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image-handler/image-handler10.0/venv/lib/site-packages/sympy/simplify/tests/test_hyperexpand.py

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from sympy.core.random import randrange
from sympy.simplify.hyperexpand import (ShiftA, ShiftB, UnShiftA, UnShiftB,
MeijerShiftA, MeijerShiftB, MeijerShiftC, MeijerShiftD,
MeijerUnShiftA, MeijerUnShiftB, MeijerUnShiftC,
MeijerUnShiftD,
ReduceOrder, reduce_order, apply_operators,
devise_plan, make_derivative_operator, Formula,
hyperexpand, Hyper_Function, G_Function,
reduce_order_meijer,
build_hypergeometric_formula)
from sympy.concrete.summations import Sum
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.numbers import I
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.combinatorial.factorials import binomial
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.hyper import (hyper, meijerg)
from sympy.abc import z, a, b, c
from sympy.testing.pytest import XFAIL, raises, slow, ON_CI, skip
from sympy.core.random import verify_numerically as tn
from sympy.core.numbers import (Rational, pi)
from sympy.functions.elementary.exponential import (exp, exp_polar, log)
from sympy.functions.elementary.hyperbolic import atanh
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (asin, cos, sin)
from sympy.functions.special.bessel import besseli
from sympy.functions.special.error_functions import erf
from sympy.functions.special.gamma_functions import (gamma, lowergamma)
def test_branch_bug():
assert hyperexpand(hyper((Rational(-1, 3), S.Half), (Rational(2, 3), Rational(3, 2)), -z)) == \
-z**S('1/3')*lowergamma(exp_polar(I*pi)/3, z)/5 \
+ sqrt(pi)*erf(sqrt(z))/(5*sqrt(z))
assert hyperexpand(meijerg([Rational(7, 6), 1], [], [Rational(2, 3)], [Rational(1, 6), 0], z)) == \
2*z**S('2/3')*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) - 2*lowergamma(
Rational(2, 3), z)/z**S('2/3'))*gamma(Rational(2, 3))/gamma(Rational(5, 3))
def test_hyperexpand():
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
assert hyperexpand(hyper([], [], z)) == exp(z)
assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z)
assert hyperexpand(hyper([], [S.Half], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [S('3/2')], -z**2/4)) == sin(z)
assert hyperexpand(hyper([S('1/2'), S('1/2')], [S('3/2')], z**2)*z) \
== asin(z)
assert isinstance(Sum(binomial(2, z)*z**2, (z, 0, a)).doit(), Expr)
def can_do(ap, bq, numerical=True, div=1, lowerplane=False):
r = hyperexpand(hyper(ap, bq, z))
if r.has(hyper):
return False
if not numerical:
return True
repl = {}
randsyms = r.free_symbols - {z}
while randsyms:
# Only randomly generated parameters are checked.
for n, ai in enumerate(randsyms):
repl[ai] = randcplx(n)/div
if not any(b.is_Integer and b <= 0 for b in Tuple(*bq).subs(repl)):
break
[a, b, c, d] = [2, -1, 3, 1]
if lowerplane:
[a, b, c, d] = [2, -2, 3, -1]
return tn(
hyper(ap, bq, z).subs(repl),
r.replace(exp_polar, exp).subs(repl),
z, a=a, b=b, c=c, d=d)
def test_roach():
# Kelly B. Roach. Meijer G Function Representations.
# Section "Gallery"
assert can_do([S.Half], [Rational(9, 2)])
assert can_do([], [1, Rational(5, 2), 4])
assert can_do([Rational(-1, 2), 1, 2], [3, 4])
assert can_do([Rational(1, 3)], [Rational(-2, 3), Rational(-1, 2), S.Half, 1])
assert can_do([Rational(-3, 2), Rational(-1, 2)], [Rational(-5, 2), 1])
assert can_do([Rational(-3, 2), ], [Rational(-1, 2), S.Half]) # shine-integral
assert can_do([Rational(-3, 2), Rational(-1, 2)], [2]) # elliptic integrals
@XFAIL
def test_roach_fail():
assert can_do([Rational(-1, 2), 1], [Rational(1, 4), S.Half, Rational(3, 4)]) # PFDD
assert can_do([Rational(3, 2)], [Rational(5, 2), 5]) # struve function
assert can_do([Rational(-1, 2), S.Half, 1], [Rational(3, 2), Rational(5, 2)]) # polylog, pfdd
assert can_do([1, 2, 3], [S.Half, 4]) # XXX ?
assert can_do([S.Half], [Rational(-1, 3), Rational(-1, 2), Rational(-2, 3)]) # PFDD ?
# For the long table tests, see end of file
def test_polynomial():
from sympy.core.numbers import oo
assert hyperexpand(hyper([], [-1], z)) is oo
assert hyperexpand(hyper([-2], [-1], z)) is oo
assert hyperexpand(hyper([0, 0], [-1], z)) == 1
assert can_do([-5, -2, randcplx(), randcplx()], [-10, randcplx()])
assert hyperexpand(hyper((-1, 1), (-2,), z)) == 1 + z/2
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)* \
lowergamma(a - 1, z) - 1
# TODO [a+1, aRational(-1, 2)], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([S.Half, 2], [Rational(3, 2)], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([S.Half, S.Half], [Rational(5, 2)], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2))
assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([Rational(-1, 2) - 1, 1, 2], [S.Half, 3], z)) == \
sqrt(z)*(z*Rational(6, 7) - Rational(6, 5))*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + S.Half)/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [S.Half, a], [Rational(3, 2), a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + S.Half)*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
# TODO [a], [a - S.Half, 2*a]
def test_hyperexpand_parametric():
assert hyperexpand(hyper([a, S.Half + a], [S.Half], z)) \
== (1 + sqrt(z))**(-2*a)/2 + (1 - sqrt(z))**(-2*a)/2
assert hyperexpand(hyper([a, Rational(-1, 2) + a], [2*a], z)) \
== 2**(2*a - 1)*((-z + 1)**S.Half + 1)**(-2*a + 1)
def test_shifted_sum():
from sympy.simplify.simplify import simplify
assert simplify(hyperexpand(z**4*hyper([2], [3, S('3/2')], -z**2))) \
== z*sin(2*z) + (-z**2 + S.Half)*cos(2*z) - S.Half
def _randrat():
""" Steer clear of integers. """
return S(randrange(25) + 10)/50
def randcplx(offset=-1):
""" Polys is not good with real coefficients. """
return _randrat() + I*_randrat() + I*(1 + offset)
@slow
def test_formulae():
from sympy.simplify.hyperexpand import FormulaCollection
formulae = FormulaCollection().formulae
for formula in formulae:
h = formula.func(formula.z)
rep = {}
for n, sym in enumerate(formula.symbols):
rep[sym] = randcplx(n)
