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 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 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_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_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_hyper_unpolarify():
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
def test_polynomial():
from diofant import oo
assert hyperexpand(hyper([], [-1], z)) == oo
assert hyperexpand(hyper([-2], [-1], z)) == 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_hyper_unpolarify():
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
def test_hyper_unpolarify():
from diofant 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
def test_hyper_rewrite_sum():
_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_hyper_rewrite_sum():
_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_limits():
k, x = symbols('k, x')
assert hyper((1,), (Rational(4, 3), Rational(5, 3)), k**2).series(k) == \
hyper((1,), (Rational(4, 3), Rational(5, 3)), 0) + \
9*k**2*hyper((2,), (Rational(7, 3), Rational(8, 3)), 0)/20 + \
81*k**4*hyper((3,), (Rational(10, 3), Rational(11, 3)), 0)/1120 + \
O(k**6) # issue sympy/sympy#6350
assert limit(meijerg((), (), (1,), (0,), -x), x, 0) == \
meijerg(((), ()), ((1,), (0,)), 0) # issue sympy/sympy#6052
def test_limits():
k, x = symbols('k, x')
assert hyper((1,), (Rational(4, 3), Rational(5, 3)), k**2).series(k) == \
hyper((1,), (Rational(4, 3), Rational(5, 3)), 0) + \
9*k**2*hyper((2,), (Rational(7, 3), Rational(8, 3)), 0)/20 + \
81*k**4*hyper((3,), (Rational(10, 3), Rational(11, 3)), 0)/1120 + \
O(k**6) # issue sympy/sympy#6350
assert limit(meijerg((), (), (1,), (0,), -x), x, 0) == \
meijerg(((), ()), ((1,), (0,)), 0) # issue sympy/sympy#6052
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([], [Rational(3, 2)], -z**2/4)) == sin(z)
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], z**2)*z) \
== asin(z)
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([], [Rational(1, 2)], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [Rational(3, 2)], -z**2/4)) == sin(z)
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], z**2)*z) \
== asin(z)
def test_airybi():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybi(z), airybi)
assert airybi(0) == 3**Rational(5, 6) / (3 * gamma(Rational(2, 3)))
assert airybi(oo) == oo
assert airybi(-oo) == 0
assert diff(airybi(z), z) == airybiprime(z)
assert series(airybi(z), z, 0,
3) == (cbrt(3) * gamma(Rational(1, 3)) / (2 * pi) +
3**Rational(2, 3) * z * gamma(Rational(2, 3)) /
(2 * pi) + O(z**3))
l = Limit(
airybi(I / x) /
(exp(Rational(2, 3) *
(I / x)**Rational(3, 2)) * sqrt(pi * sqrt(I / x))), x, 0)
assert l.doit() == l
assert airybi(z).rewrite(hyper) == (root(3, 6) * z * hyper(
(), (Rational(4, 3), ), z**3 / 9) / gamma(Rational(1, 3)) +
3**Rational(5, 6) * hyper(
(), (Rational(2, 3), ), z**3 / 9) /
(3 * gamma(Rational(2, 3))))
assert isinstance(airybi(z).rewrite(besselj), airybi)
assert (airybi(t).rewrite(besselj) == sqrt(3) * sqrt(-t) *
(besselj(-1 / 3, 2 * (-t)**Rational(3, 2) / 3) -
besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3)
assert airybi(z).rewrite(besseli) == (
sqrt(3) * (z * besseli(Rational(1, 3), 2 * z**Rational(3, 2) / 3) /
cbrt(z**Rational(3, 2)) + cbrt(z**Rational(3, 2)) *
besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3)) / 3)
assert airybi(p).rewrite(besseli) == (
sqrt(3) * sqrt(p) *
(besseli(-Rational(1, 3), 2 * p**Rational(3, 2) / 3) +
besseli(Rational(1, 3), 2 * p**Rational(3, 2) / 3)) / 3)
assert airybi(p).rewrite(besselj) == airybi(p)
assert expand_func(airybi(
2 *
cbrt(3 * z**5))) == (sqrt(3) * (1 - cbrt(z**5) / z**Rational(5, 3)) *
airyai(2 * cbrt(3) * z**Rational(5, 3)) / 2 +
(1 + cbrt(z**5) / z**Rational(5, 3)) *
airybi(2 * cbrt(3) * z**Rational(5, 3)) / 2)
assert expand_func(airybi(x * y)) == airybi(x * y)
assert expand_func(airybi(log(x))) == airybi(log(x))
assert expand_func(airybi(2 * root(3 * z**5, 5))) == airybi(
2 * root(3 * z**5, 5))
assert airybi(x).taylor_term(-1, x) == 0
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_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_airyai():
z = Symbol('z', extended_real=False)
r = Symbol('r', extended_real=True)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyai(z), airyai)
assert airyai(0) == cbrt(3)/(3*gamma(Rational(2, 3)))
assert airyai(oo) == 0
assert airyai(-oo) == 0
assert diff(airyai(z), z) == airyaiprime(z)
assert airyai(z).series(z, 0, 3) == (
3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) - root(3, 6)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
l = Limit(airyai(I/x)/(exp(-Rational(2, 3)*(I/x)**Rational(3, 2))*sqrt(pi*sqrt(I/x))/2), x, 0)
assert l.doit() == l # cover _airyais._eval_aseries
assert airyai(z).rewrite(hyper) == (
-3**Rational(2, 3)*z*hyper((), (Rational(4, 3),), z**3/9)/(3*gamma(Rational(1, 3))) +
cbrt(3)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
assert isinstance(airyai(z).rewrite(besselj), airyai)
assert airyai(t).rewrite(besselj) == (
sqrt(-t)*(besselj(-Rational(1, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airyai(z).rewrite(besseli) == (
-z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(3*cbrt(z**Rational(3, 2))) +
cbrt(z**Rational(3, 2))*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3)/3)
