def test_intractable():
assert limit(1/gamma(x), x, oo) == 0
assert limit(1/loggamma(x), x, oo) == 0
assert limit(gamma(x)/loggamma(x), x, oo) == oo
assert limit(exp(gamma(x))/gamma(x), x, oo) == oo
assert limit(gamma(3 + 1/x), x, oo) == 2
assert limit(gamma(Rational(1, 7) + 1/x), x, oo) == gamma(Rational(1, 7))
assert limit(log(x**x)/log(gamma(x)), x, oo) == 1
assert limit(log(gamma(gamma(x)))/exp(x), x, oo) == oo
assert limit(acosh(1 + 1/x)*sqrt(x), x, oo) == sqrt(2)
# issue sympy/sympy#10804
assert limit(2*airyai(x)*root(x, 4) *
exp(2*x**Rational(3, 2)/3), x, oo) == 1/sqrt(pi)
assert limit(airybi(x)*root(x, 4) *
exp(-2*x**Rational(3, 2)/3), x, oo) == 1/sqrt(pi)
assert limit(airyai(1/x), x, oo) == (3**Rational(5, 6) *
gamma(Rational(1, 3))/(6*pi))
assert limit(airybi(1/x), x, oo) == cbrt(3)*gamma(Rational(1, 3))/(2*pi)
assert limit(airyai(2 + 1/x), x, oo) == airyai(2)
assert limit(airybi(2 + 1/x), x, oo) == airybi(2)
# issue sympy/sympy#10976
assert limit(erf(m/x)/erf(1/x), x, oo) == m
assert limit(Max(x**2, x, exp(x))/x, x, oo) == oo
def test_specfun():
for f in [besselj, bessely, besseli, besselk]:
assert octave_code(f(n, x)) == f.__name__ + '(n, x)'
assert octave_code(hankel1(n, x)) == 'besselh(n, 1, x)'
assert octave_code(hankel2(n, x)) == 'besselh(n, 2, x)'
assert octave_code(airyai(x)) == 'airy(0, x)'
assert octave_code(airyaiprime(x)) == 'airy(1, x)'
assert octave_code(airybi(x)) == 'airy(2, x)'
assert octave_code(airybiprime(x)) == 'airy(3, x)'
assert octave_code(uppergamma(n, x)) == "gammainc(x, n, 'upper')"
assert octave_code(lowergamma(n, x)) == "gammainc(x, n, 'lower')"
assert octave_code(jn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*besselj(n + 1/2, x)/2'
assert octave_code(yn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*bessely(n + 1/2, x)/2'
assert octave_code(Chi(x)) == 'coshint(x)'
assert octave_code(Ci(x)) == 'cosint(x)'
assert octave_code(laguerre(n, x)) == 'laguerreL(n, x)'
assert octave_code(li(x)) == 'logint(x)'
assert octave_code(loggamma(x)) == 'gammaln(x)'
assert octave_code(polygamma(n, x)) == 'psi(n, x)'
assert octave_code(Shi(x)) == 'sinhint(x)'
assert octave_code(Si(x)) == 'sinint(x)'
assert octave_code(LambertW(x)) == 'lambertw(x)'
assert octave_code(LambertW(x, n)) == 'lambertw(n, x)'
assert octave_code(zeta(x)) == 'zeta(x)'
assert octave_code(zeta(
x, y)) == '% Not supported in Octave:\n% zeta\nzeta(x, y)'
def test_airy_base():
z = Symbol('z')
x = Symbol('x', extended_real=True)
y = Symbol('y', extended_real=True)
assert conjugate(airyai(z)) == airyai(conjugate(z))
assert airyai(x).is_extended_real
assert airyai(z).is_extended_real is None
assert airyai(x+I*y).as_real_imag() == (
airyai(x - I*x*abs(y)/abs(x))/2 + airyai(x + I*x*abs(y)/abs(x))/2,
I*x*(airyai(x - I*x*abs(y)/abs(x)) -