# NOTE hyperexpand returns truly branched functions. We know we are
# on the main sheet, but numerical evaluation can still go wrong
# (e.g. if exp_polar cannot be evalf'd).
# Just replace all exp_polar by exp, this usually works.
# first test if the closed-form is actually correct
h = h.subs(rep)
closed_form = formula.closed_form.subs(rep).rewrite('nonrepsmall')
z = formula.z
assert tn(h, closed_form.replace(exp_polar, exp), z)
# now test the computed matrix
cl = (formula.C * formula.B)[0].subs(rep).rewrite('nonrepsmall')
assert tn(closed_form.replace(
exp_polar, exp), cl.replace(exp_polar, exp), z)
deriv1 = z*formula.B.applyfunc(lambda t: t.rewrite(
'nonrepsmall')).diff(z)
deriv2 = formula.M * formula.B
for d1, d2 in zip(deriv1, deriv2):
assert tn(d1.subs(rep).replace(exp_polar, exp),
d2.subs(rep).rewrite('nonrepsmall').replace(exp_polar, exp), z)
def test_meijerg_formulae():
from sympy.simplify.hyperexpand import MeijerFormulaCollection
formulae = MeijerFormulaCollection().formulae
for sig in formulae:
for formula in formulae[sig]:
g = meijerg(formula.func.an, formula.func.ap,
formula.func.bm, formula.func.bq,
formula.z)
rep = {}
for sym in formula.symbols:
rep[sym] = randcplx()
# first test if the closed-form is actually correct
g = g.subs(rep)
closed_form = formula.closed_form.subs(rep)
z = formula.z
assert tn(g, closed_form, z)
# now test the computed matrix
cl = (formula.C * formula.B)[0].subs(rep)
assert tn(closed_form, cl, z)
deriv1 = z*formula.B.diff(z)
deriv2 = formula.M * formula.B
for d1, d2 in zip(deriv1, deriv2):
assert tn(d1.subs(rep), d2.subs(rep), z)
def op(f):
return z*f.diff(z)
def test_plan():
assert devise_plan(Hyper_Function([0], ()),
Hyper_Function([0], ()), z) == []
with raises(ValueError):
devise_plan(Hyper_Function([1], ()), Hyper_Function((), ()), z)
with raises(ValueError):
devise_plan(Hyper_Function([2], [1]), Hyper_Function([2], [2]), z)
with raises(ValueError):
devise_plan(Hyper_Function([2], []), Hyper_Function([S("1/2")], []), z)
# We cannot use pi/(10000 + n) because polys is insanely slow.
a1, a2, b1 = (randcplx(n) for n in range(3))
b1 += 2*I
h = hyper([a1, a2], [b1], z)
h2 = hyper((a1 + 1, a2), [b1], z)
assert tn(apply_operators(h,
devise_plan(Hyper_Function((a1 + 1, a2), [b1]),
Hyper_Function((a1, a2), [b1]), z), op),
h2, z)
h2 = hyper((a1 + 1, a2 - 1), [b1], z)
assert tn(apply_operators(h,
devise_plan(Hyper_Function((a1 + 1, a2 - 1), [b1]),
Hyper_Function((a1, a2), [b1]), z), op),
h2, z)
def test_plan_derivatives():
a1, a2, a3 = 1, 2, S('1/2')
b1, b2 = 3, S('5/2')
h = Hyper_Function((a1, a2, a3), (b1, b2))
h2 = Hyper_Function((a1 + 1, a2 + 1, a3 + 2), (b1 + 1, b2 + 1))
ops = devise_plan(h2, h, z)
f = Formula(h, z, h(z), [])
deriv = make_derivative_operator(f.M, z)
assert tn((apply_operators(f.C, ops, deriv)*f.B)[0], h2(z), z)
h2 = Hyper_Function((a1, a2 - 1, a3 - 2), (b1 - 1, b2 - 1))
ops = devise_plan(h2, h, z)
assert tn((apply_operators(f.C, ops, deriv)*f.B)[0], h2(z), z)
def test_reduction_operators():
a1, a2, b1 = (randcplx(n) for n in range(3))
h = hyper([a1], [b1], z)
assert ReduceOrder(2, 0) is None
assert ReduceOrder(2, -1) is None
assert ReduceOrder(1, S('1/2')) is None
h2 = hyper((a1, a2), (b1, a2), z)
assert tn(ReduceOrder(a2, a2).apply(h, op), h2, z)
h2 = hyper((a1, a2 + 1), (b1, a2), z)
assert tn(ReduceOrder(a2 + 1, a2).apply(h, op), h2, z)
h2 = hyper((a2 + 4, a1), (b1, a2), z)
assert tn(ReduceOrder(a2 + 4, a2).apply(h, op), h2, z)
# test several step order reduction
ap = (a2 + 4, a1, b1 + 1)
bq = (a2, b1, b1)
func, ops = reduce_order(Hyper_Function(ap, bq))
assert func.ap == (a1,)
assert func.bq == (b1,)
assert tn(apply_operators(h, ops, op), hyper(ap, bq, z), z)
def test_shift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
raises(ValueError, lambda: ShiftA(0))
raises(ValueError, lambda: ShiftB(1))
assert tn(ShiftA(a1).apply(h, op), hyper((a1 + 1, a2), (b1, b2, b3), z), z)
assert tn(ShiftA(a2).apply(h, op), hyper((a1, a2 + 1), (b1, b2, b3), z), z)