assert airyai(p).rewrite(besseli) == (
sqrt(p)*(besseli(-Rational(1, 3), 2*p**Rational(3, 2)/3) -
besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
assert expand_func(airyai(2*cbrt(3*z**5))) == (
-sqrt(3)*(-1 + cbrt(z**5)/z**Rational(5, 3))*airybi(2*cbrt(3)*z**Rational(5, 3))/6 +
(1 + cbrt(z**5)/z**Rational(5, 3))*airyai(2*cbrt(3)*z**Rational(5, 3))/2)
assert expand_func(airyai(x*y)) == airyai(x*y)
assert expand_func(airyai(log(x))) == airyai(log(x))
assert expand_func(airyai(2*root(3*z**5, 5))) == airyai(2*root(3*z**5, 5))
assert (airyai(r).as_real_imag() ==
airyai(r).as_real_imag(deep=False) == (airyai(r), 0))
assert airyai(x).as_real_imag() == airyai(x).as_real_imag(deep=False)
assert (airyai(x).as_real_imag() ==
(airyai(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 +
airyai(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2,
I*(airyai(re(x) - I*re(x)*abs(im(x))/abs(re(x))) -
airyai(re(x) + I*re(x)*abs(im(x))/abs(re(x)))) *
re(x)*abs(im(x))/(2*im(x)*abs(re(x)))))
assert airyai(x).taylor_term(-1, x) == 0
def test_airyai():
z = Symbol('z', extended_real=False)
r = Symbol('r', extended_real=True)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyai(z), airyai)
assert airyai(0) == cbrt(3)/(3*gamma(Rational(2, 3)))
assert airyai(oo) == 0
assert airyai(-oo) == 0
assert diff(airyai(z), z) == airyaiprime(z)
assert series(airyai(z), z, 0, 3) == (
3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) - root(3, 6)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
l = Limit(airyai(I/x)/(exp(-Rational(2, 3)*(I/x)**Rational(3, 2))*sqrt(pi*sqrt(I/x))/2), x, 0)
assert l.doit() == l # cover _airyais._eval_aseries
assert airyai(z).rewrite(hyper) == (
-3**Rational(2, 3)*z*hyper((), (Rational(4, 3),), z**3/9)/(3*gamma(Rational(1, 3))) +
cbrt(3)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
assert isinstance(airyai(z).rewrite(besselj), airyai)
assert airyai(t).rewrite(besselj) == (
sqrt(-t)*(besselj(-Rational(1, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airyai(z).rewrite(besseli) == (
-z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(3*cbrt(z**Rational(3, 2))) +
cbrt(z**Rational(3, 2))*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3)/3)
assert airyai(p).rewrite(besseli) == (
sqrt(p)*(besseli(-Rational(1, 3), 2*p**Rational(3, 2)/3) -
besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
assert expand_func(airyai(2*cbrt(3*z**5))) == (
-sqrt(3)*(-1 + cbrt(z**5)/z**Rational(5, 3))*airybi(2*cbrt(3)*z**Rational(5, 3))/6 +
(1 + cbrt(z**5)/z**Rational(5, 3))*airyai(2*cbrt(3)*z**Rational(5, 3))/2)
assert expand_func(airyai(x*y)) == airyai(x*y)
assert expand_func(airyai(log(x))) == airyai(log(x))
assert expand_func(airyai(2*root(3*z**5, 5))) == airyai(2*root(3*z**5, 5))
assert (airyai(r).as_real_imag() ==
airyai(r).as_real_imag(deep=False) == (airyai(r), 0))
assert airyai(x).as_real_imag() == airyai(x).as_real_imag(deep=False)
assert (airyai(x).as_real_imag() ==
(airyai(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 +
airyai(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2,
I*(airyai(re(x) - I*re(x)*abs(im(x))/abs(re(x))) -
airyai(re(x) + I*re(x)*abs(im(x))/Abs(re(x)))) *
re(x)*abs(im(x))/(2*im(x)*abs(re(x)))))
assert airyai(x).taylor_term(-1, x) == 0
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_airybiprime():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybiprime(z), airybiprime)
assert airybiprime(0) == root(3, 6) / gamma(Rational(1, 3))
assert airybiprime(oo) == oo
assert airybiprime(-oo) == 0
assert diff(airybiprime(z), z) == z * airybi(z)
assert series(airybiprime(z), z, 0,
3) == (root(3, 6) / gamma(Rational(1, 3)) +
3**Rational(5, 6) * z**2 /
(6 * gamma(Rational(2, 3))) + O(z**3))
assert airybiprime(z).rewrite(hyper) == (
3**Rational(5, 6) * z**2 * hyper((), (Rational(5, 3), ), z**3 / 9) /
(6 * gamma(Rational(2, 3))) + root(3, 6) * hyper(
(), (Rational(1, 3), ), z**3 / 9) / gamma(Rational(1, 3)))
assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
assert (airybiprime(t).rewrite(besselj) == -sqrt(3) * t *
(besselj(-Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3) +
besselj(Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3)
assert airybiprime(z).rewrite(besseli) == (
sqrt(3) * (z**2 * besseli(Rational(2, 3), 2 * z**Rational(3, 2) / 3) /
(z**Rational(3, 2))**Rational(2, 3) +
(z**Rational(3, 2))**Rational(2, 3) *
besseli(-Rational(2, 3), 2 * z**Rational(3, 2) / 3)) / 3)
assert airybiprime(p).rewrite(besseli) == (
sqrt(3) * p * (besseli(-Rational(2, 3), 2 * p**Rational(3, 2) / 3) +
besseli(Rational(2, 3), 2 * p**Rational(3, 2) / 3)) / 3)
assert airybiprime(p).rewrite(besselj) == airybiprime(p)
assert expand_func(airybiprime(
2 *
cbrt(3 * z**5))) == (sqrt(3) * (z**Rational(5, 3) / cbrt(z**5) - 1) *
airyaiprime(2 * cbrt(3) * z**Rational(5, 3)) / 2 +
(z**Rational(5, 3) / cbrt(z**5) + 1) *
airybiprime(2 * cbrt(3) * z**Rational(5, 3)) / 2)
assert expand_func(airybiprime(x * y)) == airybiprime(x * y)
assert expand_func(airybiprime(log(x))) == airybiprime(log(x))
assert expand_func(airybiprime(2 * root(3 * z**5, 5))) == airybiprime(
2 * root(3 * z**5, 5))
assert airybiprime(-2).evalf(50) == Float(
'0.27879516692116952268509756941098324140300059345163131', dps=50)
