) * * = * * * 1 * * n * ! * * * * * * ( * * * * d * * d * x * ...

Laguerre polynomials might refer to

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:*
*
*
* x
*
* y
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* 1
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* y
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* {\displaystyle xy''+(1-x)y'+ny=0}
* which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
* Sometimes the name Laguerre polynomials is used for solutions of
*
*
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* 0
*
* .
*
*
* {\displaystyle xy''+(\alpha +1-x)y'+ny=0~.}
* where n is still a non-negative integer.
* Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).
* More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer.
* The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form
*
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* ∫
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* .
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*
* {\displaystyle \int _{0}^{\infty }f(x)e^{-x}\,dx.}
* These polynomials, usually denoted L0, L1, ..., are a polynomial sequence which may be defined by the Rodrigues formula,
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