def test_hyper():
pytest.raises(TypeError, lambda: hyper(1, 2, z))
assert hyper((1, 2), (1, ), z) == hyper(Tuple(1, 2), Tuple(1), z)
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
# 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
assert hyper([polar_lift(z)], [polar_lift(k)], polar_lift(x)) == \
hyper([z], [k], polar_lift(x))
assert hyper((1, 2, 3), [3, 4], 1).is_number
assert not hyper((1, 2, 3), [3, x], 1).is_number
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, Rational(1, 2)) is None
h2 = hyper((a1, a2), (b1, a2), z)
assert tn(ReduceOrder(a2, a2).apply(h, op), h2, z)
assert str(ReduceOrder(a2,
a2)).find('<Reduce order by cancelling upper ') == 0
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_hyper():
pytest.raises(TypeError, lambda: hyper(1, 2, z))
assert hyper((1, 2), (1,), z) == hyper(Tuple(1, 2), Tuple(1), z)
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
# 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
assert hyper([polar_lift(z)], [polar_lift(k)], polar_lift(x)) == \
hyper([z], [k], polar_lift(x))
assert hyper((1, 2, 3), [3, 4], 1).is_number
assert not hyper((1, 2, 3), [3, x], 1).is_number
def test_formulae():
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_K():
assert K(0) == pi/2
assert K(Rational(1, 2)) == 8*pi**Rational(3, 2)/gamma(-Rational(1, 4))**2
assert K(1) == zoo
assert K(-1) == gamma(Rational(1, 4))**2/(4*sqrt(2*pi))
assert K(oo) == 0
assert K(-oo) == 0
assert K(I*oo) == 0
assert K(-I*oo) == 0
assert K(zoo) == 0
assert K(z).diff(z) == (E(z) - (1 - z)*K(z))/(2*z*(1 - z))
assert td(K(z), z)
pytest.raises(ArgumentIndexError, lambda: K(z).fdiff(2))
zi = Symbol('z', extended_real=False)
assert K(zi).conjugate() == K(zi.conjugate())
zr = Symbol('z', extended_real=True, negative=True)
assert K(zr).conjugate() == K(zr)
assert K(z).rewrite(hyper) == \
(pi/2)*hyper((S.Half, S.Half), (S.One,), z)
assert tn(K(z), (pi/2)*hyper((S.Half, S.Half), (S.One,), z))
assert K(z).rewrite(meijerg) == \
meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2
assert tn(K(z), meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2)
assert K(z).series(z) == pi/2 + pi*z/8 + 9*pi*z**2/128 + \
25*pi*z**3/512 + 1225*pi*z**4/32768 + 3969*pi*z**5/131072 + O(z**6)
def test_meijerg_formulae():
from diofant.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 test_formulae():
from diofant.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 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=Rational(1, 2), d=Rational(1, 2)):
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=Rational(1, 2), d=Rational(1, 2)):
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}) - expr.subs({z: 1 - 1e-15})).evalf(strict=False) > 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
def test_plan():
assert devise_plan(Hyper_Function([0], ()),
Hyper_Function([0], ()), z) == []
with pytest.raises(ValueError):
devise_plan(Hyper_Function([1], ()), Hyper_Function((), ()), z)
with pytest.raises(ValueError):
devise_plan(Hyper_Function([2], [1]), Hyper_Function([2], [2]), z)
with pytest.raises(ValueError):
devise_plan(Hyper_Function([2], []), Hyper_Function([Rational(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 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=Rational(1, 2), d=Rational(1, 2)):
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=Rational(1, 2), d=Rational(1, 2)):
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}) - expr.subs({z: 1 - 1e-15})).evalf(strict=False) > 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
def test_conjugate():