def test_airyaiprime():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyaiprime(z), airyaiprime)
assert airyaiprime(0) == -3**Rational(2, 3) / (3 * gamma(Rational(1, 3)))
assert airyaiprime(oo) == 0
assert diff(airyaiprime(z), z) == z * airyai(z)
assert series(airyaiprime(z), z, 0,
3) == (-3**Rational(2, 3) / (3 * gamma(Rational(1, 3))) +
cbrt(3) * z**2 / (6 * gamma(Rational(2, 3))) +
O(z**3))
assert airyaiprime(z).rewrite(hyper) == (
cbrt(3) * z**2 * hyper((), (Rational(5, 3), ), z**3 / 9) /
(6 * gamma(Rational(2, 3))) - 3**Rational(2, 3) * hyper(
(), (Rational(1, 3), ), z**3 / 9) / (3 * gamma(Rational(1, 3))))
assert isinstance(airyaiprime(z).rewrite(besselj), airyaiprime)
assert (airyaiprime(t).rewrite(besselj) == t *
(besselj(-Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3) -
besselj(Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3)
assert airyaiprime(z).rewrite(besseli) == (
z**2 * besseli(Rational(2, 3), 2 * z**Rational(3, 2) / 3) /
(3 * (z**Rational(3, 2))**Rational(2, 3)) -
(z**Rational(3, 2))**Rational(2, 3) *
besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3) / 3)
assert airyaiprime(p).rewrite(besseli) == (
p * (-besseli(-Rational(2, 3), 2 * p**Rational(3, 2) / 3) +
besseli(Rational(2, 3), 2 * p**Rational(3, 2) / 3)) / 3)
assert airyaiprime(p).rewrite(besselj) == airyaiprime(p)
assert expand_func(airyaiprime(
2 *
cbrt(3 * z**5))) == (sqrt(3) * (z**Rational(5, 3) / cbrt(z**5) - 1) *
airybiprime(2 * cbrt(3) * z**Rational(5, 3)) / 6 +
(z**Rational(5, 3) / cbrt(z**5) + 1) *
airyaiprime(2 * cbrt(3) * z**Rational(5, 3)) / 2)
assert expand_func(airyaiprime(x * y)) == airyaiprime(x * y)
assert expand_func(airyaiprime(log(x))) == airyaiprime(log(x))
assert expand_func(airyaiprime(2 * root(3 * z**5, 5))) == airyaiprime(
2 * root(3 * z**5, 5))
assert airyaiprime(-2).evalf(50) == Float(
'0.61825902074169104140626429133247528291577794512414753', dps=50)
def test_branch_bug():
assert hyperexpand(hyper((-Rational(1, 3), Rational(1, 2)), (Rational(2, 3), Rational(3, 2)), -z)) == \
-z**Rational(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**Rational(2, 3)*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) -
2*lowergamma(Rational(2, 3), z)/z**Rational(2, 3))*gamma(Rational(2, 3))/gamma(Rational(5, 3))
def test_li():
z = Symbol("z")
zr = Symbol("z", extended_real=True)
zp = Symbol("z", positive=True)
zn = Symbol("z", negative=True)
assert li(0) == 0
assert li(1) == -oo
assert li(oo) == oo
assert isinstance(li(z), li)
assert diff(li(z), z) == 1/log(z)
assert conjugate(li(z)) == li(conjugate(z))
assert conjugate(li(-zr)) == li(-zr)
assert conjugate(li(-zp)) == conjugate(li(-zp))
assert conjugate(li(zn)) == conjugate(li(zn))
assert li(z).rewrite(Li) == Li(z) + li(2)
assert li(z).rewrite(Ei) == Ei(log(z))
assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) +
log(log(z))/2 - expint(1, -log(z)))
assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(hyper) == (log(z)*hyper((1, 1), (2, 2), log(z)) -
log(1/log(z))/2 + log(log(z))/2 + EulerGamma)
assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 -
meijerg(((), (1,)), ((0, 0), ()), -log(z)))
def test_erf():
assert erf(nan) == nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(I * oo) == oo * I
assert erf(-I * oo) == -oo * I
assert erf(-2) == -erf(2)
assert erf(-x * y) == -erf(x * y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_extended_real is False
assert erf(w).is_extended_real is True
assert erf(z).is_extended_real is None
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi)
assert erf(1 / x).as_leading_term(x) == erf(1 / x)
assert erf(z).rewrite('uppergamma') == sqrt(z**2) * erf(sqrt(z**2)) / z
assert erf(z).rewrite('erfc') == S.One - erfc(z)
assert erf(z).rewrite('erfi') == -I * erfi(I * z)
assert erf(z).rewrite('fresnels') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('hyper') == 2 * z * hyper([S.Half], [3 * S.Half],
-z**2) / sqrt(pi)
assert erf(z).rewrite('meijerg') == z * meijerg([S.Half], [], [0],
[-S.Half], z**2) / sqrt(pi)
assert erf(z).rewrite(
'expint') == sqrt(z**2) / z - z * expint(S.Half, z**2) / sqrt(S.Pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi)
assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1
assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x,
oo) == -1
assert erf(x).as_real_imag() == \
((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
def test_jacobi():
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a / 2 - b / 2 + x * (a / 2 + b / 2 + 1)
assert (jacobi(2, a, b, x) == a**2 / 8 - a * b / 4 - a / 8 + b**2 / 8 -
b / 8 + x**2 * (a**2 / 8 + a * b / 4 + 7 * a / 8 + b**2 / 8 +
7 * b / 8 + Rational(3, 2)) + x *
(a**2 / 4 + 3 * a / 4 - b**2 / 4 - 3 * b / 4) - S.Half)
assert jacobi(n, a, a, x) == RisingFactorial(a + 1, n) * gegenbauer(
n, a + Rational(1, 2), x) / RisingFactorial(2 * a + 1, n)
assert jacobi(n, a, -a,
x) == ((-1)**a * (-x + 1)**(-a / 2) * (x + 1)**(a / 2) *
assoc_legendre(n, a, x) * factorial(-a + n) *
gamma(a + n + 1) / (factorial(a + n) * gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b / 2) * (x + 1)**(-b / 2) *
assoc_legendre(n, b, x) *
gamma(-b + n + 1) / gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, S.Half, S.Half, x) == RisingFactorial(Rational(
3, 2), n) * chebyshevu(n, x) / factorial(n + 1)
