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| from sympy.core.containers import Tuple | |
| from sympy.core.function import Derivative | |
| from sympy.core.numbers import (I, Rational, oo, pi) | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import symbols | |
| from sympy.functions.elementary.exponential import (exp, log) | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import cos | |
| from sympy.functions.special.gamma_functions import gamma | |
| from sympy.functions.special.hyper import (appellf1, hyper, meijerg) | |
| from sympy.series.order import O | |
| from sympy.abc import x, z, k | |
| from sympy.series.limits import limit | |
| from sympy.testing.pytest import raises, slow | |
| from sympy.core.random import ( | |
| random_complex_number as randcplx, | |
| verify_numerically as tn, | |
| test_derivative_numerically as td) | |
| def test_TupleParametersBase(): | |
| # test that our implementation of the chain rule works | |
| p = hyper((), (), z**2) | |
| assert p.diff(z) == p*2*z | |
| def test_hyper(): | |
| raises(TypeError, lambda: hyper(1, 2, z)) | |
| assert hyper((2, 1), (1,), z) == hyper(Tuple(1, 2), Tuple(1), z) | |
| assert hyper((2, 1, 2), (1, 2, 1, 3), z) == hyper((2,), (1, 3), z) | |
| u = hyper((2, 1, 2), (1, 2, 1, 3), z, evaluate=False) | |
| assert u.ap == Tuple(1, 2, 2) | |
| assert u.bq == Tuple(1, 1, 2, 3) | |
| h = hyper((1, 2), (3, 4, 5), z) | |
| assert h.ap == Tuple(1, 2) | |
| assert h.bq == Tuple(3, 4, 5) | |
| assert h.argument == z | |
| assert h.is_commutative is True | |
| h = hyper((2, 1), (4, 3, 5), z) | |
| assert h.ap == Tuple(1, 2) | |
| assert h.bq == Tuple(3, 4, 5) | |
| assert h.argument == z | |
| assert h.is_commutative is True | |
| # just a few checks to make sure that all arguments go where they should | |
| assert tn(hyper(Tuple(), Tuple(), z), exp(z), z) | |
| assert tn(z*hyper((1, 1), Tuple(2), -z), log(1 + z), z) | |
| # differentiation | |
| h = hyper( | |
| (randcplx(), randcplx(), randcplx()), (randcplx(), randcplx()), z) | |
| assert td(h, z) | |
| a1, a2, b1, b2, b3 = symbols('a1:3, b1:4') | |
| assert hyper((a1, a2), (b1, b2, b3), z).diff(z) == \ | |
| a1*a2/(b1*b2*b3) * hyper((a1 + 1, a2 + 1), (b1 + 1, b2 + 1, b3 + 1), z) | |
| # differentiation wrt parameters is not supported | |
| assert hyper([z], [], z).diff(z) == Derivative(hyper([z], [], z), z) | |
| # hyper is unbranched wrt parameters | |
| from sympy.functions.elementary.complexes import polar_lift | |
| assert hyper([polar_lift(z)], [polar_lift(k)], polar_lift(x)) == \ | |
| hyper([z], [k], polar_lift(x)) | |
| # hyper does not automatically evaluate anyway, but the test is to make | |
| # sure that the evaluate keyword is accepted | |
| assert hyper((1, 2), (1,), z, evaluate=False).func is hyper | |
| def test_expand_func(): | |
| # evaluation at 1 of Gauss' hypergeometric function: | |
| from sympy.abc import a, b, c | |
| from sympy.core.function import expand_func | |
| a1, b1, c1 = randcplx(), randcplx(), randcplx() + 5 | |
| assert expand_func(hyper([a, b], [c], 1)) == \ | |
| gamma(c)*gamma(-a - b + c)/(gamma(-a + c)*gamma(-b + c)) | |
| assert abs(expand_func(hyper([a1, b1], [c1], 1)).n() | |
| - hyper([a1, b1], [c1], 1).n()) < 1e-10 | |
| # hyperexpand wrapper for hyper: | |
| assert expand_func(hyper([], [], z)) == exp(z) | |
