# Source code for sympy.functions.special.hyper

"""Hypergeometric and Meijer G-functions"""

from sympy import S
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.containers import Tuple
from sympy.core.sympify import sympify
from sympy.core.mul import Mul

# TODO should __new__ accept **options?
# TODO should constructors should check if parameters are sensible?

# TODO when pull request #399 is in, this should be no longer necessary
def _make_tuple(v):
"""
Turn an iterable argument V into a Tuple.

Examples:
>>> from sympy.functions.special.hyper import _make_tuple as mt
>>> from sympy.core.containers import Tuple
>>> mt([1, 2, 3])
(1, 2, 3)
>>> mt((4, 5))
(4, 5)
>>> mt((7, 8, 9))
(7, 8, 9)
"""
return Tuple(*[sympify(x) for x in v])

class TupleParametersBase(Function):
""" Base class that takes care of differentiation, when some of
the arguments are actually tuples. """
def _eval_derivative(self, s):
if self.args[0].has(s) or self.args[1].has(s):
raise NotImplementedError('differentiation with respect to ' \
'a parameter')
return self.fdiff(3)*self.args[2].diff(s)

[docs]class hyper(TupleParametersBase): r""" The (generalized) hypergeometric function is defined by a series where the ratios of successive terms are a rational function of the summation index. When convergent, it is continued analytically to the largest possible domain. The hypergeometric function depends on two vectors of parameters, called the numerator parameters :math:a_p, and the denominator parameters :math:b_q. It also has an argument :math:z. The series definition is .. math :: {}_pF_q\left.\left(\begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \right| z \right) = \sum_{n=0}^\infty \frac{(a_1)_n \dots (a_p)_n}{(b_1)_n \dots (b_q)_n} \frac{z^n}{n!}, where :math:(a)_n = (a)(a+1)\dots(a+n-1) denotes the rising factorial. If one of the :math:b_q is a non-positive integer then the series is undefined unless one of the a_p is a larger (i.e. smaller in magnitude) non-positive integer. If none of the :math:b_q is a non-positive integer and one of the :math:a_p is a non-positive integer, then the series reduces to a polynomial. To simplify the following discussion, we assume that none of the :math:a_p or :math:b_q is a non-positive integer. For more details, see the references. The series converges for all :math:z if :math:p \le q, and thus defines an entire single-valued function in this case. If :math:p = q+1 the series converges for :math:|z| < 1, and can be continued analytically into a half-plane. If :math:p > q+1 the series is divergent for all :math:z. Note: The hypergeometric function constructor currently does *not* check if the parameters actually yield a well-defined function. **Examples** The parameters :math:a_p and :math:b_q can be passed as arbitrary iterables, for example: >>> from sympy.functions import hyper >>> from sympy.abc import x, n, a >>> hyper((1, 2, 3), [3, 4], x) hyper((1, 2, 3), (3, 4), x) There is also pretty printing (it looks better using unicode): >>> from sympy import pprint >>> pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False) _ |_ /1, 2, 3 | \ | | | x| 3 2 \ 3, 4 | / The parameters must always be iterables, even if they are vectors of length one or zero: >>> hyper((1, ), [], x) hyper((1,), (), x) But of course they may be variables (but if they depend on x then you should not expect much implemented functionality): >>> hyper((n, a), (n**2,), x) hyper((n, a), (n**2,), x) The hypergeometric function generalises many named special functions. The function hyperexpand() tries to express a hypergeometric function using named special functions. For example: >>> from sympy import hyperexpand >>> hyperexpand(hyper([], [], x)) exp(x) You can also use expand_func: >>> from sympy import expand_func >>> expand_func(x*hyper([1, 1], [2], -x)) log(x + 1) More examples: >>> from sympy import S >>> hyperexpand(hyper([], [S(1)/2], -x**2/4)) cos(x) >>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2)) asin(x) We can also sometimes hyperexpand parametric functions: >>> from sympy.abc import a >>> hyperexpand(hyper([-a], [], x)) (-x + 1)**a See Also: - :func:sympy.simplify.hyperexpand **References** - Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 - http://en.wikipedia.org/wiki/Generalized_hypergeometric_function """ nargs = 3 def __new__(cls, ap, bq, z): # TODO should we check convergence conditions? return Function.