assert jacobi(n, -S.Half, -S.Half, x) == RisingFactorial(
Rational(1, 2), n) * chebyshevt(n, x) / factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n * jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n) * gamma(a + n + 1) * hyper(
(-b - n, -n), (a + 1, ), -1) / (factorial(n) * gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n) / factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo * RisingFactorial(a + b + m + 1, m)
assert conjugate(jacobi(m, a, b, x)) == \
jacobi(m, conjugate(a), conjugate(b), conjugate(x))
assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n, a, b, x), x) == \
(a/2 + b/2 + n/2 + Rational(1, 2))*jacobi(n - 1, a + 1, b + 1, x)
# XXX see issue sympy/sympy#5539
assert str(jacobi(n, a, b, x).diff(a)) == \
("Sum((jacobi(n, a, b, x) + (a + b + 2*_k + 1)*RisingFactorial(b + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)))/(a + b + n + _k + 1), (_k, 0, n - 1))")
assert str(jacobi(n, a, b, x).diff(b)) == \
("Sum(((-1)**(n - _k)*(a + b + 2*_k + 1)*RisingFactorial(a + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)) + jacobi(n, a, b, x))/(a + b + n + "
"_k + 1), (_k, 0, n - 1))")
assert jacobi_normalized(n, a, b, x) == \
(jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
/ ((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
pytest.raises(ValueError, lambda: jacobi(-2.1, a, b, x))
pytest.raises(ValueError,
lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
pytest.raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
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([], [Rational(1, 2)], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [Rational(3, 2)], -z**2/4)) == sin(z)
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], z**2)*z) \
== asin(z)
# Test place option
f = meijerg(((0, 1), ()), ((Rational(1, 2),), (0,)), z**2)
assert hyperexpand(f) == sqrt(pi)/sqrt(1 + z**(-2))
assert hyperexpand(f, place=0) == sqrt(pi)*z/sqrt(z**2 + 1)
assert hyperexpand(f, place=zoo) == sqrt(pi)/sqrt(1 + z**(-2))
def test_branch_bug():
assert hyperexpand(hyper((-Rational(1, 3), Rational(1, 2)), (Rational(2, 3), Rational(3, 2)), -z)) == \
-cbrt(z)*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**Rational(2, 3)*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) -
2*lowergamma(Rational(2, 3), z)/z**Rational(2, 3))*gamma(Rational(2, 3))/gamma(Rational(5, 3))
def test_catalan():
n = Symbol('n', integer=True)
m = Symbol('n', integer=True, positive=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs({n: i}) == c
assert catalan(n).rewrite(Product).subs({n: i}).doit() == c
assert catalan(x) == catalan(x)
assert catalan(2 *
x).rewrite(binomial) == binomial(4 * x, 2 * x) / (2 * x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8 / (3 * pi)
assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\
8 / (3 * pi)
assert catalan(3 * x).rewrite(gamma) == 4**(
3 * x) * gamma(3 * x + Rational(1, 2)) / (sqrt(pi) * gamma(3 * x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2, ), 1)
assert catalan(n).rewrite(factorial) == factorial(
2 * n) / (factorial(n + 1) * factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(0, x + Rational(1, 2)) -
polygamma(0, x + 2) + log(4)) * catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(Rational(1, 2)).evalf()
assert str(c) == '0.848826363156775'
c = catalan(I).evalf(3)
assert sstr((re(c), im(c))) == '(0.398, -0.0209)'
def test_erfc():
assert erfc(nan) == nan
assert erfc(oo) == 0
assert erfc(-oo) == 2
assert erfc(0) == 1
assert erfc(I * oo) == -oo * I
assert erfc(-I * oo) == oo * I
assert erfc(-x) == Integer(2) - erfc(x)
assert erfc(erfcinv(x)) == x
assert erfc(erfinv(x)) == 1 - x
assert erfc(I).is_extended_real is False
assert erfc(w).is_extended_real is True
assert erfc(z).is_extended_real is None
assert conjugate(erfc(z)) == erfc(conjugate(z))
assert erfc(x).as_leading_term(x) == 1
assert erfc(1 / x).as_leading_term(x) == erfc(1 / x)
assert erfc(z).rewrite('erf') == 1 - erf(z)
assert erfc(z).rewrite('erfi') == 1 + I * erfi(I * z)
assert erfc(z).rewrite('fresnels') == 1 - (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erfc(z).rewrite('fresnelc') == 1 - (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erfc(z).rewrite('hyper') == 1 - 2 * z * hyper(
[Rational(1, 2)], [Rational(3, 2)], -z**2) / sqrt(pi)
assert erfc(z).rewrite('meijerg') == 1 - z * meijerg(
[Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2) / sqrt(pi)
assert erfc(z).rewrite(
'uppergamma') == 1 - sqrt(z**2) * erf(sqrt(z**2)) / z
assert erfc(z).rewrite('expint') == 1 - sqrt(z**2) / z + z * expint(
Rational(1, 2), z**2) / sqrt(pi)
assert erfc(x).as_real_imag() == \
((erfc(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 +
erfc(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2,
I*(erfc(re(x) - I*re(x)*abs(im(x))/abs(re(x))) -
erfc(re(x) + I*re(x)*abs(im(x))/abs(re(x)))) *
re(x)*abs(im(x))/(2*im(x)*abs(re(x)))))
assert erfc(x).as_real_imag(deep=False) == erfc(x).as_real_imag()
assert erfc(w).as_real_imag() == (erfc(w), 0)
assert erfc(w).as_real_imag(deep=False) == erfc(w).as_real_imag()
assert erfc(I).as_real_imag() == (1, -erfi(1))
pytest.raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
assert erfc(x).taylor_term(3, x, *(-2 * x / sqrt(pi),
0)) == 2 * x**3 / 3 / sqrt(pi)
assert erfc(x).limit(x, oo) == 0