n, z, x = Symbol('n'), Symbol('z', extended_real=False), Symbol(
'x', extended_real=True)
y, t = Symbol('y', extended_real=True,
positive=True), Symbol('t', negative=True)
for f in [besseli, besselj, besselk, bessely, jn, yn, hankel1, hankel2]:
assert f(n, -1).conjugate() != f(conjugate(n), -1)
assert f(n, x).conjugate() != f(conjugate(n), x)
assert f(n, t).conjugate() != f(conjugate(n), t)
rz = randcplx(b=0.5)
for f in [besseli, besselj, besselk, bessely, jn, yn]:
assert f(n, 1 + I).conjugate() == f(conjugate(n), 1 - I)
assert f(n, 0).conjugate() == f(conjugate(n), 0)
assert f(n, 1).conjugate() == f(conjugate(n), 1)
assert f(n, z).conjugate() == f(conjugate(n), conjugate(z))
assert f(n, y).conjugate() == f(conjugate(n), y)
assert tn(f(n, rz).conjugate(), f(conjugate(n), conjugate(rz)))
assert hankel1(n, 1 + I).conjugate() == hankel2(conjugate(n), 1 - I)
assert hankel1(n, 0).conjugate() == hankel2(conjugate(n), 0)
assert hankel1(n, 1).conjugate() == hankel2(conjugate(n), 1)
assert hankel1(n, y).conjugate() == hankel2(conjugate(n), y)
assert hankel1(n, z).conjugate() == hankel2(conjugate(n), conjugate(z))
assert tn(hankel1(n, rz).conjugate(), hankel2(conjugate(n), conjugate(rz)))
assert hankel2(n, 1 + I).conjugate() == hankel1(conjugate(n), 1 - I)
assert hankel2(n, 0).conjugate() == hankel1(conjugate(n), 0)
assert hankel2(n, 1).conjugate() == hankel1(conjugate(n), 1)
assert hankel2(n, y).conjugate() == hankel1(conjugate(n), y)
assert hankel2(n, z).conjugate() == hankel1(conjugate(n), conjugate(z))
assert tn(hankel2(n, rz).conjugate(), hankel1(conjugate(n), conjugate(rz)))
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.
from diofant.abc import 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_plan():
assert devise_plan(Hyper_Function([0], ()),
Hyper_Function([0], ()), z) == []
with pytest.raises(ValueError):
devise_plan(Hyper_Function([1], ()), Hyper_Function((), ()), z)
with pytest.raises(ValueError):
devise_plan(Hyper_Function([2], [1]), Hyper_Function([2], [2]), z)
with pytest.raises(ValueError):
devise_plan(Hyper_Function([2], []), Hyper_Function([Rational(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_conjugate():
n, z, x = Symbol('n'), Symbol('z', extended_real=False), Symbol('x', extended_real=True)
y, t = Symbol('y', extended_real=True, positive=True), Symbol('t', negative=True)
for f in [besseli, besselj, besselk, bessely, jn, yn, hankel1, hankel2]:
assert f(n, -1).conjugate() != f(conjugate(n), -1)
assert f(n, x).conjugate() != f(conjugate(n), x)
assert f(n, t).conjugate() != f(conjugate(n), t)
rz = randcplx(b=0.5)
for f in [besseli, besselj, besselk, bessely, jn, yn]:
assert f(n, 1 + I).conjugate() == f(conjugate(n), 1 - I)
assert f(n, 0).conjugate() == f(conjugate(n), 0)
assert f(n, 1).conjugate() == f(conjugate(n), 1)
assert f(n, z).conjugate() == f(conjugate(n), conjugate(z))
assert f(n, y).conjugate() == f(conjugate(n), y)
assert tn(f(n, rz).conjugate(), f(conjugate(n), conjugate(rz)))
assert hankel1(n, 1 + I).conjugate() == hankel2(conjugate(n), 1 - I)
assert hankel1(n, 0).conjugate() == hankel2(conjugate(n), 0)
assert hankel1(n, 1).conjugate() == hankel2(conjugate(n), 1)
assert hankel1(n, y).conjugate() == hankel2(conjugate(n), y)
assert hankel1(n, z).conjugate() == hankel2(conjugate(n), conjugate(z))
assert tn(hankel1(n, rz).conjugate(), hankel2(conjugate(n), conjugate(rz)))
assert hankel2(n, 1 + I).conjugate() == hankel1(conjugate(n), 1 - I)
assert hankel2(n, 0).conjugate() == hankel1(conjugate(n), 0)
assert hankel2(n, 1).conjugate() == hankel1(conjugate(n), 1)
assert hankel2(n, y).conjugate() == hankel1(conjugate(n), y)
