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75 lines
2.6 KiB
ReStructuredText
75 lines
2.6 KiB
ReStructuredText
Hypergeometric functions
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The functions listed in :doc:`expintegrals`, :doc:`bessel` and
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:doc:`orthogonal`, and many other functions as well, are merely particular
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instances of the generalized hypergeometric function `\,_pF_q`. The functions
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listed in the following section enable efficient direct evaluation of the
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underlying hypergeometric series, as well as linear combinations, limits with
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respect to parameters, and analytic continuations thereof. Extensions to
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twodimensional series are also provided. See also the basic or q-analog of the
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hypergeometric series in :doc:`qfunctions`.
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For convenience, most of the hypergeometric series of low order are provided as
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standalone functions. They can equivalently be evaluated using
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:func:`~mpmath.hyper`. As will be demonstrated in the respective docstrings,
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all the ``hyp#f#`` functions implement analytic continuations and/or asymptotic
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expansions with respect to the argument `z`, thereby permitting evaluation for
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`z` anywhere in the complex plane. Functions of higher degree can be computed
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via :func:`~mpmath.hyper`, but generally only in rapidly convergent instances.
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Most hypergeometric and hypergeometric-derived functions accept optional
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keyword arguments to specify options for :func:`~mpmath.hypercomb` or
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:func:`~mpmath.hyper`. Some useful options are *maxprec*, *maxterms*,
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*zeroprec*, *accurate_small*, *hmag*, *force_series*, *asymp_tol* and
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*eliminate*. These options give control over what to do in case of slow
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convergence, extreme loss of accuracy or evaluation at zeros (these two cases
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cannot generally be distinguished from each other automatically), and singular
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parameter combinations.
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Common hypergeometric series
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............................
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.. autofunction:: mpmath.hyp0f1
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.. autofunction:: mpmath.hyp1f1
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.. autofunction:: mpmath.hyp1f2
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.. autofunction:: mpmath.hyp2f0
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.. autofunction:: mpmath.hyp2f1
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.. autofunction:: mpmath.hyp2f2
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.. autofunction:: mpmath.hyp2f3
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.. autofunction:: mpmath.hyp3f2
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Generalized hypergeometric functions
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....................................
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.. autofunction:: mpmath.hyper
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.. autofunction:: mpmath.hypercomb
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Meijer G-function
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.................
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.. autofunction:: mpmath.meijerg
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Fox H-function
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.................
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.. autofunction:: mpmath.foxh
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Bilateral hypergeometric series
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...............................
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.. autofunction:: mpmath.bihyper
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Hypergeometric functions of two variables
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.........................................
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.. autofunction:: mpmath.hyper2d
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.. autofunction:: mpmath.appellf1
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.. autofunction:: mpmath.appellf2
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.. autofunction:: mpmath.appellf3
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.. autofunction:: mpmath.appellf4
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