assert tn(ShiftB(b1).apply(h, op), hyper((a1, a2), (b1 - 1, b2, b3), z), z)
assert tn(ShiftB(b2).apply(h, op), hyper((a1, a2), (b1, b2 - 1, b3), z), z)
assert tn(ShiftB(b3).apply(h, op), hyper((a1, a2), (b1, b2, b3 - 1), z), z)
def test_ushift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
raises(ValueError, lambda: UnShiftA((1,), (), 0, z))
raises(ValueError, lambda: UnShiftB((), (-1,), 0, z))
raises(ValueError, lambda: UnShiftA((1,), (0, -1, 1), 0, z))
raises(ValueError, lambda: UnShiftB((0, 1), (1,), 0, z))
s = UnShiftA((a1, a2), (b1, b2, b3), 0, z)
assert tn(s.apply(h, op), hyper((a1 - 1, a2), (b1, b2, b3), z), z)
s = UnShiftA((a1, a2), (b1, b2, b3), 1, z)
assert tn(s.apply(h, op), hyper((a1, a2 - 1), (b1, b2, b3), z), z)
s = UnShiftB((a1, a2), (b1, b2, b3), 0, z)
assert tn(s.apply(h, op), hyper((a1, a2), (b1 + 1, b2, b3), z), z)
s = UnShiftB((a1, a2), (b1, b2, b3), 1, z)
assert tn(s.apply(h, op), hyper((a1, a2), (b1, b2 + 1, b3), z), z)
s = UnShiftB((a1, a2), (b1, b2, b3), 2, z)
assert tn(s.apply(h, op), hyper((a1, a2), (b1, b2, b3 + 1), z), z)
def can_do_meijer(a1, a2, b1, b2, numeric=True):
"""
This helper function tries to hyperexpand() the meijer g-function
corresponding to the parameters a1, a2, b1, b2.
It returns False if this expansion still contains g-functions.
If numeric is True, it also tests the so-obtained formula numerically
(at random values) and returns False if the test fails.
Else it returns True.
"""
from sympy.core.function import expand
from sympy.functions.elementary.complexes import unpolarify
r = hyperexpand(meijerg(a1, a2, b1, b2, z))
if r.has(meijerg):
return False
# NOTE hyperexpand() returns a truly branched function, whereas numerical
# evaluation only works on the main branch. Since we are evaluating on
# the main branch, this should not be a problem, but expressions like
# exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get
# rid of them. The expand heuristically does this...
r = unpolarify(expand(r, force=True, power_base=True, power_exp=False,
mul=False, log=False, multinomial=False, basic=False))
if not numeric:
return True
repl = {}
for n, ai in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[ai] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
@slow
def test_meijerg_expand():
from sympy.simplify.gammasimp import gammasimp
from sympy.simplify.simplify import simplify
# from mpmath docs
assert hyperexpand(meijerg([[], []], [[0], []], -z)) == exp(z)
assert hyperexpand(meijerg([[1, 1], []], [[1], [0]], z)) == \
log(z + 1)
assert hyperexpand(meijerg([[1, 1], []], [[1], [1]], z)) == \
z/(z + 1)
assert hyperexpand(meijerg([[], []], [[S.Half], [0]], (z/2)**2)) \
== sin(z)/sqrt(pi)
assert hyperexpand(meijerg([[], []], [[0], [S.Half]], (z/2)**2)) \
== cos(z)/sqrt(pi)
assert can_do_meijer([], [a], [a - 1, a - S.Half], [])
assert can_do_meijer([], [], [a/2], [-a/2], False) # branches...
assert can_do_meijer([a], [b], [a], [b, a - 1])
# wikipedia
assert hyperexpand(meijerg([1], [], [], [0], z)) == \
Piecewise((0, abs(z) < 1), (1, abs(1/z) < 1),
(meijerg([1], [], [], [0], z), True))
assert hyperexpand(meijerg([], [1], [0], [], z)) == \
Piecewise((1, abs(z) < 1), (0, abs(1/z) < 1),
(meijerg([], [1], [0], [], z), True))
# The Special Functions and their Approximations
assert can_do_meijer([], [], [a + b/2], [a, a - b/2, a + S.Half])
assert can_do_meijer(
[], [], [a], [b], False) # branches only agree for small z
assert can_do_meijer([], [S.Half], [a], [-a])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, a + S.Half], [b, b + S.Half])
assert can_do_meijer([], [], [a, -a], [0, S.Half], False) # dito
assert can_do_meijer([], [], [a, a + S.Half, b, b + S.Half], [])
assert can_do_meijer([S.Half], [], [0], [a, -a])
assert can_do_meijer([S.Half], [], [a], [0, -a], False) # dito
assert can_do_meijer([], [a - S.Half], [a, b], [a - S.Half], False)
assert can_do_meijer([], [a + S.Half], [a + b, a - b, a], [], False)
assert can_do_meijer([a + S.Half], [], [b, 2*a - b, a], [], False)