assert erfc(x).diff(x) == -2 * exp(-x**2) / sqrt(pi)
def test_meijerg_with_Floats():
# see sympy/sympy#10681
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_meijerg_expand():
from diofant import combsimp, 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([[], []], [[Rational(1, 2)], [0]], (z/2)**2)) \
== sin(z)/sqrt(pi)
assert hyperexpand(meijerg([[], []], [[0], [Rational(1, 2)]], (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/2 - 1/(2*z), abs(1/z) < 1),
(meijerg([0, 2], [], [], [-1, 1], z), True))
# Test that the simplest possible answer is returned:
assert combsimp(simplify(hyperexpand(
meijerg([1], [1 - a], [-a/2, -a/2 + Rational(1, 2)], [], 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
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 test_erfi():
assert erfi(nan) == nan
assert erfi(+oo) == +oo
assert erfi(-oo) == -oo
assert erfi(0) == 0
assert erfi(I * oo) == I
assert erfi(-I * oo) == -I
assert erfi(-x) == -erfi(x)
assert erfi(I * erfinv(x)) == I * x
assert erfi(I * erfcinv(x)) == I * (1 - x)
assert erfi(I * erf2inv(0, x)) == I * x
assert erfi(I).is_extended_real is False
assert erfi(w).is_extended_real is True
assert erfi(z).is_extended_real is None
assert conjugate(erfi(z)) == erfi(conjugate(z))
assert erfi(z).rewrite('erf') == -I * erf(I * z)
assert erfi(z).rewrite('erfc') == I * erfc(I * z) - I
assert erfi(z).rewrite('fresnels') == (1 - I) * (
fresnelc(z * (1 + I) / sqrt(pi)) - I * fresnels(z *
(1 + I) / sqrt(pi)))
assert erfi(z).rewrite('fresnelc') == (1 - I) * (
fresnelc(z * (1 + I) / sqrt(pi)) - I * fresnels(z *
(1 + I) / sqrt(pi)))
assert erfi(z).rewrite('hyper') == 2 * z * hyper(
[Rational(1, 2)], [Rational(3, 2)], z**2) / sqrt(pi)
assert erfi(z).rewrite('meijerg') == z * meijerg(
[Rational(1, 2)], [], [0], [Rational(-1, 2)], -z**2) / sqrt(pi)
assert erfi(z).rewrite('uppergamma') == (
sqrt(-z**2) / z * (uppergamma(Rational(1, 2), -z**2) / sqrt(pi) - 1))
assert erfi(z).rewrite('expint') == sqrt(-z**2) / z - z * expint(
Rational(1, 2), -z**2) / sqrt(pi)
assert erfi(x).as_real_imag() == \
((erfi(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 +
erfi(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2,
I*(erfi(re(x) - I*re(x)*abs(im(x))/abs(re(x))) -
erfi(re(x) + I*re(x)*abs(im(x))/abs(re(x)))) *
re(x)*abs(im(x))/(2*im(x)*abs(re(x)))))
assert erfi(x).as_real_imag(deep=False) == erfi(x).as_real_imag()
assert erfi(w).as_real_imag() == (erfi(w), 0)
assert erfi(w).as_real_imag(deep=False) == erfi(w).as_real_imag()
assert erfi(I).as_real_imag() == (0, erf(1))
pytest.raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
assert erfi(x).taylor_term(3, x, *(2 * x / sqrt(pi),
0)) == 2 * x**3 / 3 / sqrt(pi)
assert erfi(x).limit(x, oo) == oo
def test_airyai():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyai(z), airyai)
assert airyai(0) == 3**Rational(1, 3) / (3 * gamma(Rational(2, 3)))
assert airyai(oo) == 0
assert airyai(-oo) == 0
assert diff(airyai(z), z) == airyaiprime(z)
assert series(airyai(z), z, 0,
3) == (3**Rational(5, 6) * gamma(Rational(1, 3)) / (6 * pi) -
3**Rational(1, 6) * z * gamma(Rational(2, 3)) /
(2 * pi) + O(z**3))
assert airyai(z).rewrite(hyper) == (-3**Rational(2, 3) * z * hyper(
(), (Rational(4, 3), ),
z**Integer(3) / 9) / (3 * gamma(Rational(1, 3))) +
3**Rational(1, 3) * hyper(
(), (Rational(2, 3), ),
z**Integer(3) / 9) /
(3 * gamma(Rational(2, 3))))
assert isinstance(airyai(z).rewrite(besselj), airyai)
assert airyai(t).rewrite(besselj) == (
sqrt(-t) * (besselj(-Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3) +
besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3)
assert airyai(z).rewrite(besseli) == (
-z * besseli(Rational(1, 3), 2 * z**Rational(3, 2) / 3) /
(3 * (z**Rational(3, 2))**Rational(1, 3)) +
(z**Rational(3, 2))**Rational(1, 3) *
besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3) / 3)
assert airyai(p).rewrite(besseli) == (
sqrt(p) * (besseli(-Rational(1, 3), 2 * p**Rational(3, 2) / 3) -
besseli(Rational(1, 3), 2 * p**Rational(3, 2) / 3)) / 3)
assert expand_func(airyai(2 * (3 * z**5)**Rational(1, 3))) == (
-sqrt(3) * (-1 + (z**5)**Rational(1, 3) / z**Rational(5, 3)) *
airybi(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 6 +
(1 + (z**5)**Rational(1, 3) / z**Rational(5, 3)) *
airyai(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 2)
def test_jacobi():
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1)
assert (jacobi(2, a, b, x) == a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 +
x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + Rational(3, 2)) +
x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - Rational(1, 2))
assert jacobi(n, a, a, x) == RisingFactorial(
a + 1, n)*gegenbauer(n, a + Rational(1, 2), x)/RisingFactorial(2*a + 1, n)
assert jacobi(n, a, -a, x) == ((-1)**a*(-x + 1)**(-a/2)*(x + 1)**(a/2)*assoc_legendre(n, a, x) *
factorial(-a + n)*gamma(a + n + 1)/(factorial(a + n)*gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b/2)*(x + 1)**(-b/2)*assoc_legendre(n, b, x) *
gamma(-b + n + 1)/gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, Rational(1, 2), Rational(1, 2), x) == RisingFactorial(
Rational(3, 2), n)*chebyshevu(n, x)/factorial(n + 1)
assert jacobi(n, Rational(-1, 2), Rational(-1, 2), x) == RisingFactorial(