assert hankel2(n, z).conjugate() == hankel1(conjugate(n), conjugate(z))
assert tn(hankel2(n, rz).conjugate(), hankel1(conjugate(n), conjugate(rz)))
def test_rewrite():
assert besselj(
n, z).rewrite(jn) == sqrt(2 * z / pi) * jn(n - Rational(1, 2), z)
assert bessely(
n, z).rewrite(yn) == sqrt(2 * z / pi) * yn(n - Rational(1, 2), z)
assert besseli(n, z).rewrite(besselj) == \
exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z)
assert besselj(n, z).rewrite(besseli) == \
exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z)
assert besselj(2, z).rewrite(bessely) == besselj(2, z)
assert bessely(2, z).rewrite(besselj) == bessely(2, z)
assert bessely(2, z).rewrite(besseli) == bessely(2, z)
assert besselk(2, z).rewrite(besseli) == besselk(2, z)
assert besselk(2, z).rewrite(besselj) == besselk(2, z)
assert besselk(2, z).rewrite(bessely) == besselk(2, z)
nu = randcplx()
assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z)
assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z)
assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)
assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z)
assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z)
assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z)
def test_rewrite():
assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - Rational(1, 2), z)
assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - Rational(1, 2), z)
assert besseli(n, z).rewrite(besselj) == \
exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z)
assert besselj(n, z).rewrite(besseli) == \
exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z)
assert besselj(2, z).rewrite(bessely) == besselj(2, z)
assert bessely(2, z).rewrite(besselj) == bessely(2, z)
assert bessely(2, z).rewrite(besseli) == bessely(2, z)
assert besselk(2, z).rewrite(besseli) == besselk(2, z)
assert besselk(2, z).rewrite(besselj) == besselk(2, z)
assert besselk(2, z).rewrite(bessely) == besselk(2, z)
nu = randcplx()
assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z)
assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z)
assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)
assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z)
assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z)
assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z)
def test_rewriting():
assert dirichlet_eta(x).rewrite(zeta) == (1 - 2**(1 - x))*zeta(x)
assert zeta(x).rewrite(dirichlet_eta) == dirichlet_eta(x)/(1 - 2**(1 - x))
assert tn(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
assert tn(zeta(x), zeta(x).rewrite(dirichlet_eta), x)
assert zeta(x, a).rewrite(lerchphi) == lerchphi(1, x, a)
assert polylog(s, z).rewrite(lerchphi) == lerchphi(z, s, 1)*z
assert lerchphi(1, x, a).rewrite(zeta) == zeta(x, a)
assert z*lerchphi(z, s, 1).rewrite(polylog) == polylog(s, 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)
pytest.raises(ValueError, lambda: ShiftA(0))
pytest.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_shift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
pytest.raises(ValueError, lambda: ShiftA(0))
pytest.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_plan_derivatives():
a1, a2, a3 = 1, 2, Rational(1, 2)
b1, b2 = 3, Rational(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_plan_derivatives():
a1, a2, a3 = 1, 2, Rational(1, 2)
b1, b2 = 3, Rational(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_elliptic_e():
assert elliptic_e(z, 0) == z
assert elliptic_e(0, m) == 0
assert elliptic_e(i * pi / 2, m) == i * elliptic_e(m)
assert elliptic_e(z, oo) == zoo
assert elliptic_e(z, -oo) == zoo