# This for example is actually zero.
assert can_do_meijer([], [], [], [a, b])
# Testing a bug:
assert hyperexpand(meijerg([0, 2], [], [], [-1, 1], z)) == \
Piecewise((0, abs(z) < 1),
(z*(1 - 1/z**2)/2, abs(1/z) < 1),
(meijerg([0, 2], [], [], [-1, 1], z), True))
# Test that the simplest possible answer is returned:
assert gammasimp(simplify(hyperexpand(
meijerg([1], [1 - a], [-a/2, -a/2 + S.Half], [], 1/z)))) == \
-2*sqrt(pi)*(sqrt(z + 1) + 1)**a/a
# Test that hyper is returned
assert hyperexpand(meijerg([1], [], [a], [0, 0], z)) == hyper(
(a,), (a + 1, a + 1), z*exp_polar(I*pi))*z**a*gamma(a)/gamma(a + 1)**2
# Test place option
f = meijerg(((0, 1), ()), ((S.Half,), (0,)), z**2)
assert hyperexpand(f) == sqrt(pi)/sqrt(1 + z**(-2))
assert hyperexpand(f, place=0) == sqrt(pi)*z/sqrt(z**2 + 1)
def test_meijerg_lookup():
from sympy.functions.special.error_functions import (Ci, Si)
from sympy.functions.special.gamma_functions import uppergamma
assert hyperexpand(meijerg([a], [], [b, a], [], z)) == \
z**b*exp(z)*gamma(-a + b + 1)*uppergamma(a - b, z)
assert hyperexpand(meijerg([0], [], [0, 0], [], z)) == \
exp(z)*uppergamma(0, z)
assert can_do_meijer([a], [], [b, a + 1], [])
assert can_do_meijer([a], [], [b + 2, a], [])
assert can_do_meijer([a], [], [b - 2, a], [])
assert hyperexpand(meijerg([a], [], [a, a, a - S.Half], [], z)) == \
-sqrt(pi)*z**(a - S.Half)*(2*cos(2*sqrt(z))*(Si(2*sqrt(z)) - pi/2)
- 2*sin(2*sqrt(z))*Ci(2*sqrt(z))) == \
hyperexpand(meijerg([a], [], [a, a - S.Half, a], [], z)) == \
hyperexpand(meijerg([a], [], [a - S.Half, a, a], [], z))
assert can_do_meijer([a - 1], [], [a + 2, a - Rational(3, 2), a + 1], [])
@XFAIL
def test_meijerg_expand_fail():
# These basically test hyper([], [1/2 - a, 1/2 + 1, 1/2], z),
# which is *very* messy. But since the meijer g actually yields a
# sum of bessel functions, things can sometimes be simplified a lot and
# are then put into tables...
assert can_do_meijer([], [], [a + S.Half], [a, a - b/2, a + b/2])
assert can_do_meijer([], [], [0, S.Half], [a, -a])
assert can_do_meijer([], [], [3*a - S.Half, a, -a - S.Half], [a - S.Half])
assert can_do_meijer([], [], [0, a - S.Half, -a - S.Half], [S.Half])
assert can_do_meijer([], [], [a, b + S.Half, b], [2*b - a])
assert can_do_meijer([], [], [a, b + S.Half, b, 2*b - a])
assert can_do_meijer([S.Half], [], [-a, a], [0])
@slow
def test_meijerg():
# carefully set up the parameters.
# NOTE: this used to fail sometimes. I believe it is fixed, but if you
# hit an inexplicable test failure here, please let me know the seed.
a1, a2 = (randcplx(n) - 5*I - n*I for n in range(2))
b1, b2 = (randcplx(n) + 5*I + n*I for n in range(2))
b3, b4, b5, a3, a4, a5 = (randcplx() for n in range(6))
g = meijerg([a1], [a3, a4], [b1], [b3, b4], z)
assert ReduceOrder.meijer_minus(3, 4) is None
assert ReduceOrder.meijer_plus(4, 3) is None
g2 = meijerg([a1, a2], [a3, a4], [b1], [b3, b4, a2], z)
assert tn(ReduceOrder.meijer_plus(a2, a2).apply(g, op), g2, z)
g2 = meijerg([a1, a2], [a3, a4], [b1], [b3, b4, a2 + 1], z)
assert tn(ReduceOrder.meijer_plus(a2, a2 + 1).apply(g, op), g2, z)
g2 = meijerg([a1, a2 - 1], [a3, a4], [b1], [b3, b4, a2 + 2], z)
assert tn(ReduceOrder.meijer_plus(a2 - 1, a2 + 2).apply(g, op), g2, z)
g2 = meijerg([a1], [a3, a4, b2 - 1], [b1, b2 + 2], [b3, b4], z)
assert tn(ReduceOrder.meijer_minus(
b2 + 2, b2 - 1).apply(g, op), g2, z, tol=1e-6)
# test several-step reduction
an = [a1, a2]
bq = [b3, b4, a2 + 1]
ap = [a3, a4, b2 - 1]
bm = [b1, b2 + 1]
niq, ops = reduce_order_meijer(G_Function(an, ap, bm, bq))
assert niq.an == (a1,)
assert set(niq.ap) == {a3, a4}
assert niq.bm == (b1,)
assert set(niq.bq) == {b3, b4}
assert tn(apply_operators(g, ops, op), meijerg(an, ap, bm, bq, z), z)
def test_meijerg_shift_operators():
# carefully set up the parameters. XXX this still fails sometimes
a1, a2, a3, a4, a5, b1, b2, b3, b4, b5 = (randcplx(n) for n in range(10))
g = meijerg([a1], [a3, a4], [b1], [b3, b4], z)
assert tn(MeijerShiftA(b1).apply(g, op),
meijerg([a1], [a3, a4], [b1 + 1], [b3, b4], z), z)
assert tn(MeijerShiftB(a1).apply(g, op),
meijerg([a1 - 1], [a3, a4], [b1], [b3, b4], z), z)
assert tn(MeijerShiftC(b3).apply(g, op),
meijerg([a1], [a3, a4], [b1], [b3 + 1, b4], z), z)
assert tn(MeijerShiftD(a3).apply(g, op),
meijerg([a1], [a3 - 1, a4], [b1], [b3, b4], z), z)
s = MeijerUnShiftA([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1], [a3, a4], [b1 - 1], [b3, b4], z), z)