| assert expand_func(hyper([1, 2, 3], [], z)) == hyper([1, 2, 3], [], z) | |
| assert expand_func(meijerg([[1, 1], []], [[1], [0]], z)) == log(z + 1) | |
| assert expand_func(meijerg([[1, 1], []], [[], []], z)) == \ | |
| meijerg([[1, 1], []], [[], []], z) | |
| def replace_dummy(expr, sym): | |
| from sympy.core.symbol import Dummy | |
| dum = expr.atoms(Dummy) | |
| if not dum: | |
| return expr | |
| assert len(dum) == 1 | |
| return expr.xreplace({dum.pop(): sym}) | |
| def test_hyper_rewrite_sum(): | |
| from sympy.concrete.summations import Sum | |
| from sympy.core.symbol import Dummy | |
| from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial) | |
| _k = Dummy("k") | |
| assert replace_dummy(hyper((1, 2), (1, 3), x).rewrite(Sum), _k) == \ | |
| Sum(x**_k / factorial(_k) * RisingFactorial(2, _k) / | |
| RisingFactorial(3, _k), (_k, 0, oo)) | |
| assert hyper((1, 2, 3), (-1, 3), z).rewrite(Sum) == \ | |
| hyper((1, 2, 3), (-1, 3), z) | |
| def test_radius_of_convergence(): | |
| assert hyper((1, 2), [3], z).radius_of_convergence == 1 | |
| assert hyper((1, 2), [3, 4], z).radius_of_convergence is oo | |
| assert hyper((1, 2, 3), [4], z).radius_of_convergence == 0 | |
| assert hyper((0, 1, 2), [4], z).radius_of_convergence is oo | |
| assert hyper((-1, 1, 2), [-4], z).radius_of_convergence == 0 | |
| assert hyper((-1, -2, 2), [-1], z).radius_of_convergence is oo | |
| assert hyper((-1, 2), [-1, -2], z).radius_of_convergence == 0 | |
| assert hyper([-1, 1, 3], [-2, 2], z).radius_of_convergence == 1 | |
| assert hyper([-1, 1], [-2, 2], z).radius_of_convergence is oo | |
| assert hyper([-1, 1, 3], [-2], z).radius_of_convergence == 0 | |
| assert hyper((-1, 2, 3, 4), [], z).radius_of_convergence is oo | |
| assert hyper([1, 1], [3], 1).convergence_statement == True | |
| assert hyper([1, 1], [2], 1).convergence_statement == False | |
| assert hyper([1, 1], [2], -1).convergence_statement == True | |
| assert hyper([1, 1], [1], -1).convergence_statement == False | |
| def test_meijer(): | |
| raises(TypeError, lambda: meijerg(1, z)) | |
| raises(TypeError, lambda: meijerg(((1,), (2,)), (3,), (4,), z)) | |
| assert meijerg(((1, 2), (3,)), ((4,), (5,)), z) == \ | |
| meijerg(Tuple(1, 2), Tuple(3), Tuple(4), Tuple(5), z) | |
| g = meijerg((1, 2), (3, 4, 5), (6, 7, 8, 9), (10, 11, 12, 13, 14), z) | |
| assert g.an == Tuple(1, 2) | |
| assert g.ap == Tuple(1, 2, 3, 4, 5) | |
| assert g.aother == Tuple(3, 4, 5) | |
| assert g.bm == Tuple(6, 7, 8, 9) | |
| assert g.bq == Tuple(6, 7, 8, 9, 10, 11, 12, 13, 14) | |
| assert g.bother == Tuple(10, 11, 12, 13, 14) | |
| assert g.argument == z | |
| assert g.nu == 75 | |
| assert g.delta == -1 | |
| assert g.is_commutative is True | |
| assert g.is_number is False | |
| #issue 13071 | |
| assert meijerg([[],[]], [[S.Half],[0]], 1).is_number is True | |
| assert meijerg([1, 2], [3], [4], [5], z).delta == S.Half | |
| # just a few checks to make sure that all arguments go where they should | |
| assert tn(meijerg(Tuple(), Tuple(), Tuple(0), Tuple(), -z), exp(z), z) | |
| assert tn(sqrt(pi)*meijerg(Tuple(), Tuple(), | |
| Tuple(0), Tuple(S.Half), z**2/4), cos(z), z) | |
| assert tn(meijerg(Tuple(1, 1), Tuple(), Tuple(1), Tuple(0), z), | |
| log(1 + z), z) | |
| # test exceptions | |
| raises(ValueError, lambda: meijerg(((3, 1), (2,)), ((oo,), (2, 0)), x)) | |