assert elliptic_e(0) == pi / 2
assert elliptic_e(1) == 1
assert elliptic_e(oo) == I * oo
assert elliptic_e(-oo) == oo
assert elliptic_e(zoo) == zoo
assert elliptic_e(-z, m) == -elliptic_e(z, m)
assert elliptic_e(z, m).diff(z) == sqrt(1 - m * sin(z)**2)
assert elliptic_e(
z, m).diff(m) == (elliptic_e(z, m) - elliptic_f(z, m)) / (2 * m)
assert elliptic_e(z).diff(z) == (elliptic_e(z) - elliptic_k(z)) / (2 * z)
r = randcplx()
assert td(elliptic_e(r, m), m)
assert td(elliptic_e(z, r), z)
assert td(elliptic_e(z), z)
pytest.raises(ArgumentIndexError, lambda: elliptic_e(z, m).fdiff(3))
pytest.raises(ArgumentIndexError, lambda: elliptic_e(z).fdiff(2))
mi = Symbol('m', extended_real=False)
assert elliptic_e(z, mi).conjugate() == elliptic_e(z.conjugate(),
mi.conjugate())
assert elliptic_e(mi).conjugate() == elliptic_e(mi.conjugate())
mr = Symbol('m', extended_real=True, negative=True)
assert elliptic_e(z, mr).conjugate() == elliptic_e(z.conjugate(), mr)
assert elliptic_e(mr).conjugate() == elliptic_e(mr)
assert elliptic_e(z, m).conjugate() == conjugate(elliptic_e(z, m))
assert elliptic_e(z).conjugate() == conjugate(elliptic_e(z))
assert elliptic_e(z).rewrite(hyper) == (pi / 2) * hyper(
(Rational(-1, 2), Rational(1, 2)), (1, ), z)
assert elliptic_e(z, m).rewrite(hyper) == elliptic_e(z, m)
assert tn(elliptic_e(z), (pi / 2) * hyper(
(Rational(-1, 2), Rational(1, 2)), (1, ), z))
assert elliptic_e(z).rewrite(meijerg) == \
-meijerg(((Rational(1, 2), Rational(3, 2)), []), ((0,), (0,)), -z)/4
assert elliptic_e(z, m).rewrite(meijerg) == elliptic_e(z, m)
assert tn(
elliptic_e(z), -meijerg(((Rational(1, 2), Rational(3, 2)), []),
((0, ), (0, )), -z) / 4)
assert elliptic_e(z, m).series(z) == \
z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
assert elliptic_e(z).series(z) == pi/2 - pi*z/8 - 3*pi*z**2/128 - \
5*pi*z**3/512 - 175*pi*z**4/32768 - 441*pi*z**5/131072 + O(z**6)
def test_partial_simp2():
# Now test that formulae are partially simplified.
c, d, e = (randcplx() for _ in range(3))
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_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)
def test_partial_simp2():
# Now test that formulae are partially simplified.
c, d, e = (randcplx() for _ in range(3))
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_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)
def test_rewriting():
assert dirichlet_eta(x).rewrite(zeta) == (1 - 2**(1 - x))*zeta(x)
assert zeta(x).rewrite(dirichlet_eta) == dirichlet_eta(x)/(1 - 2**(1 - x))
assert zeta(z, 2).rewrite(dirichlet_eta) == zeta(z, 2)
assert zeta(z, 2).rewrite('tractable') == zeta(z, 2)
assert tn(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
assert tn(zeta(x), zeta(x).rewrite(dirichlet_eta), x)
assert zeta(x, a).rewrite(lerchphi) == lerchphi(1, x, a)
assert polylog(s, z).rewrite(lerchphi) == lerchphi(z, s, 1)*z
assert lerchphi(1, x, a).rewrite(zeta) == zeta(x, a)
assert z*lerchphi(z, s, 1).rewrite(polylog) == polylog(s, z)
assert lerchphi(z, s, a).rewrite(zeta) == lerchphi(z, s, a)
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 diofant import unpolarify, expand
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, a in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[a] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
def can_do(ap, bq, numerical=True, div=1, lowerplane=False):
from diofant import exp_polar, exp
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, a in enumerate(randsyms):
repl[a] = 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 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.
