chore: import upstream snapshot with attribution

2026-07-13 12:32:53 +08:00
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+Hypergeometric functions
+------------------------
+
+The functions listed in :doc:`expintegrals`, :doc:`bessel` and
+:doc:`orthogonal`, and many other functions as well, are merely particular
+instances of the generalized hypergeometric function `\,_pF_q`.  The functions
+listed in the following section enable efficient direct evaluation of the
+underlying hypergeometric series, as well as linear combinations, limits with
+respect to parameters, and analytic continuations thereof.  Extensions to
+twodimensional series are also provided.  See also the basic or q-analog of the
+hypergeometric series in :doc:`qfunctions`.
+
+For convenience, most of the hypergeometric series of low order are provided as
+standalone functions.  They can equivalently be evaluated using
+:func:`~mpmath.hyper`.  As will be demonstrated in the respective docstrings,
+all the ``hyp#f#`` functions implement analytic continuations and/or asymptotic
+expansions with respect to the argument `z`, thereby permitting evaluation for
+`z` anywhere in the complex plane.  Functions of higher degree can be computed
+via :func:`~mpmath.hyper`, but generally only in rapidly convergent instances.
+
+Most hypergeometric and hypergeometric-derived functions accept optional
+keyword arguments to specify options for :func:`~mpmath.hypercomb` or
+:func:`~mpmath.hyper`.  Some useful options are *maxprec*, *maxterms*,
+*zeroprec*, *accurate_small*, *hmag*, *force_series*, *asymp_tol* and
+*eliminate*.  These options give control over what to do in case of slow
+convergence, extreme loss of accuracy or evaluation at zeros (these two cases
+cannot generally be distinguished from each other automatically), and singular
+parameter combinations.
+
+Common hypergeometric series
+............................
+
+.. autofunction:: mpmath.hyp0f1
+.. autofunction:: mpmath.hyp1f1
+.. autofunction:: mpmath.hyp1f2
+.. autofunction:: mpmath.hyp2f0
+.. autofunction:: mpmath.hyp2f1
+.. autofunction:: mpmath.hyp2f2
+.. autofunction:: mpmath.hyp2f3
+.. autofunction:: mpmath.hyp3f2
+
+
+Generalized hypergeometric functions
+....................................
+
+.. autofunction:: mpmath.hyper
+.. autofunction:: mpmath.hypercomb
+
+
+Meijer G-function
+.................
+
+.. autofunction:: mpmath.meijerg
+
+Fox H-function
+.................
+
+.. autofunction:: mpmath.foxh
+
+
+Bilateral hypergeometric series
+...............................
+
+.. autofunction:: mpmath.bihyper
+
+
+Hypergeometric functions of two variables
+.........................................
+
+.. autofunction:: mpmath.hyper2d
+.. autofunction:: mpmath.appellf1
+.. autofunction:: mpmath.appellf2
+.. autofunction:: mpmath.appellf3
+.. autofunction:: mpmath.appellf4