Rational(1, 2), n)*chebyshevt(n, x)/factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n*jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n)*gamma(a + n + 1)*hyper(
(-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n)/factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo*RisingFactorial(a + b + m + 1, m)
assert jacobi(n, a, b, oo) == jacobi(n, a, b, oo, evaluate=False)
assert conjugate(jacobi(m, a, b, x)) == \
jacobi(m, conjugate(a), conjugate(b), conjugate(x))
assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n, a, b, x), x) == \
(a/2 + b/2 + n/2 + Rational(1, 2))*jacobi(n - 1, a + 1, b + 1, x)
# XXX see issue sympy/sympy#5539
assert str(jacobi(n, a, b, x).diff(a)) == \
("Sum((jacobi(n, a, b, x) + (a + b + 2*_k + 1)*RisingFactorial(b + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)))/(a + b + n + _k + 1), (_k, 0, n - 1))")
assert str(jacobi(n, a, b, x).diff(b)) == \
("Sum(((-1)**(n - _k)*(a + b + 2*_k + 1)*RisingFactorial(a + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)) + jacobi(n, a, b, x))/(a + b + n + "
"_k + 1), (_k, 0, n - 1))")
assert jacobi_normalized(n, a, b, x) == \
(jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
/ ((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
pytest.raises(ValueError, lambda: jacobi(-2.1, a, b, x))
pytest.raises(ValueError, lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
pytest.raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
def test_airybi():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybi(z), airybi)
assert airybi(0) == 3**Rational(5, 6)/(3*gamma(Rational(2, 3)))
assert airybi(oo) == oo
assert airybi(-oo) == 0
assert diff(airybi(z), z) == airybiprime(z)
assert series(airybi(z), z, 0, 3) == (
cbrt(3)*gamma(Rational(1, 3))/(2*pi) + 3**Rational(2, 3)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
l = Limit(airybi(I/x)/(exp(Rational(2, 3)*(I/x)**Rational(3, 2))*sqrt(pi*sqrt(I/x))), x, 0)
assert l.doit() == l
assert airybi(z).rewrite(hyper) == (
root(3, 6)*z*hyper((), (Rational(4, 3),), z**3/9)/gamma(Rational(1, 3)) +
3**Rational(5, 6)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
assert isinstance(airybi(z).rewrite(besselj), airybi)
assert (airybi(t).rewrite(besselj) ==
sqrt(3)*sqrt(-t)*(besselj(-1/3, 2*(-t)**Rational(3, 2)/3) -
besselj(Rational(1, 3),
2*(-t)**Rational(3, 2)/3))/3)
assert airybi(z).rewrite(besseli) == (
sqrt(3)*(z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/cbrt(z**Rational(3, 2)) +
cbrt(z**Rational(3, 2))*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3))/3)
assert airybi(p).rewrite(besseli) == (
sqrt(3)*sqrt(p)*(besseli(-Rational(1, 3), 2*p**Rational(3, 2)/3) +
besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
assert airybi(p).rewrite(besselj) == airybi(p)
assert expand_func(airybi(2*cbrt(3*z**5))) == (
sqrt(3)*(1 - cbrt(z**5)/z**Rational(5, 3))*airyai(2*cbrt(3)*z**Rational(5, 3))/2 +
(1 + cbrt(z**5)/z**Rational(5, 3))*airybi(2*cbrt(3)*z**Rational(5, 3))/2)
assert expand_func(airybi(x*y)) == airybi(x*y)
assert expand_func(airybi(log(x))) == airybi(log(x))
assert expand_func(airybi(2*root(3*z**5, 5))) == airybi(2*root(3*z**5, 5))
assert airybi(x).taylor_term(-1, x) == 0
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_lerchphi():
from diofant import combsimp, exp_polar, polylog, log, lerchphi
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 combsimp(
hyperexpand(meijerg([0, 1 - a], [], [0], [-a],
exp_polar(-I * pi) * z))) == lerchphi(z, 1, a)
assert combsimp(
hyperexpand(
meijerg([0, 1 - a, 1 - a], [], [0], [-a, -a],
exp_polar(-I * pi) * z))) == lerchphi(z, 2, a)
assert combsimp(
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 + Rational(1, 2)], [a + 1, Rational(1, 2)], 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 diofant import Abs
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2), Rational(1, 2), 1],
[Rational(3, 2), Rational(3, 2), Rational(3, 2)], Rational(1, 4))) == \
Abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, Rational(1, 2)))
def test_erfc():
assert erfc(nan) == nan
assert erfc(oo) == 0
assert erfc(-oo) == 2
assert erfc(0) == 1
assert erfc(I*oo) == -oo*I
assert erfc(-I*oo) == oo*I
assert erfc(-x) == Integer(2) - erfc(x)
assert erfc(erfcinv(x)) == x
assert erfc(erfinv(x)) == 1 - x
assert erfc(I).is_extended_real is False
assert erfc(w).is_extended_real is True
assert erfc(z).is_extended_real is None
assert conjugate(erfc(z)) == erfc(conjugate(z))
assert erfc(x).as_leading_term(x) == 1
assert erfc(1/x).as_leading_term(x) == erfc(1/x)
assert erfc(z).rewrite('erf') == 1 - erf(z)
assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z)
assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([Rational(1, 2)], [Rational(3, 2)], -z**2)/sqrt(pi)
assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*erf(sqrt(z**2))/z
assert erfc(z).rewrite('expint') == 1 - sqrt(z**2)/z + z*expint(Rational(1, 2), z**2)/sqrt(pi)
assert erfc(x).as_real_imag() == \
((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
assert erfc(x).as_real_imag(deep=False) == erfc(x).as_real_imag()
assert erfc(w).as_real_imag() == (erfc(w), 0)
assert erfc(w).as_real_imag(deep=False) == erfc(w).as_real_imag()
assert erfc(I).as_real_imag() == (1, -erfi(1))
pytest.raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