"""
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, a in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[a] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
def test_E():
assert E(z, 0) == z
assert E(0, m) == 0
assert E(i*pi/2, m) == i*E(m)
assert E(z, oo) == zoo
assert E(z, -oo) == zoo
assert E(0) == pi/2
assert E(1) == 1
assert E(oo) == I*oo
assert E(-oo) == oo
assert E(zoo) == zoo
assert E(-z, m) == -E(z, m)
assert E(z, m).diff(z) == sqrt(1 - m*sin(z)**2)
assert E(z, m).diff(m) == (E(z, m) - F(z, m))/(2*m)
assert E(z).diff(z) == (E(z) - K(z))/(2*z)
r = randcplx()
assert td(E(r, m), m)
assert td(E(z, r), z)
assert td(E(z), z)
pytest.raises(ArgumentIndexError, lambda: E(z, m).fdiff(3))
pytest.raises(ArgumentIndexError, lambda: E(z).fdiff(2))
mi = Symbol('m', extended_real=False)
assert E(z, mi).conjugate() == E(z.conjugate(), mi.conjugate())
assert E(mi).conjugate() == E(mi.conjugate())
mr = Symbol('m', extended_real=True, negative=True)
assert E(z, mr).conjugate() == E(z.conjugate(), mr)
assert E(mr).conjugate() == E(mr)
assert E(z).rewrite(hyper) == (pi/2)*hyper((-S.Half, S.Half), (S.One,), z)
assert E(z, m).rewrite(hyper) == E(z, m)
assert tn(E(z), (pi/2)*hyper((-S.Half, S.Half), (S.One,), z))
assert E(z).rewrite(meijerg) == \
-meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4
assert E(z, m).rewrite(meijerg) == E(z, m)
assert tn(E(z), -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4)
assert E(z, m).series(z) == \
z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
assert E(z).series(z) == pi/2 - pi*z/8 - 3*pi*z**2/128 - \
5*pi*z**3/512 - 175*pi*z**4/32768 - 441*pi*z**5/131072 + O(z**6)
def test_lowergamma():
assert lowergamma(x, y).diff(y) == y**(x - 1)*exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert td(lowergamma(x, randcplx()), x)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
- meijerg([], [1, 1], [0, 0, x], [], y)
pytest.raises(ArgumentIndexError, lambda: lowergamma(x, y).fdiff(3))
assert lowergamma(Rational(1, 2), x) == sqrt(pi)*erf(sqrt(x))
assert not lowergamma(Rational(1, 2) - 3, x).has(lowergamma)
assert not lowergamma(Rational(1, 2) + 3, x).has(lowergamma)
assert lowergamma(Rational(1, 2), x, evaluate=False).has(lowergamma)
assert tn(lowergamma(Rational(1, 2) + 3, x, evaluate=False),
lowergamma(Rational(1, 2) + 3, x), x)
assert tn(lowergamma(Rational(1, 2) - 3, x, evaluate=False),
lowergamma(Rational(1, 2) - 3, x), x)
assert lowergamma(0, 1) == zoo
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(Rational(1, 3), lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y))
assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))
assert lowergamma(
x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(
k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
assert (x*lowergamma(x, 1)/gamma(x + 1)).limit(x, 0) == 1
def test_ushift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
pytest.raises(ValueError, lambda: UnShiftA((1,), (), 0, z))
pytest.raises(ValueError, lambda: UnShiftB((), (-1,), 0, z))
pytest.raises(ValueError, lambda: UnShiftA((1,), (0, -1, 1), 0, z))
pytest.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 test_ushift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
pytest.raises(ValueError, lambda: UnShiftA((1,), (), 0, z))
pytest.raises(ValueError, lambda: UnShiftB((), (-1,), 0, z))
pytest.raises(ValueError, lambda: UnShiftA((1,), (0, -1, 1), 0, z))
pytest.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 test_lowergamma():
from diofant import meijerg, exp_polar, I, expint
assert lowergamma(x, y).diff(y) == y**(x - 1) * exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert td(lowergamma(x, randcplx()), x)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
- meijerg([], [1, 1], [0, 0, x], [], y)
assert lowergamma(S.Half, x) == sqrt(pi) * erf(sqrt(x))
assert not lowergamma(S.Half - 3, x).has(lowergamma)
assert not lowergamma(S.Half + 3, x).has(lowergamma)
assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
assert tn(lowergamma(S.Half + 3, x, evaluate=False),
lowergamma(S.Half + 3, x), x)
assert tn(lowergamma(S.Half - 3, x, evaluate=False),
lowergamma(S.Half - 3, x), x)
