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78 lines
2.1 KiB
ReStructuredText
78 lines
2.1 KiB
ReStructuredText
Orthogonal polynomials
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An orthogonal polynomial sequence is a sequence of polynomials `P_0(x), P_1(x),
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\ldots` of degree `0, 1, \ldots`, which are mutually orthogonal in the sense
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that
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.. math ::
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\int_S P_n(x) P_m(x) w(x) dx =
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\begin{cases}
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c_n \ne 0 & \text{if $m = n$} \\
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0 & \text{if $m \ne n$}
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\end{cases}
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where `S` is some domain (e.g. an interval `[a,b] \in \mathbb{R}`) and `w(x)`
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is a fixed *weight function*. A sequence of orthogonal polynomials is
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determined completely by `w`, `S`, and a normalization convention (e.g. `c_n =
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1`). Applications of orthogonal polynomials include function approximation and
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solution of differential equations.
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Orthogonal polynomials are sometimes defined using the differential equations
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they satisfy (as functions of `x`) or the recurrence relations they satisfy
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with respect to the order `n`. Other ways of defining orthogonal polynomials
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include differentiation formulas and generating functions. The standard
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orthogonal polynomials can also be represented as hypergeometric series (see
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:doc:`hypergeometric`), more specifically using the Gauss hypergeometric
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function `\,_2F_1` in most cases. The following functions are generally
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implemented using hypergeometric functions since this is computationally
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efficient and easily generalizes.
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For more information, see the `Wikipedia article on orthogonal polynomials
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<http://en.wikipedia.org/wiki/Orthogonal_polynomials>`_.
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Legendre functions
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..................
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.. autofunction:: mpmath.legendre
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.. autofunction:: mpmath.legenp
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.. autofunction:: mpmath.legenq
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Chebyshev polynomials
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.....................
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.. autofunction:: mpmath.chebyt
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.. autofunction:: mpmath.chebyu
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Jacobi polynomials
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..................
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.. autofunction:: mpmath.jacobi
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Gegenbauer polynomials
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......................
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.. autofunction:: mpmath.gegenbauer
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Hermite polynomials
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...................
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.. autofunction:: mpmath.hermite
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Laguerre polynomials
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....................
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.. autofunction:: mpmath.laguerre
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Spherical harmonics
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...................
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.. autofunction:: mpmath.spherharm
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