assert erfc(x).taylor_term(3, x, *(-2*x/sqrt(pi), 0)) == 2*x**3/3/sqrt(pi)
assert erfc(x).limit(x, oo) == 0
assert erfc(x).diff(x) == -2*exp(-x**2)/sqrt(pi)
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_airybiprime():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybiprime(z), airybiprime)
assert airybiprime(0) == root(3, 6)/gamma(Rational(1, 3))
assert airybiprime(oo) == oo
assert airybiprime(-oo) == 0
assert diff(airybiprime(z), z) == z*airybi(z)
assert series(airybiprime(z), z, 0, 3) == (
root(3, 6)/gamma(Rational(1, 3)) + 3**Rational(5, 6)*z**2/(6*gamma(Rational(2, 3))) + O(z**3))
assert airybiprime(z).rewrite(hyper) == (
3**Rational(5, 6)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) +
root(3, 6)*hyper((), (Rational(1, 3),), z**3/9)/gamma(Rational(1, 3)))
assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
assert (airybiprime(t).rewrite(besselj) ==
-sqrt(3)*t*(besselj(-Rational(2, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(2, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airybiprime(z).rewrite(besseli) == (
sqrt(3)*(z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(z**Rational(3, 2))**Rational(2, 3) +
(z**Rational(3, 2))**Rational(2, 3)*besseli(-Rational(2, 3), 2*z**Rational(3, 2)/3))/3)
assert airybiprime(p).rewrite(besseli) == (
sqrt(3)*p*(besseli(-Rational(2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3)
assert airybiprime(p).rewrite(besselj) == airybiprime(p)
assert expand_func(airybiprime(2*cbrt(3*z**5))) == (
sqrt(3)*(z**Rational(5, 3)/cbrt(z**5) - 1)*airyaiprime(2*cbrt(3)*z**Rational(5, 3))/2 +
(z**Rational(5, 3)/cbrt(z**5) + 1)*airybiprime(2*cbrt(3)*z**Rational(5, 3))/2)
assert expand_func(airybiprime(x*y)) == airybiprime(x*y)
assert expand_func(airybiprime(log(x))) == airybiprime(log(x))
assert expand_func(airybiprime(2*root(3*z**5, 5))) == airybiprime(2*root(3*z**5, 5))
assert airybiprime(-2).evalf(50) == Float('0.27879516692116952268509756941098324140300059345163131', dps=50)
def test_erfi():
assert erfi(nan) == nan
assert erfi(+oo) == +oo
assert erfi(-oo) == -oo
assert erfi(0) == 0
assert erfi(I*oo) == I
assert erfi(-I*oo) == -I
assert erfi(-x) == -erfi(x)
assert erfi(I*erfinv(x)) == I*x
assert erfi(I*erfcinv(x)) == I*(1 - x)
assert erfi(I*erf2inv(0, x)) == I*x
assert erfi(I).is_extended_real is False
assert erfi(w).is_extended_real is True
assert erfi(z).is_extended_real is None
assert conjugate(erfi(z)) == erfi(conjugate(z))
assert erfi(z).rewrite('erf') == -I*erf(I*z)
assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I
assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('hyper') == 2*z*hyper([Rational(1, 2)], [Rational(3, 2)], z**2)/sqrt(pi)
assert erfi(z).rewrite('meijerg') == z*meijerg([Rational(1, 2)], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi)
assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(Rational(1, 2),
-z**2)/sqrt(pi) - 1))
assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(Rational(1, 2), -z**2)/sqrt(pi)
assert erfi(x).as_real_imag() == \
((erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
assert erfi(x).as_real_imag(deep=False) == erfi(x).as_real_imag()
assert erfi(w).as_real_imag() == (erfi(w), 0)
assert erfi(w).as_real_imag(deep=False) == erfi(w).as_real_imag()
assert erfi(I).as_real_imag() == (0, erf(1))
pytest.raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
assert erfi(x).taylor_term(3, x, *(2*x/sqrt(pi), 0)) == 2*x**3/3/sqrt(pi)
assert erfi(x).limit(x, oo) == oo
def test_airyaiprime():
z = Symbol('z', extended_real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyaiprime(z), airyaiprime)
assert airyaiprime(0) == -3**Rational(2, 3)/(3*gamma(Rational(1, 3)))
assert airyaiprime(oo) == 0
assert diff(airyaiprime(z), z) == z*airyai(z)
assert series(airyaiprime(z), z, 0, 3) == (
-3**Rational(2, 3)/(3*gamma(Rational(1, 3))) + cbrt(3)*z**2/(6*gamma(Rational(2, 3))) + O(z**3))
assert airyaiprime(z).rewrite(hyper) == (
cbrt(3)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) -
3**Rational(2, 3)*hyper((), (Rational(1, 3),), z**3/9)/(3*gamma(Rational(1, 3))))
assert isinstance(airyaiprime(z).rewrite(besselj), airyaiprime)
assert (airyaiprime(t).rewrite(besselj) ==
t*(besselj(-Rational(2, 3), 2*(-t)**Rational(3, 2)/3) -
besselj(Rational(2, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airyaiprime(z).rewrite(besseli) == (
z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(3*(z**Rational(3, 2))**Rational(2, 3)) -
(z**Rational(3, 2))**Rational(2, 3)*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3)/3)
assert airyaiprime(p).rewrite(besseli) == (
p*(-besseli(-Rational(2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3)
assert airyaiprime(p).rewrite(besselj) == airyaiprime(p)
assert expand_func(airyaiprime(2*cbrt(3*z**5))) == (
sqrt(3)*(z**Rational(5, 3)/cbrt(z**5) - 1)*airybiprime(2*cbrt(3)*z**Rational(5, 3))/6 +
(z**Rational(5, 3)/cbrt(z**5) + 1)*airyaiprime(2*cbrt(3)*z**Rational(5, 3))/2)
assert expand_func(airyaiprime(x*y)) == airyaiprime(x*y)
assert expand_func(airyaiprime(log(x))) == airyaiprime(log(x))
assert expand_func(airyaiprime(2*root(3*z**5, 5))) == airyaiprime(2*root(3*z**5, 5))
assert airyaiprime(-2).evalf(50) == Float('0.61825902074169104140626429133247528291577794512414753', dps=50)
def test_lerchphi():