airyai(x + I*x*abs(y)/abs(x)))*abs(y)/(2*y*abs(x)))
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_intractable():
assert gruntz(1/gamma(x), x) == 0
assert gruntz(1/loggamma(x), x) == 0
assert gruntz(gamma(x)/loggamma(x), x) == oo
assert gruntz(exp(gamma(x))/gamma(x), x) == oo
assert gruntz(gamma(3 + 1/x), x) == 2
assert gruntz(gamma(Rational(1, 7) + 1/x), x) == gamma(Rational(1, 7))
assert gruntz(log(x**x)/log(gamma(x)), x) == 1
assert gruntz(log(gamma(gamma(x)))/exp(x), x) == oo
# issue sympy/sympy#10804
assert gruntz(2*airyai(x)*root(x, 4) *
exp(2*x**Rational(3, 2)/3), x) == 1/sqrt(pi)
assert gruntz(airybi(x)*root(x, 4) *
exp(-2*x**Rational(3, 2)/3), x) == 1/sqrt(pi)
assert gruntz(airyai(1/x), x) == (3**Rational(5, 6) *
gamma(Rational(1, 3))/(6*pi))
assert gruntz(airybi(1/x), x) == cbrt(3)*gamma(Rational(1, 3))/(2*pi)
assert gruntz(airyai(2 + 1/x), x) == airyai(2)
assert gruntz(airybi(2 + 1/x), x) == airybi(2)
def test_intractable():
assert gruntz(1/gamma(x), x) == 0
assert gruntz(1/loggamma(x), x) == 0
assert gruntz(gamma(x)/loggamma(x), x) == oo
assert gruntz(exp(gamma(x))/gamma(x), x) == oo
assert gruntz(gamma(3 + 1/x), x) == 2
assert gruntz(gamma(Rational(1, 7) + 1/x), x) == gamma(Rational(1, 7))
assert gruntz(log(x**x)/log(gamma(x)), x) == 1
assert gruntz(log(gamma(gamma(x)))/exp(x), x) == oo
# issue sympy/sympy#10804
assert gruntz(2*airyai(x)*root(x, 4) *
exp(2*x**Rational(3, 2)/3), x) == 1/sqrt(pi)
assert gruntz(airybi(x)*root(x, 4) *
exp(-2*x**Rational(3, 2)/3), x) == 1/sqrt(pi)
assert gruntz(airyai(1/x), x) == (3**Rational(5, 6) *
gamma(Rational(1, 3))/(6*pi))
assert gruntz(airybi(1/x), x) == cbrt(3)*gamma(Rational(1, 3))/(2*pi)
assert gruntz(airyai(2 + 1/x), x) == airyai(2)
assert gruntz(airybi(2 + 1/x), x) == airybi(2)
def test_diff():
assert besselj(n, z).diff(z) == besselj(n - 1, z)/2 - besselj(n + 1, z)/2
assert bessely(n, z).diff(z) == bessely(n - 1, z)/2 - bessely(n + 1, z)/2
assert besseli(n, z).diff(z) == besseli(n - 1, z)/2 + besseli(n + 1, z)/2
assert besselk(n, z).diff(z) == -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
assert hankel1(n, z).diff(z) == hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
assert hankel2(n, z).diff(z) == hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2
pytest.raises(ArgumentIndexError, lambda: besselj(n, z).fdiff(3))
pytest.raises(ArgumentIndexError, lambda: jn(n, z).fdiff(3))
pytest.raises(ArgumentIndexError, lambda: airyai(z).fdiff(2))
pytest.raises(ArgumentIndexError, lambda: airybi(z).fdiff(2))
pytest.raises(ArgumentIndexError, lambda: airyaiprime(z).fdiff(2))
pytest.raises(ArgumentIndexError, lambda: airybiprime(z).fdiff(2))
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_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
