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_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