def test_specfun():
n = Symbol('n')
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(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'
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_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_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_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_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_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_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_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_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