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 combsimp(hyperexpand(meijerg([0, 1 - a], [], [0],
[-a], exp_polar(-I*pi)*z))) == lerchphi(z, 1, a)
assert combsimp(hyperexpand(meijerg([0, 1 - a, 1 - a], [], [0],
[-a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 2, a)
assert combsimp(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 + Rational(1, 2)], [a + 1, Rational(1, 2)], 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
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2), Rational(1, 2), 1],
[Rational(3, 2), Rational(3, 2), Rational(3, 2)], Rational(1, 4))) == \
abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, Rational(1, 2)))
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_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_expand_func():
# evaluation at 1 of Gauss' hypergeometric function:
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))
- hyper([a1, b1], [c1], 1)).evalf(strict=False) < 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 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, a+Rational(-1, 2)], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([Rational(1, 2), 2], [Rational(3, 2)], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [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], [Rational(1, 2), 3], z)) == \
sqrt(z)*(6*z/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 + Rational(1, 2), 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + Rational(1, 2))/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [Rational(1, 2), a], [Rational(3, 2), a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + Rational(1, 2))*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_li():
z = Symbol("z")
zr = Symbol("z", extended_real=True)
zp = Symbol("z", positive=True)
zn = Symbol("z", negative=True)
assert li(0) == 0
assert li(1) == -oo
assert li(oo) == oo
assert isinstance(li(z), li)
assert diff(li(z), z) == 1/log(z)
pytest.raises(ArgumentIndexError, lambda: li(z).fdiff(2))
assert conjugate(li(z)) == li(conjugate(z))
assert conjugate(li(-zr)) == li(-zr)
assert conjugate(li(-zp)) == conjugate(li(-zp))
assert conjugate(li(zn)) == conjugate(li(zn))
assert li(z).rewrite(Li) == Li(z) + li(2)
assert li(z).rewrite(Ei) == Ei(log(z))
assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) +
log(log(z))/2 - expint(1, -log(z)))
assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(hyper) == (log(z)*hyper((1, 1), (2, 2), log(z)) -
log(1/log(z))/2 + log(log(z))/2 + EulerGamma)
assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 -
meijerg(((), (1,)), ((0, 0), ()), -log(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_hyperexpand_parametric():
assert hyperexpand(hyper([a, Rational(1, 2) + a], [Rational(1, 2)], 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)*(sqrt(-z + 1) + 1)**(-2*a + 1)
def test_sympyissue_6052():
G0 = meijerg((), (), (1,), (0,), 0)
assert hyperexpand(G0) == 0
assert hyperexpand(hyper((), (2,), 0)) == 1
def test_diofantissue_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)
def test_diofantissue_241():
e = hyper((2, 3, 5, 9, 1), (1, 4, 6, 10), 1)
assert hyperexpand(e) == Rational(108, 7)
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_polynomial():
assert hyperexpand(hyper([], [-1], z)) == oo
assert hyperexpand(hyper([-2], [-1], z)) == 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_shifted_sum():
assert simplify(hyperexpand(z**4*hyper([2], [3, Rational(3, 2)], -z**2))) \
== z*sin(2*z) + (-z**2 + Rational(1, 2))*cos(2*z) - Rational(1, 2)
def test_meijerg_expand():
# 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([[], []], [[Rational(1, 2)], [0]], (z/2)**2)) \
== sin(z)/sqrt(pi)
assert hyperexpand(meijerg([[], []], [[0], [Rational(1, 2)]], (z/2)**2)) \
== cos(z)/sqrt(pi)
assert can_do_meijer([], [a], [a - 1, a - Rational(1, 2)], [])
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 + Rational(1, 2)])
assert can_do_meijer(
[], [], [a], [b], False) # branches only agree for small z
assert can_do_meijer([], [Rational(1, 2)], [a], [-a])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, a + Rational(1, 2)], [b, b + Rational(1, 2)])
assert can_do_meijer([], [], [a, -a], [0, Rational(1, 2)], False) # dito
assert can_do_meijer([], [], [a, a + Rational(1, 2), b, b + Rational(1, 2)], [])
assert can_do_meijer([Rational(1, 2)], [], [0], [a, -a])
assert can_do_meijer([Rational(1, 2)], [], [a], [0, -a], False) # dito
assert can_do_meijer([], [a - Rational(1, 2)], [a, b], [a - Rational(1, 2)], False)
assert can_do_meijer([], [a + Rational(1, 2)], [a + b, a - b, a], [], False)
assert can_do_meijer([a + Rational(1, 2)], [], [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/2 - 1/(2*z), abs(1/z) < 1),
(meijerg([0, 2], [], [], [-1, 1], z), True))
# Test that the simplest possible answer is returned:
assert combsimp(simplify(hyperexpand(
meijerg([1], [1 - a], [-a/2, -a/2 + Rational(1, 2)], [], 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
assert can_do_meijer([], [], [a + Rational(1, 2)], [a, a - b/2, a + b/2])
assert can_do_meijer([], [], [3*a - Rational(1, 2), a, -a - Rational(1, 2)], [a - Rational(1, 2)])
assert can_do_meijer([], [], [0, a - Rational(1, 2), -a - Rational(1, 2)], [Rational(1, 2)])
assert can_do_meijer([Rational(1, 2)], [], [-a, a], [0])