| raises(ValueError, lambda: meijerg(((3, 1), (2,)), ((1,), (2, 0)), x)) | |
| # differentiation | |
| g = meijerg((randcplx(),), (randcplx() + 2*I,), Tuple(), | |
| (randcplx(), randcplx()), z) | |
| assert td(g, z) | |
| g = meijerg(Tuple(), (randcplx(),), Tuple(), | |
| (randcplx(), randcplx()), z) | |
| assert td(g, z) | |
| g = meijerg(Tuple(), Tuple(), Tuple(randcplx()), | |
| Tuple(randcplx(), randcplx()), z) | |
| assert td(g, z) | |
| a1, a2, b1, b2, c1, c2, d1, d2 = symbols('a1:3, b1:3, c1:3, d1:3') | |
| assert meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z).diff(z) == \ | |
| (meijerg((a1 - 1, a2), (b1, b2), (c1, c2), (d1, d2), z) | |
| + (a1 - 1)*meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z))/z | |
| assert meijerg([z, z], [], [], [], z).diff(z) == \ | |
| Derivative(meijerg([z, z], [], [], [], z), z) | |
| # meijerg is unbranched wrt parameters | |
| from sympy.functions.elementary.complexes import polar_lift as pl | |
| assert meijerg([pl(a1)], [pl(a2)], [pl(b1)], [pl(b2)], pl(z)) == \ | |
| meijerg([a1], [a2], [b1], [b2], pl(z)) | |
| # integrand | |
| from sympy.abc import a, b, c, d, s | |
| assert meijerg([a], [b], [c], [d], z).integrand(s) == \ | |
| z**s*gamma(c - s)*gamma(-a + s + 1)/(gamma(b - s)*gamma(-d + s + 1)) | |
| def test_meijerg_derivative(): | |
| assert meijerg([], [1, 1], [0, 0, x], [], z).diff(x) == \ | |
| log(z)*meijerg([], [1, 1], [0, 0, x], [], z) \ | |
| + 2*meijerg([], [1, 1, 1], [0, 0, x, 0], [], z) | |
| y = randcplx() | |
| a = 5 # mpmath chokes with non-real numbers, and Mod1 with floats | |
| assert td(meijerg([x], [], [], [], y), x) | |
| assert td(meijerg([x**2], [], [], [], y), x) | |
| assert td(meijerg([], [x], [], [], y), x) | |
| assert td(meijerg([], [], [x], [], y), x) | |
| assert td(meijerg([], [], [], [x], y), x) | |
| assert td(meijerg([x], [a], [a + 1], [], y), x) | |
| assert td(meijerg([x], [a + 1], [a], [], y), x) | |
| assert td(meijerg([x, a], [], [], [a + 1], y), x) | |
| assert td(meijerg([x, a + 1], [], [], [a], y), x) | |
| b = Rational(3, 2) | |
| assert td(meijerg([a + 2], [b], [b - 3, x], [a], y), x) | |
| def test_meijerg_period(): | |
| assert meijerg([], [1], [0], [], x).get_period() == 2*pi | |
| assert meijerg([1], [], [], [0], x).get_period() == 2*pi | |
| assert meijerg([], [], [0], [], x).get_period() == 2*pi # exp(x) | |
| assert meijerg( | |
| [], [], [0], [S.Half], x).get_period() == 2*pi # cos(sqrt(x)) | |
| assert meijerg( | |
| [], [], [S.Half], [0], x).get_period() == 4*pi # sin(sqrt(x)) | |
| assert meijerg([1, 1], [], [1], [0], x).get_period() is oo # log(1 + x) | |
| def test_hyper_unpolarify(): | |
| from sympy.functions.elementary.exponential import exp_polar | |
| a = exp_polar(2*pi*I)*x | |
| b = x | |
| assert hyper([], [], a).argument == b | |
| assert hyper([0], [], a).argument == a | |
| assert hyper([0], [0], a).argument == b | |
| assert hyper([0, 1], [0], a).argument == a | |
| assert hyper([0, 1], [0], exp_polar(2*pi*I)).argument == 1 | |
| def test_hyperrep(): | |
| from sympy.functions.special.hyper import (HyperRep, HyperRep_atanh, | |
| HyperRep_power1, HyperRep_power2, HyperRep_log1, HyperRep_asin1, | |
| HyperRep_asin2, HyperRep_sqrts1, HyperRep_sqrts2, HyperRep_log2, | |
| HyperRep_cosasin, HyperRep_sinasin) | |