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(Rational(1, 3), lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4 * pi * I) * x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x),
conjugate(y))
assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))
assert lowergamma(
x, y).rewrite(expint) == -y**x * expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(
k, y).rewrite(expint) == -y**k * expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
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)
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)
def test_branching():
assert besselj(polar_lift(k), x) == besselj(k, x)
assert besseli(polar_lift(k), x) == besseli(k, x)
n = Symbol('n', integer=True)
assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
assert besselj(n, polar_lift(x)) == besselj(n, x)
assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
assert besseli(n, polar_lift(x)) == besseli(n, x)
def tn(func, s):
c = uniform(1, 5)
expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
return abs(expr - expr2).evalf(strict=False) < 1e-10
nu = Symbol('nu')
assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
assert tn(besselj, 2)
assert tn(besselj, pi)
assert tn(besselj, I)
assert tn(besseli, 2)
assert tn(besseli, pi)
assert tn(besseli, I)
def test_uppergamma():
from diofant import meijerg, exp_polar, I, expint
assert uppergamma(4, 0) == 6
assert uppergamma(x, oo) == 0
assert uppergamma(x, y).diff(y) == -y**(x - 1) * exp(-y)
assert td(uppergamma(randcplx(), y), y)
assert uppergamma(x, y).diff(x) == \
uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
assert td(uppergamma(x, randcplx()), x)
pytest.raises(ArgumentIndexError, lambda: uppergamma(x, y).fdiff(3))
assert uppergamma(S.Half, x) == sqrt(pi) * (1 - erf(sqrt(x)))
assert not uppergamma(S.Half - 3, x).has(uppergamma)
assert not uppergamma(S.Half + 3, x).has(uppergamma)
assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
assert tn(uppergamma(S.Half + 3, x, evaluate=False),
uppergamma(S.Half + 3, x), x)
assert tn(uppergamma(S.Half - 3, x, evaluate=False),
uppergamma(S.Half - 3, x), x)
assert tn_branch(-3, uppergamma)
assert tn_branch(-4, uppergamma)
assert tn_branch(Rational(1, 3), uppergamma)
assert tn_branch(pi, uppergamma)
assert uppergamma(3, exp_polar(4 * pi * I) * x) == uppergamma(3, x)
assert uppergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + \
gamma(y)*(1 - exp(4*pi*I*y))
assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
assert uppergamma(-2, x) == expint(3, x) / x**2
assert conjugate(uppergamma(x, y)) == uppergamma(conjugate(x),
conjugate(y))
assert conjugate(uppergamma(x, 0)) == gamma(conjugate(x))
assert conjugate(uppergamma(x, -oo)) == conjugate(uppergamma(x, -oo))
assert uppergamma(x, y).rewrite(expint) == y**x * expint(-x + 1, y)
assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
def test_branching():
from diofant import exp_polar, polar_lift, Symbol, I, exp
assert besselj(polar_lift(k), x) == besselj(k, x)
assert besseli(polar_lift(k), x) == besseli(k, x)
n = Symbol('n', integer=True)
assert besselj(n, exp_polar(2 * pi * I) * x) == besselj(n, x)
assert besselj(n, polar_lift(x)) == besselj(n, x)
assert besseli(n, exp_polar(2 * pi * I) * x) == besseli(n, x)
assert besseli(n, polar_lift(x)) == besseli(n, x)
def tn(func, s):
from random import uniform
c = uniform(1, 5)
expr = func(s, c * exp_polar(I * pi)) - func(s, c * exp_polar(-I * pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps * I) - func(s + eps, -c - eps * I)
return abs(expr.n() - expr2.n()).n() < 1e-10
nu = Symbol('nu')
assert besselj(nu,
exp_polar(2 * pi * I) *
x) == exp(2 * pi * I * nu) * besselj(nu, x)
assert besseli(nu,
exp_polar(2 * pi * I) *
x) == exp(2 * pi * I * nu) * besseli(nu, x)
assert tn(besselj, 2)
assert tn(besselj, pi)
assert tn(besselj, I)
assert tn(besseli, 2)
assert tn(besseli, pi)
assert tn(besseli, I)
def test_branching():
assert besselj(polar_lift(k), x) == besselj(k, x)
assert besseli(polar_lift(k), x) == besseli(k, x)
n = Symbol('n', integer=True)
assert besselj(n, exp_polar(2 * pi * I) * x) == besselj(n, x)
assert besselj(n, polar_lift(x)) == besselj(n, x)
assert besseli(n, exp_polar(2 * pi * I) * x) == besseli(n, x)
assert besseli(n, polar_lift(x)) == besseli(n, x)