s = MeijerUnShiftC([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1], [a3, a4], [b1], [b3 - 1, b4], z), z)
s = MeijerUnShiftB([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1 + 1], [a3, a4], [b1], [b3, b4], z), z)
s = MeijerUnShiftD([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1], [a3 + 1, a4], [b1], [b3, b4], z), z)
@slow
def test_meijerg_confluence():
def t(m, a, b):
from sympy.core.sympify import sympify
a, b = sympify([a, b])
m_ = m
m = hyperexpand(m)
if not m == Piecewise((a, abs(z) < 1), (b, abs(1/z) < 1), (m_, True)):
return False
if not (m.args[0].args[0] == a and m.args[1].args[0] == b):
return False
z0 = randcplx()/10
if abs(m.subs(z, z0).n() - a.subs(z, z0).n()).n() > 1e-10:
return False
if abs(m.subs(z, 1/z0).n() - b.subs(z, 1/z0).n()).n() > 1e-10:
return False
return True
assert t(meijerg([], [1, 1], [0, 0], [], z), -log(z), 0)
assert t(meijerg(
[], [3, 1], [0, 0], [], z), -z**2/4 + z - log(z)/2 - Rational(3, 4), 0)
assert t(meijerg([], [3, 1], [-1, 0], [], z),
z**2/12 - z/2 + log(z)/2 + Rational(1, 4) + 1/(6*z), 0)
assert t(meijerg([], [1, 1, 1, 1], [0, 0, 0, 0], [], z), -log(z)**3/6, 0)
assert t(meijerg([1, 1], [], [], [0, 0], z), 0, -log(1/z))
assert t(meijerg([1, 1], [2, 2], [1, 1], [0, 0], z),
-z*log(z) + 2*z, -log(1/z) + 2)
assert t(meijerg([S.Half], [1, 1], [0, 0], [Rational(3, 2)], z), log(z)/2 - 1, 0)
def u(an, ap, bm, bq):
m = meijerg(an, ap, bm, bq, z)
m2 = hyperexpand(m, allow_hyper=True)
if m2.has(meijerg) and not (m2.is_Piecewise and len(m2.args) == 3):
return False
return tn(m, m2, z)
assert u([], [1], [0, 0], [])
assert u([1, 1], [], [], [0])
assert u([1, 1], [2, 2, 5], [1, 1, 6], [0, 0])
assert u([1, 1], [2, 2, 5], [1, 1, 6], [0])
def test_meijerg_with_Floats():
# see issue #10681
from sympy.polys.domains.realfield import RR
f = meijerg(((3.0, 1), ()), ((Rational(3, 2),), (0,)), z)
a = -2.3632718012073
g = a*z**Rational(3, 2)*hyper((-0.5, Rational(3, 2)), (Rational(5, 2),), z*exp_polar(I*pi))
assert RR.almosteq((hyperexpand(f)/g).n(), 1.0, 1e-12)
def test_lerchphi():
from sympy.functions.special.zeta_functions import (lerchphi, polylog)
from sympy.simplify.gammasimp import gammasimp
assert hyperexpand(hyper([1, a], [a + 1], z)/a) == lerchphi(z, 1, a)
assert hyperexpand(
hyper([1, a, a], [a + 1, a + 1], z)/a**2) == lerchphi(z, 2, a)
assert hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z)/a**3) == \
lerchphi(z, 3, a)
assert hyperexpand(hyper([1] + [a]*10, [a + 1]*10, z)/a**10) == \
lerchphi(z, 10, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a], [], [0],
[-a], exp_polar(-I*pi)*z))) == lerchphi(z, 1, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a], [], [0],
[-a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 2, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a, 1 - a], [], [0],
[-a, -a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 3, a)
assert hyperexpand(z*hyper([1, 1], [2], z)) == -log(1 + -z)
assert hyperexpand(z*hyper([1, 1, 1], [2, 2], z)) == polylog(2, z)
assert hyperexpand(z*hyper([1, 1, 1, 1], [2, 2, 2], z)) == polylog(3, z)
assert hyperexpand(hyper([1, a, 1 + S.Half], [a + 1, S.Half], z)) == \
-2*a/(z - 1) + (-2*a**2 + a)*lerchphi(z, 1, a)
# Now numerical tests. These make sure reductions etc are carried out
# correctly
# a rational function (polylog at negative integer order)
assert can_do([2, 2, 2], [1, 1])
# NOTE these contain log(1-x) etc ... better make sure we have |z| < 1
# reduction of order for polylog
assert can_do([1, 1, 1, b + 5], [2, 2, b], div=10)
# reduction of order for lerchphi
# XXX lerchphi in mpmath is flaky
assert can_do(
[1, a, a, a, b + 5], [a + 1, a + 1, a + 1, b], numerical=False)
# test a bug
from sympy.functions.elementary.complexes import Abs
assert hyperexpand(hyper([S.Half, S.Half, S.Half, 1],
[Rational(3, 2), Rational(3, 2), Rational(3, 2)], Rational(1, 4))) == \
Abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, S.Half))
def test_partial_simp():
# First test that hypergeometric function formulae work.
a, b, c, d, e = (randcplx() for _ in range(5))
for func in [Hyper_Function([a, b, c], [d, e]),
Hyper_Function([], [a, b, c, d, e])]:
f = build_hypergeometric_formula(func)
z = f.z
assert f.closed_form == func(z)
deriv1 = f.B.diff(z)*z
deriv2 = f.M*f.B
for func1, func2 in zip(deriv1, deriv2):
assert tn(func1, func2, z)