| # First test the base class works. | |
| from sympy.functions.elementary.exponential import exp_polar | |
| from sympy.functions.elementary.piecewise import Piecewise | |
| a, b, c, d, z = symbols('a b c d z') | |
| class myrep(HyperRep): | |
| def _expr_small(cls, x): | |
| return a | |
| def _expr_small_minus(cls, x): | |
| return b | |
| def _expr_big(cls, x, n): | |
| return c*n | |
| def _expr_big_minus(cls, x, n): | |
| return d*n | |
| assert myrep(z).rewrite('nonrep') == Piecewise((0, abs(z) > 1), (a, True)) | |
| assert myrep(exp_polar(I*pi)*z).rewrite('nonrep') == \ | |
| Piecewise((0, abs(z) > 1), (b, True)) | |
| assert myrep(exp_polar(2*I*pi)*z).rewrite('nonrep') == \ | |
| Piecewise((c, abs(z) > 1), (a, True)) | |
| assert myrep(exp_polar(3*I*pi)*z).rewrite('nonrep') == \ | |
| Piecewise((d, abs(z) > 1), (b, True)) | |
| assert myrep(exp_polar(4*I*pi)*z).rewrite('nonrep') == \ | |
| Piecewise((2*c, abs(z) > 1), (a, True)) | |
| assert myrep(exp_polar(5*I*pi)*z).rewrite('nonrep') == \ | |
| Piecewise((2*d, abs(z) > 1), (b, True)) | |
| assert myrep(z).rewrite('nonrepsmall') == a | |
| assert myrep(exp_polar(I*pi)*z).rewrite('nonrepsmall') == b | |
| def t(func, hyp, z): | |
| """ Test that func is a valid representation of hyp. """ | |
| # First test that func agrees with hyp for small z | |
| if not tn(func.rewrite('nonrepsmall'), hyp, z, | |
| a=Rational(-1, 2), b=Rational(-1, 2), c=S.Half, d=S.Half): | |
| return False | |
| # Next check that the two small representations agree. | |
| if not tn( | |
| func.rewrite('nonrepsmall').subs( | |
| z, exp_polar(I*pi)*z).replace(exp_polar, exp), | |
| func.subs(z, exp_polar(I*pi)*z).rewrite('nonrepsmall'), | |
| z, a=Rational(-1, 2), b=Rational(-1, 2), c=S.Half, d=S.Half): | |
| return False | |
| # Next check continuity along exp_polar(I*pi)*t | |
| expr = func.subs(z, exp_polar(I*pi)*z).rewrite('nonrep') | |
| if abs(expr.subs(z, 1 + 1e-15).n() - expr.subs(z, 1 - 1e-15).n()) > 1e-10: | |
| return False | |
| # Finally check continuity of the big reps. | |
| def dosubs(func, a, b): | |
| rv = func.subs(z, exp_polar(a)*z).rewrite('nonrep') | |
| return rv.subs(z, exp_polar(b)*z).replace(exp_polar, exp) | |
| for n in [0, 1, 2, 3, 4, -1, -2, -3, -4]: | |
| expr1 = dosubs(func, 2*I*pi*n, I*pi/2) | |
| expr2 = dosubs(func, 2*I*pi*n + I*pi, -I*pi/2) | |
| if not tn(expr1, expr2, z): | |
| return False | |
| expr1 = dosubs(func, 2*I*pi*(n + 1), -I*pi/2) | |
| expr2 = dosubs(func, 2*I*pi*n + I*pi, I*pi/2) | |
| if not tn(expr1, expr2, z): | |
| return False | |
| return True | |
| # Now test the various representatives. | |
| a = Rational(1, 3) | |
| assert t(HyperRep_atanh(z), hyper([S.Half, 1], [Rational(3, 2)], z), z) | |
| assert t(HyperRep_power1(a, z), hyper([-a], [], z), z) | |
| assert t(HyperRep_power2(a, z), hyper([a, a - S.Half], [2*a], z), z) | |
| assert t(HyperRep_log1(z), -z*hyper([1, 1], [2], z), z) | |
| assert t(HyperRep_asin1(z), hyper([S.Half, S.Half], [Rational(3, 2)], z), z) | |
| assert t(HyperRep_asin2(z), hyper([1, 1], [Rational(3, 2)], z), z) | |
| assert t(HyperRep_sqrts1(a, z), hyper([-a, S.Half - a], [S.Half], z), z) | |
| assert t(HyperRep_sqrts2(a, z), | |
| -2*z/(2*a + 1)*hyper([-a - S.Half, -a], [S.Half], z).diff(z), z) | |
| assert t(HyperRep_log2(z), -z/4*hyper([Rational(3, 2), 1, 1], [2, 2], z), z) | |