def tn(func, s):
c = uniform(1, 5)
expr = func(s, c * exp_polar(I * pi)) - func(s, c * exp_polar(-I * pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps * I) - func(s + eps, -c - eps * I)
return abs(expr - expr2).evalf(strict=False) < 1e-10
nu = Symbol('nu')
assert besselj(nu,
exp_polar(2 * pi * I) *
x) == exp(2 * pi * I * nu) * besselj(nu, x)
assert besseli(nu,
exp_polar(2 * pi * I) *
x) == exp(2 * pi * I * nu) * besseli(nu, x)
assert tn(besselj, 2)
assert tn(besselj, pi)
assert tn(besselj, I)
assert tn(besseli, 2)
assert tn(besseli, pi)
assert tn(besseli, I)
def test_elliptic_k():
assert elliptic_k(0) == pi / 2
assert elliptic_k(Rational(
1, 2)) == 8 * pi**Rational(3, 2) / gamma(-Rational(1, 4))**2
assert elliptic_k(1) == zoo
assert elliptic_k(-1) == gamma(Rational(1, 4))**2 / (4 * sqrt(2 * pi))
assert elliptic_k(oo) == 0
assert elliptic_k(-oo) == 0
assert elliptic_k(I * oo) == 0
assert elliptic_k(-I * oo) == 0
assert elliptic_k(zoo) == 0
assert elliptic_k(z).diff(z) == (elliptic_e(z) -
(1 - z) * elliptic_k(z)) / (2 * z *
(1 - z))
assert td(elliptic_k(z), z)
pytest.raises(ArgumentIndexError, lambda: elliptic_k(z).fdiff(2))
zi = Symbol('z', extended_real=False)
assert elliptic_k(zi).conjugate() == elliptic_k(zi.conjugate())
zr = Symbol('z', extended_real=True, negative=True)
assert elliptic_k(zr).conjugate() == elliptic_k(zr)
assert elliptic_k(z).conjugate() == conjugate(elliptic_k(z),
evaluate=False)
assert elliptic_k(z).rewrite(hyper) == \
(pi/2)*hyper((Rational(1, 2), Rational(1, 2)), (1,), z)
assert tn(elliptic_k(z), (pi / 2) * hyper((Rational(1, 2), Rational(1, 2)),
(1, ), z))
assert elliptic_k(z).rewrite(meijerg) == \
meijerg(((Rational(1, 2), Rational(1, 2)), []), ((0,), (0,)), -z)/2
assert tn(
elliptic_k(z),
meijerg(((Rational(1, 2), Rational(1, 2)), []),
((0, ), (0, )), -z) / 2)
assert elliptic_k(z).series(z) == pi/2 + pi*z/8 + 9*pi*z**2/128 + \
25*pi*z**3/512 + 1225*pi*z**4/32768 + 3969*pi*z**5/131072 + O(z**6)
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():
# 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)
assert str(MeijerShiftA(b1)).find('<Increment upper b=') == 0
assert str(MeijerShiftB(a1)).find('<Decrement upper a=') == 0
assert str(MeijerShiftC(b3)).find('<Increment lower b=') == 0
assert str(MeijerShiftD(a3)).find('<Decrement lower a=') == 0
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)
assert str(s).find('<Decrement upper b index #') == 0
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)
assert str(s).find('<Decrement lower b index #') == 0
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)
assert str(s).find('<Increment upper a index #') == 0
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)
assert str(s).find('<Increment lower a index #') == 0
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, Rational(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_meijerg_formulae():
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 can_do(ap, bq, numerical=True, div=1, lowerplane=False):
from diofant import exp_polar, exp
r = hyperexpand(hyper(ap, bq, z))
if r.has(hyper):
return False
if not numerical:
return True
repl = {}
for n, a in enumerate(r.free_symbols - {z}):
repl[a] = randcplx(n)/div
[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 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, a in enumerate(randsyms):
repl[a] = 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_meijer():
pytest.raises(TypeError, lambda: meijerg(1, z))
pytest.raises(TypeError, lambda: meijerg(((1,), (2,)), (3,), (4,), z))
pytest.raises(TypeError, lambda: meijerg((1, 2, 3), (4, 5), 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 meijerg([1, 2], [3], [4], [5], z).delta == Rational(1, 2)
# 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(Rational(1, 2)), z**2/4), cos(z), z)
assert tn(meijerg(Tuple(1, 1), Tuple(), Tuple(1), Tuple(0), z),
log(1 + z), z)
# test exceptions
pytest.raises(ValueError, lambda: meijerg(((3, 1), (2,)),
((oo,), (2, 0)), x))
pytest.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
assert meijerg([polar_lift(a1)], [polar_lift(a2)], [polar_lift(b1)],
[polar_lift(b2)], polar_lift(z)) == meijerg([a1], [a2],
[b1], [b2],
polar_lift(z))