# Now test that formulae are partially simplified.
a, b, z = symbols('a b z')
assert hyperexpand(hyper([3, a], [1, b], z)) == \
(-a*b/2 + a*z/2 + 2*a)*hyper([a + 1], [b], z) \
+ (a*b/2 - 2*a + 1)*hyper([a], [b], z)
assert tn(
hyperexpand(hyper([3, d], [1, e], z)), hyper([3, d], [1, e], z), z)
assert hyperexpand(hyper([3], [1, a, b], z)) == \
hyper((), (a, b), z) \
+ z*hyper((), (a + 1, b), z)/(2*a) \
- z*(b - 4)*hyper((), (a + 1, b + 1), z)/(2*a*b)
assert tn(
hyperexpand(hyper([3], [1, d, e], z)), hyper([3], [1, d, e], z), z)
def test_hyperexpand_special():
assert hyperexpand(hyper([a, b], [c], 1)) == \
gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b)
assert hyperexpand(hyper([a, b], [1 + a - b], -1)) == \
gamma(1 + a/2)*gamma(1 + a - b)/gamma(1 + a)/gamma(1 + a/2 - b)
assert hyperexpand(hyper([a, b], [1 + b - a], -1)) == \
gamma(1 + b/2)*gamma(1 + b - a)/gamma(1 + b)/gamma(1 + b/2 - a)
assert hyperexpand(meijerg([1 - z - a/2], [1 - z + a/2], [b/2], [-b/2], 1)) == \
gamma(1 - 2*z)*gamma(z + a/2 + b/2)/gamma(1 - z + a/2 - b/2) \
/gamma(1 - z - a/2 + b/2)/gamma(1 - z + a/2 + b/2)
assert hyperexpand(hyper([a], [b], 0)) == 1
assert hyper([a], [b], 0) != 0
def test_Mod1_behavior():
from sympy.core.symbol import Symbol
from sympy.simplify.simplify import simplify
n = Symbol('n', integer=True)
# Note: this should not hang.
assert simplify(hyperexpand(meijerg([1], [], [n + 1], [0], z))) == \
lowergamma(n + 1, z)
@slow
def test_prudnikov_misc():
assert can_do([1, (3 + I)/2, (3 - I)/2], [Rational(3, 2), 2])
assert can_do([S.Half, a - 1], [Rational(3, 2), a + 1], lowerplane=True)
assert can_do([], [b + 1])
assert can_do([a], [a - 1, b + 1])
assert can_do([a], [a - S.Half, 2*a])
assert can_do([a], [a - S.Half, 2*a + 1])
assert can_do([a], [a - S.Half, 2*a - 1])
assert can_do([a], [a + S.Half, 2*a])
assert can_do([a], [a + S.Half, 2*a + 1])
assert can_do([a], [a + S.Half, 2*a - 1])
assert can_do([S.Half], [b, 2 - b])
assert can_do([S.Half], [b, 3 - b])
assert can_do([1], [2, b])
assert can_do([a, a + S.Half], [2*a, b, 2*a - b + 1])
assert can_do([a, a + S.Half], [S.Half, 2*a, 2*a + S.Half])
assert can_do([a], [a + 1], lowerplane=True) # lowergamma
def test_prudnikov_1():
# A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
# Integrals and Series: More Special Functions, Vol. 3,.
# Gordon and Breach Science Publisher
# 7.3.1
assert can_do([a, -a], [S.Half])
assert can_do([a, 1 - a], [S.Half])
assert can_do([a, 1 - a], [Rational(3, 2)])
assert can_do([a, 2 - a], [S.Half])
assert can_do([a, 2 - a], [Rational(3, 2)])
assert can_do([a, 2 - a], [Rational(3, 2)])
assert can_do([a, a + S.Half], [2*a - 1])
assert can_do([a, a + S.Half], [2*a])
assert can_do([a, a + S.Half], [2*a + 1])
assert can_do([a, a + S.Half], [S.Half])
assert can_do([a, a + S.Half], [Rational(3, 2)])
assert can_do([a, a/2 + 1], [a/2])
assert can_do([1, b], [2])
assert can_do([1, b], [b + 1], numerical=False) # Lerch Phi
# NOTE: branches are complicated for |z| > 1
assert can_do([a], [2*a])
assert can_do([a], [2*a + 1])
assert can_do([a], [2*a - 1])
@slow
def test_prudnikov_2():
h = S.Half
assert can_do([-h, -h], [h])
assert can_do([-h, h], [3*h])
assert can_do([-h, h], [5*h])
assert can_do([-h, h], [7*h])
assert can_do([-h, 1], [h])
for p in [-h, h]:
for n in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]:
for m in [-h, h, 3*h, 5*h, 7*h]:
assert can_do([p, n], [m])
for n in [1, 2, 3, 4]:
for m in [1, 2, 3, 4]:
assert can_do([p, n], [m])
@slow
def test_prudnikov_3():
if ON_CI:
# See https://github.com/sympy/sympy/pull/12795
skip("Too slow for CI.")