| assert t(HyperRep_cosasin(a, z), hyper([-a, a], [S.Half], z), z) | |
| assert t(HyperRep_sinasin(a, z), 2*a*z*hyper([1 - a, 1 + a], [Rational(3, 2)], z), z) | |
| def test_meijerg_eval(): | |
| from sympy.functions.elementary.exponential import exp_polar | |
| from sympy.functions.special.bessel import besseli | |
| from sympy.abc import l | |
| a = randcplx() | |
| arg = x*exp_polar(k*pi*I) | |
| expr1 = pi*meijerg([[], [(a + 1)/2]], [[a/2], [-a/2, (a + 1)/2]], arg**2/4) | |
| expr2 = besseli(a, arg) | |
| # Test that the two expressions agree for all arguments. | |
| for x_ in [0.5, 1.5]: | |
| for k_ in [0.0, 0.1, 0.3, 0.5, 0.8, 1, 5.751, 15.3]: | |
| assert abs((expr1 - expr2).n(subs={x: x_, k: k_})) < 1e-10 | |
| assert abs((expr1 - expr2).n(subs={x: x_, k: -k_})) < 1e-10 | |
| # Test continuity independently | |
| eps = 1e-13 | |
| expr2 = expr1.subs(k, l) | |
| for x_ in [0.5, 1.5]: | |
| for k_ in [0.5, Rational(1, 3), 0.25, 0.75, Rational(2, 3), 1.0, 1.5]: | |
| assert abs((expr1 - expr2).n( | |
| subs={x: x_, k: k_ + eps, l: k_ - eps})) < 1e-10 | |
| assert abs((expr1 - expr2).n( | |
| subs={x: x_, k: -k_ + eps, l: -k_ - eps})) < 1e-10 | |
| expr = (meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(-I*pi)/4) | |
| + meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(I*pi)/4)) \ | |
| /(2*sqrt(pi)) | |
| assert (expr - pi/exp(1)).n(chop=True) == 0 | |
| def test_limits(): | |
| k, x = symbols('k, x') | |
| assert hyper((1,), (Rational(4, 3), Rational(5, 3)), k**2).series(k) == \ | |
| 1 + 9*k**2/20 + 81*k**4/1120 + O(k**6) # issue 6350 | |
| # https://github.com/sympy/sympy/issues/11465 | |
| assert limit(1/hyper((1, ), (1, ), x), x, 0) == 1 | |
| def test_appellf1(): | |
| a, b1, b2, c, x, y = symbols('a b1 b2 c x y') | |
| assert appellf1(a, b2, b1, c, y, x) == appellf1(a, b1, b2, c, x, y) | |
| assert appellf1(a, b1, b1, c, y, x) == appellf1(a, b1, b1, c, x, y) | |
| assert appellf1(a, b1, b2, c, S.Zero, S.Zero) is S.One | |
| f = appellf1(a, b1, b2, c, S.Zero, S.Zero, evaluate=False) | |
| assert f.func is appellf1 | |
| assert f.doit() is S.One | |
| def test_derivative_appellf1(): | |
| from sympy.core.function import diff | |
| a, b1, b2, c, x, y, z = symbols('a b1 b2 c x y z') | |
| assert diff(appellf1(a, b1, b2, c, x, y), x) == a*b1*appellf1(a + 1, b2, b1 + 1, c + 1, y, x)/c | |
| assert diff(appellf1(a, b1, b2, c, x, y), y) == a*b2*appellf1(a + 1, b1, b2 + 1, c + 1, x, y)/c | |
| assert diff(appellf1(a, b1, b2, c, x, y), z) == 0 | |
| assert diff(appellf1(a, b1, b2, c, x, y), a) == Derivative(appellf1(a, b1, b2, c, x, y), a) | |
| def test_eval_nseries(): | |
| a1, b1, a2, b2 = symbols('a1 b1 a2 b2') | |
| assert hyper((1,2), (1,2,3), x**2)._eval_nseries(x, 7, None) == \ | |
| 1 + x**2/3 + x**4/24 + x**6/360 + O(x**7) | |
| assert exp(x)._eval_nseries(x,7,None) == \ | |
| hyper((a1, b1), (a1, b1), x)._eval_nseries(x, 7, None) | |
| assert hyper((a1, a2), (b1, b2), x)._eval_nseries(z, 7, None) ==\ | |
| hyper((a1, a2), (b1, b2), x) + O(z**7) | |
| assert hyper((-S(1)/2, S(1)/2), (1,), 4*x/(x + 1)).nseries(x) == \ | |
| 1 - x + x**2/4 - 3*x**3/4 - 15*x**4/64 - 93*x**5/64 + O(x**6) | |
| assert (pi/2*hyper((-S(1)/2, S(1)/2), (1,), 4*x/(x + 1))).nseries(x) == \ | |
| pi/2 - pi*x/2 + pi*x**2/8 - 3*pi*x**3/8 - 15*pi*x**4/128 - 93*pi*x**5/128 + O(x**6) | |