__new__(cls, _make_tuple(ap), _make_tuple(bq), z) def fdiff(self, argindex=3): if argindex != 3: raise ArgumentIndexError(self, argindex) nap = Tuple(*[a + 1 for a in self.ap]) nbq = Tuple(*[b + 1 for b in self.bq]) fac = Mul(*self.ap)/Mul(*self.bq) return fac*hyper(nap, nbq, self.argument) def _eval_expand_func(self, deep=True, **hints): from sympy import gamma, hyperexpand if len(self.ap) == 2 and len(self.bq) == 1 and self.argument == 1: a, b = self.ap c = self.bq[0] return gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b) return hyperexpand(self) @property def argument(self): """ Argument of the hypergeometric function. """ return self.args[2] @property def ap(self): """ Numerator parameters of the hypergeometric function. """ return self.args[0] @property def bq(self): """ Denominator parameters of the hypergeometric function. """ return self.args[1] @property def eta(self): """ A quantity related to the convergence of the series. """ return sum(self.ap) - sum(self.bq) @property def radius_of_convergence(self): """ Compute the radius of convergence of the defining series. Note that even if this is not oo, the function may still be evaluated outside of the radius of convergence by analytic continuation. But if this is zero, then the function is not actually defined anywhere else. >>> from sympy.functions import hyper >>> from sympy.abc import z >>> hyper((1, 2), [3], z).radius_of_convergence 1 >>> hyper((1, 2, 3), [4], z).radius_of_convergence 0 >>> hyper((1, 2), (3, 4), z).radius_of_convergence oo """ from sympy import oo if any(a.is_integer and a <= 0 for a in self.ap + self.bq): aints = [a for a in self.ap if a.is_Integer and a <= 0] bints = [a for a in self.bq if a.is_Integer and a <= 0] if len(aints) < len(bints): return S(0) popped = False for b in bints: cancelled = False while aints: a = aints.pop() if a >= b: cancelled = True break popped = True if not cancelled: return S(0) if aints or popped: # There are still non-positive numerator parameters. # This is a polynomial. return oo if len(self.ap) == len(self.bq) + 1: return S(1) elif len(self.ap) <= len(self.bq): return oo else: return S(0) @property def convergence_statement(self): """ Return a condition on z under which the series converges. """ from sympy import And, Or, re, Ne, oo R = self.radius_of_convergence if R == 0: return False if R == oo: return True # The special functions and their approximations, page 44 e = self.eta z = self.argument c1 = And(re(e) < 0, abs(z) <= 1) c2 = And(0 <= re(e), re(e) < 1, abs(z) <= 1, Ne(z, 1)) c3 = And(re(e) >= 1, abs(z) < 1) return Or(c1, c2, c3)
[docs]class meijerg(TupleParametersBase): r""" The Meijer G-function is defined by a Mellin-Barnes type integral that resembles an inverse Mellin transform. It generalises the hypergeometric functions. The Meijer G-function depends on four sets of parameters. There are "*numerator parameters*" :math:a_1, \dots, a_n and :math:a_{n+1}, \dots, a_p, and there are "*denominator parameters*" :math:b_1, \dots, b_m and :math:b_{m+1}, \dots, b_q. Confusingly, it is traditionally denoted as follows (note the position of m, n, p, q, and how they relate to the lengths of the four parameter vectors): .. math :: G_{p,q}^{m,n} \left.\left(\begin{matrix}a_1, \dots, a_n & a_{n+1}, \dots, a_p \\ b_1, \dots, b_m & b_{m+1}, \dots, b_q \end{matrix} \right| z \right). However, in sympy the four parameter vectors are always available separately (see examples), so that there is no need to keep track of the decorating sub- and super-scripts on the G symbol. The G function is defined as the following integral: .. math :: \frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s, where :math:\Gamma(z) is the gamma function. There are three possible contours which we will not describe in detail here (see the references). If the integral converges along more than one of them the definitions agree. The contours all separate the poles of :math:\Gamma(1-a_j+s) from the poles of :math:\Gamma(b_k-s), so in particular the G function is undefined if :math:a_j - b_k \in \mathbb{Z}_{>0} for some :math:j \le n and :math:k \le m. The conditions under which one of the contours yields a convergent integral are complicated and we do not state them here, see the references. Note: Currently the Meijer G-function constructor does *not* check any convergence conditions. **Examples** You can pass the parameters either as four separate vectors: >>> from sympy.functions import meijerg >>> from sympy.abc import x, a >>> from sympy.core.containers import Tuple >>> from sympy import pprint >>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False) __1, 2 /1, 2 a, 4 | \ /__ | | x| \_|4, 1 \ 5 | / or as two nested vectors: >>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False) __1, 2 /1, 2 3, 4 | \ /__ | | x| \_|4, 1 \ 5 | / As with the hypergeometric function, the parameters may be passed as arbitrary iterables. Vectors of length zero and one also have to be passed as iterables. The parameters need not be constants, but if they depend on the argument then not much implemented functionality should be expected. All the subvectors of parameters are available: >>> from sympy import pprint >>> g = meijerg([1], [2], [3], [4], x) >>> pprint(g, use_unicode=False) __1, 1 /1 2 | \ /__ | | x| \_|2, 2 \3 4 | / >>> g.an (1,) >>> g.ap (1, 2) >>> g.aother (2,) >>> g.bm (3,) >>> g.bq (3, 4) >>> g.bother (4,) The Meijer G-function generalises the hypergeometric functions. In some cases it can be expressed in terms of hypergeometric functions, using Slater's theorem. For example: >>> from sympy import hyperexpand >>> from sympy.abc import a, b, c >>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True) x**c*gamma(-a + c + 1)*hyper((-a + c + 1,), (-b + c + 1,), -x)/gamma(-b + c + 1) Thus the Meijer G-function also subsumes many named functions as special cases. You can use expand_func or hyperexpand to (try to) rewrite a Meijer G-function in terms of named special functions. For example: >>> from sympy import expand_func, S >>> expand_func(meijerg([[],[]], [[0],[]], -x)) exp(x) >>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2)) sin(x)/pi**(1/2) See Also: - :func:sympy.simplify.hyperexpand **References** - Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 - http://en.wikipedia.org/wiki/Meijer_G-function """ nargs = 3 def __new__(cls, *args): if len(args) == 5: args = [(args[0], args[1]), (args[2], args[3]), args[4]] if len(args) != 3: raise TypeError("args must eiter be as, as', bs, bs', z or " \ "as, bs, z") def tr(p): if len(p) != 2: raise TypeError("wrong argument") return Tuple(_make_tuple(p[0]), _make_tuple(p[1])) # TODO should we check convergence conditions? return Function.__new__(cls, tr(args[0]), tr(args[1]), args[2]) def fdiff(self, argindex=3): if argindex != 3: raise ArgumentIndexError(self, argindex) if len(self.an) >= 1: a = list(self.an) a[0] -= 1 G = meijerg(a, self.aother, self.bm, self.bother, self.argument) return 1/self.argument * ((self.an[0]-1)*self + G) elif len(self.bm) >= 1: b = list(self.bm) b[0] += 1 G = meijerg(self.an, self.aother, b, self.bother, self.argument) return 1/self.argument * (self.bm[0]*self - G) else: return S.Zero def _eval_expand_func(self, deep=True, **hints): from sympy import hyperexpand return hyperexpand(self) @property def argument(self): """ Argument of the Meijer G-function. """ return self.args[2] @property def an(self): """ First set of numerator parameters. """ return self.args[0][0] @property def ap(self): """ Combined numerator parameters. """ return self.args[0][0] + self.args[0][1] @property def aother(self): """ Second set of numerator parameters. """ return self.args[0][1] @property def bm(self): """ First set of denominator parameters. """ return self.args[1][0] @property def bq(self): """ Combined denominator parameters. """ return self.args[1][0] + self.args[1][1] @property def bother(self): """ Second set of denominator parameters. """ return self.args[1][1] @property def nu(self): """ A quantity related to the convergence region of the integral, c.f. references. """ return sum(self.bq) - sum(self.ap) @property def delta(self): """ A quantity related to the convergence region of the integral, c.f. references. """ return len(self.bm) + len(self.an) - S(len(self.ap) + len(self.bq))/2