h = S.Half
assert can_do([Rational(1, 4), Rational(3, 4)], [h])
assert can_do([Rational(1, 4), Rational(3, 4)], [3*h])
assert can_do([Rational(1, 3), Rational(2, 3)], [3*h])
assert can_do([Rational(3, 4), Rational(5, 4)], [h])
assert can_do([Rational(3, 4), Rational(5, 4)], [3*h])
for p in [1, 2, 3, 4]:
for n in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4, 9*h]:
for m in [1, 3*h, 2, 5*h, 3, 7*h, 4]:
assert can_do([p, m], [n])
@slow
def test_prudnikov_4():
h = S.Half
for p in [3*h, 5*h, 7*h]:
for n in [-h, h, 3*h, 5*h, 7*h]:
for m in [3*h, 2, 5*h, 3, 7*h, 4]:
assert can_do([p, m], [n])
for n in [1, 2, 3, 4]:
for m in [2, 3, 4]:
assert can_do([p, m], [n])
@slow
def test_prudnikov_5():
h = S.Half
for p in [1, 2, 3]:
for q in range(p, 4):
for r in [1, 2, 3]:
for s in range(r, 4):
assert can_do([-h, p, q], [r, s])
for p in [h, 1, 3*h, 2, 5*h, 3]:
for q in [h, 3*h, 5*h]:
for r in [h, 3*h, 5*h]:
for s in [h, 3*h, 5*h]:
if s <= q and s <= r:
assert can_do([-h, p, q], [r, s])
for p in [h, 1, 3*h, 2, 5*h, 3]:
for q in [1, 2, 3]:
for r in [h, 3*h, 5*h]:
for s in [1, 2, 3]:
assert can_do([-h, p, q], [r, s])
@slow
def test_prudnikov_6():
h = S.Half
for m in [3*h, 5*h]:
for n in [1, 2, 3]:
for q in [h, 1, 2]:
for p in [1, 2, 3]:
assert can_do([h, q, p], [m, n])
for q in [1, 2, 3]:
for p in [3*h, 5*h]:
assert can_do([h, q, p], [m, n])
for q in [1, 2]:
for p in [1, 2, 3]:
for m in [1, 2, 3]:
for n in [1, 2, 3]:
assert can_do([h, q, p], [m, n])
assert can_do([h, h, 5*h], [3*h, 3*h])
assert can_do([h, 1, 5*h], [3*h, 3*h])
assert can_do([h, 2, 2], [1, 3])
# pages 435 to 457 contain more PFDD and stuff like this
@slow
def test_prudnikov_7():
assert can_do([3], [6])
h = S.Half
for n in [h, 3*h, 5*h, 7*h]:
assert can_do([-h], [n])
for m in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]: # HERE
for n in [-h, h, 3*h, 5*h, 7*h, 1, 2, 3, 4]:
assert can_do([m], [n])
@slow
def test_prudnikov_8():
h = S.Half
# 7.12.2
for ai in [1, 2, 3]:
for bi in [1, 2, 3]:
for ci in range(1, ai + 1):
for di in [h, 1, 3*h, 2, 5*h, 3]:
assert can_do([ai, bi], [ci, di])
for bi in [3*h, 5*h]:
for ci in [h, 1, 3*h, 2, 5*h, 3]:
for di in [1, 2, 3]:
assert can_do([ai, bi], [ci, di])
for ai in [-h, h, 3*h, 5*h]:
for bi in [1, 2, 3]:
for ci in [h, 1, 3*h, 2, 5*h, 3]:
for di in [1, 2, 3]:
assert can_do([ai, bi], [ci, di])
for bi in [h, 3*h, 5*h]:
for ci in [h, 3*h, 5*h, 3]:
for di in [h, 1, 3*h, 2, 5*h, 3]:
if ci <= bi:
assert can_do([ai, bi], [ci, di])
def test_prudnikov_9():
# 7.13.1 [we have a general formula ... so this is a bit pointless]
for i in range(9):
assert can_do([], [(S(i) + 1)/2])
for i in range(5):
assert can_do([], [-(2*S(i) + 1)/2])
@slow
def test_prudnikov_10():
# 7.14.2
h = S.Half
for p in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]:
for m in [1, 2, 3, 4]:
for n in range(m, 5):
assert can_do([p], [m, n])
for p in [1, 2, 3, 4]:
for n in [h, 3*h, 5*h, 7*h]:
for m in [1, 2, 3, 4]:
assert can_do([p], [n, m])
for p in [3*h, 5*h, 7*h]:
for m in [h, 1, 2, 5*h, 3, 7*h, 4]:
assert can_do([p], [h, m])
assert can_do([p], [3*h, m])
for m in [h, 1, 2, 5*h, 3, 7*h, 4]:
assert can_do([7*h], [5*h, m])
assert can_do([Rational(-1, 2)], [S.Half, S.Half]) # shine-integral shi
def test_prudnikov_11():
# 7.15
assert can_do([a, a + S.Half], [2*a, b, 2*a - b])
assert can_do([a, a + S.Half], [Rational(3, 2), 2*a, 2*a - S.Half])
assert can_do([Rational(1, 4), Rational(3, 4)], [S.Half, S.Half, 1])
assert can_do([Rational(5, 4), Rational(3, 4)], [Rational(3, 2), S.Half, 2])
assert can_do([Rational(5, 4), Rational(3, 4)], [Rational(3, 2), Rational(3, 2), 1])
assert can_do([Rational(5, 4), Rational(7, 4)], [Rational(3, 2), Rational(5, 2), 2])
assert can_do([1, 1], [Rational(3, 2), 2, 2]) # cosh-integral chi
def test_prudnikov_12():
# 7.16
assert can_do(
[], [a, a + S.Half, 2*a], False) # branches only agree for some z!