# integrand
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))
assert meijerg([[], []], [[Rational(1, 2)], [0]], 1).is_number
assert not meijerg([[], []], [[x], [0]], 1).is_number
def valid(a, b):
if not (tn(a, b) and a == b):
return False
return True
def test_expand():
assert expand_func(besselj(Rational(1, 2), z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(Rational(1, 2), z).rewrite(yn)) == \
-sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert expand_func(besselj(I, z)) == besselj(I, z)
# Test simplify helper
assert simplify(besselj(Rational(1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
# XXX: teach sin/cos to work around arguments like
# x*exp_polar(I*pi*n/2). Then change besselsimp -> expand_func
assert besselsimp(besselj(Rational(1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(Rational(-1, 2), z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(Rational(5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselj(-Rational(5, 2), z)) == \
-sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(1, 2), z)) == \
-(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(-1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(5, 2), z)) == \
sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(-5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(1, 2), z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(-1, 2), z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(5, 2), z)) == \
sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(-5, 2), z)) == \
sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselk(Rational(1, 2), z)) == \
besselsimp(besselk(Rational(-1, 2), z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
assert besselsimp(besselk(Rational(5, 2), z)) == \
besselsimp(besselk(Rational(-5, 2), z)) == \
sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**Rational(5, 2))
def check(eq, ans):
return tn(eq, ans) and eq == ans
rn = randcplx(a=1, b=0, d=0, c=2)
for besselx in [besselj, bessely, besseli, besselk]:
ri = Rational(2*randint(-11, 10) + 1, 2) # half integer in [-21/2, 21/2]
assert tn(besselsimp(besselx(ri, z)), besselx(ri, z))
assert check(expand_func(besseli(rn, x)),
besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x)
assert check(expand_func(besseli(-rn, x)),
besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x)
assert check(expand_func(besselj(rn, x)),
-besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x)
assert check(expand_func(besselj(-rn, x)),
-besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x)
assert check(expand_func(besselk(rn, x)),
besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x)
assert check(expand_func(besselk(-rn, x)),
besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x)
assert check(expand_func(bessely(rn, x)),
-bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x)
assert check(expand_func(bessely(-rn, x)),
-bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x)
n = Symbol('n', integer=True, positive=True)
assert expand_func(besseli(n + 2, z)) == \
besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
assert expand_func(besselj(n + 2, z)) == \
-besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
assert expand_func(besselk(n + 2, z)) == \
besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
assert expand_func(bessely(n + 2, z)) == \
-bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z
assert expand_func(besseli(n + Rational(1, 2), z).rewrite(jn)) == \
(sqrt(2)*sqrt(z)*exp(-I*pi*(n + Rational(1, 2))/2) *
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
assert expand_func(besselj(n + Rational(1, 2), z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
r = Symbol('r', extended_real=True)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
for besselx in [besselj, bessely, besseli, besselk]:
assert besselx(i, p).is_extended_real
assert besselx(i, x).is_extended_real is None
assert besselx(x, z).is_extended_real is None
for besselx in [besselj, besseli]:
assert besselx(i, r).is_extended_real
for besselx in [bessely, besselk]:
assert besselx(i, r).is_extended_real is None
def check(eq, ans):
return tn(eq, ans) and eq == ans
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)