assert can_do([], [a, a + S.Half, 2*a + 1], False) # dito
assert can_do([], [S.Half, a, a + S.Half])
assert can_do([], [Rational(3, 2), a, a + S.Half])
assert can_do([], [Rational(1, 4), S.Half, Rational(3, 4)])
assert can_do([], [S.Half, S.Half, 1])
assert can_do([], [S.Half, Rational(3, 2), 1])
assert can_do([], [Rational(3, 4), Rational(3, 2), Rational(5, 4)])
assert can_do([], [1, 1, Rational(3, 2)])
assert can_do([], [1, 2, Rational(3, 2)])
assert can_do([], [1, Rational(3, 2), Rational(3, 2)])
assert can_do([], [Rational(5, 4), Rational(3, 2), Rational(7, 4)])
assert can_do([], [2, Rational(3, 2), Rational(3, 2)])
@slow
def test_prudnikov_2F1():
h = S.Half
# Elliptic integrals
for p in [-h, h]:
for m in [h, 3*h, 5*h, 7*h]:
for n in [1, 2, 3, 4]:
assert can_do([p, m], [n])
@XFAIL
def test_prudnikov_fail_2F1():
assert can_do([a, b], [b + 1]) # incomplete beta function
assert can_do([-1, b], [c]) # Poly. also -2, -3 etc
# TODO polys
# Legendre functions:
assert can_do([a, b], [a + b + S.Half])
assert can_do([a, b], [a + b - S.Half])
assert can_do([a, b], [a + b + Rational(3, 2)])
assert can_do([a, b], [(a + b + 1)/2])
assert can_do([a, b], [(a + b)/2 + 1])
assert can_do([a, b], [a - b + 1])
assert can_do([a, b], [a - b + 2])
assert can_do([a, b], [2*b])
assert can_do([a, b], [S.Half])
assert can_do([a, b], [Rational(3, 2)])
assert can_do([a, 1 - a], [c])
assert can_do([a, 2 - a], [c])
assert can_do([a, 3 - a], [c])
assert can_do([a, a + S.Half], [c])
assert can_do([1, b], [c])
assert can_do([1, b], [Rational(3, 2)])
assert can_do([Rational(1, 4), Rational(3, 4)], [1])
# PFDD
o = S.One
assert can_do([o/8, 1], [o/8*9])
assert can_do([o/6, 1], [o/6*7])
assert can_do([o/6, 1], [o/6*13])
assert can_do([o/5, 1], [o/5*6])
assert can_do([o/5, 1], [o/5*11])
assert can_do([o/4, 1], [o/4*5])
assert can_do([o/4, 1], [o/4*9])
assert can_do([o/3, 1], [o/3*4])
assert can_do([o/3, 1], [o/3*7])
assert can_do([o/8*3, 1], [o/8*11])
assert can_do([o/5*2, 1], [o/5*7])
assert can_do([o/5*2, 1], [o/5*12])
assert can_do([o/5*3, 1], [o/5*8])
assert can_do([o/5*3, 1], [o/5*13])
assert can_do([o/8*5, 1], [o/8*13])
assert can_do([o/4*3, 1], [o/4*7])
assert can_do([o/4*3, 1], [o/4*11])
assert can_do([o/3*2, 1], [o/3*5])
assert can_do([o/3*2, 1], [o/3*8])
assert can_do([o/5*4, 1], [o/5*9])
assert can_do([o/5*4, 1], [o/5*14])
assert can_do([o/6*5, 1], [o/6*11])
assert can_do([o/6*5, 1], [o/6*17])
assert can_do([o/8*7, 1], [o/8*15])
@XFAIL
def test_prudnikov_fail_3F2():
assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [Rational(1, 3), Rational(2, 3)])
assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [Rational(2, 3), Rational(4, 3)])
assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [Rational(4, 3), Rational(5, 3)])
# page 421
assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [a*Rational(3, 2), (3*a + 1)/2])
# pages 422 ...
assert can_do([Rational(-1, 2), S.Half, S.Half], [1, 1]) # elliptic integrals
assert can_do([Rational(-1, 2), S.Half, 1], [Rational(3, 2), Rational(3, 2)])
# TODO LOTS more
# PFDD
assert can_do([Rational(1, 8), Rational(3, 8), 1], [Rational(9, 8), Rational(11, 8)])
assert can_do([Rational(1, 8), Rational(5, 8), 1], [Rational(9, 8), Rational(13, 8)])
assert can_do([Rational(1, 8), Rational(7, 8), 1], [Rational(9, 8), Rational(15, 8)])
assert can_do([Rational(1, 6), Rational(1, 3), 1], [Rational(7, 6), Rational(4, 3)])
assert can_do([Rational(1, 6), Rational(2, 3), 1], [Rational(7, 6), Rational(5, 3)])
assert can_do([Rational(1, 6), Rational(2, 3), 1], [Rational(5, 3), Rational(13, 6)])
assert can_do([S.Half, 1, 1], [Rational(1, 4), Rational(3, 4)])
# LOTS more
@XFAIL
def test_prudnikov_fail_other():
# 7.11.2
# 7.12.1
assert can_do([1, a], [b, 1 - 2*a + b]) # ???
# 7.14.2
assert can_do([Rational(-1, 2)], [S.Half, 1]) # struve
assert can_do([1], [S.Half, S.Half]) # struve
assert can_do([Rational(1, 4)], [S.Half, Rational(5, 4)]) # PFDD
assert can_do([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)]) # PFDD
assert can_do([1], [Rational(1, 4), Rational(3, 4)]) # PFDD
assert can_do([1], [Rational(3, 4), Rational(5, 4)]) # PFDD
assert can_do([1], [Rational(5, 4), Rational(7, 4)]) # PFDD
# TODO LOTS more
# 7.15.2
assert can_do([S.Half, 1], [Rational(3, 4), Rational(5, 4), Rational(3, 2)]) # PFDD
assert can_do([S.Half, 1], [Rational(7, 4), Rational(5, 4), Rational(3, 2)]) # PFDD
# 7.16.1
assert can_do([], [Rational(1, 3), S(2/3)]) # PFDD
assert can_do([], [Rational(2, 3), S(4/3)]) # PFDD
assert can_do([], [Rational(5, 3), S(4/3)]) # PFDD
# XXX this does not *evaluate* right??
assert can_do([], [a, a + S.Half, 2*a - 1])
def test_bug():
h = hyper([-1, 1], [z], -1)
assert hyperexpand(h) == (z + 1)/z
def test_omgissue_203():
h = hyper((-5, -3, -4), (-6, -6), 1)
assert hyperexpand(h) == Rational(1, 30)
h = hyper((-6, -7, -5), (-6, -6), 1)
assert hyperexpand(h) == Rational(-1, 6)
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