from itertools import product from sympy.concrete.summations import Sum from sympy.core.function import (diff, expand_func) from sympy.core.numbers import (I, Rational, oo, pi) from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.complexes import (conjugate, polar_lift) from sympy.functions.elementary.exponential import (exp, exp_polar, log) from sympy.functions.elementary.hyperbolic import (cosh, sinh) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely, hankel1, hankel2, hn1, hn2, jn, jn_zeros, yn) from sympy.functions.special.gamma_functions import (gamma, uppergamma) from sympy.functions.special.hyper import hyper from sympy.integrals.integrals import Integral from sympy.series.order import O from sympy.series.series import series from sympy.functions.special.bessel import (airyai, airybi, airyaiprime, airybiprime, marcumq) from sympy.core.random import (random_complex_number as randcplx, verify_numerically as tn, test_derivative_numerically as td, _randint) from sympy.simplify import besselsimp from sympy.testing.pytest import raises, slow from sympy.abc import z, n, k, x randint = _randint() def test_bessel_rand(): for f in [besselj, bessely, besseli, besselk, hankel1, hankel2]: assert td(f(randcplx(), z), z) for f in [jn, yn, hn1, hn2]: assert td(f(randint(-10, 10), z), z) def test_bessel_twoinputs(): for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn]: raises(TypeError, lambda: f(1)) raises(TypeError, lambda: f(1, 2, 3)) def test_besselj_leading_term(): assert besselj(0, x).as_leading_term(x) == 1 assert besselj(1, sin(x)).as_leading_term(x) == x/2 assert besselj(1, 2*sqrt(x)).as_leading_term(x) == sqrt(x) # https://github.com/sympy/sympy/issues/21701 assert (besselj(z, x)/x**z).as_leading_term(x) == 1/(2**z*gamma(z + 1)) def test_bessely_leading_term(): assert bessely(0, x).as_leading_term(x) == (2*log(x) - 2*log(2) + 2*S.EulerGamma)/pi assert bessely(1, sin(x)).as_leading_term(x) == -2/(pi*x) assert bessely(1, 2*sqrt(x)).as_leading_term(x) == -1/(pi*sqrt(x)) def test_besseli_leading_term(): assert besseli(0, x).as_leading_term(x) == 1 assert besseli(1, sin(x)).as_leading_term(x) == x/2 assert besseli(1, 2*sqrt(x)).as_leading_term(x) == sqrt(x) def test_besselk_leading_term(): assert besselk(0, x).as_leading_term(x) == -log(x) - S.EulerGamma + log(2) assert besselk(1, sin(x)).as_leading_term(x) == 1/x assert besselk(1, 2*sqrt(x)).as_leading_term(x) == 1/(2*sqrt(x)) def test_besselj_series(): assert besselj(0, x).series(x) == 1 - x**2/4 + x**4/64 + O(x**6) assert besselj(0, x**(1.1)).series(x) == 1 + x**4.4/64 - x**2.2/4 + O(x**6) assert besselj(0, x**2 + x).series(x) == 1 - x**2/4 - x**3/2\ - 15*x**4/64 + x**5/16 + O(x**6) assert besselj(0, sqrt(x) + x).series(x, n=4) == 1 - x/4 - 15*x**2/64\ + 215*x**3/2304 - x**Rational(3, 2)/2 + x**Rational(5, 2)/16\ + 23*x**Rational(7, 2)/384 + O(x**4) assert besselj(0, x/(1 - x)).series(x) == 1 - x**2/4 - x**3/2 - 47*x**4/64\ - 15*x**5/16 + O(x**6) assert besselj(0, log(1 + x)).series(x) == 1 - x**2/4 + x**3/4\ - 41*x**4/192 + 17*x**5/96 + O(x**6) assert besselj(1, sin(x)).series(x) == x/2 - 7*x**3/48 + 73*x**5/1920 + O(x**6) assert besselj(1, 2*sqrt(x)).series(x) == sqrt(x) - x**Rational(3, 2)/2\ + x**Rational(5, 2)/12 - x**Rational(7, 2)/144 + x**Rational(9, 2)/2880\ - x**Rational(11, 2)/86400 + O(x**6) assert besselj(-2, sin(x)).series(x, n=4) == besselj(2, sin(x)).series(x, n=4) def test_bessely_series(): const = 2*S.EulerGamma/pi - 2*log(2)/pi + 2*log(x)/pi assert bessely(0, x).series(x, n=4) == const + x**2*(-log(x)/(2*pi)\ + (2 - 2*S.EulerGamma)/(4*pi) + log(2)/(2*pi)) + O(x**4*log(x)) assert bessely(1, x).series(x, n=4) == -2/(pi*x) + x*(log(x)/pi - log(2)/pi - \ (1 - 2*S.EulerGamma)/(2*pi)) + x**3*(-log(x)/(8*pi) + \ (S(5)/2 - 2*S.EulerGamma)/(16*pi) + log(2)/(8*pi)) + O(x**4*log(x)) assert bessely(2, x).series(x, n=4) == -4/(pi*x**2) - 1/pi + x**2*(log(x)/(4*pi) - \ log(2)/(4*pi) - (S(3)/2 - 2*S.EulerGamma)/(8*pi)) + O(x**4*log(x)) assert bessely(3, x).series(x, n=4) == -16/(pi*x**3) - 2/(pi*x) - \ x/(4*pi) + x**3*(log(x)/(24*pi) - log(2)/(24*pi) - \ (S(11)/6 - 2*S.EulerGamma)/(48*pi)) + O(x**4*log(x)) assert bessely(0, x**(1.1)).series(x, n=4) == 2*S.EulerGamma/pi\ - 2*log(2)/pi + 2.2*log(x)/pi + x**2.2*(-0.55*log(x)/pi\ + (2 - 2*S.EulerGamma)/(4*pi) + log(2)/(2*pi)) + O(x**4*log(x)) assert bessely(0, x**2 + x).series(x, n=4) == \ const - (2 - 2*S.EulerGamma)*(-x**3/(2*pi) - x**2/(4*pi)) + 2*x/pi\ + x**2*(-log(x)/(2*pi) - 1/pi + log(2)/(2*pi))\ + x**3*(-log(x)/pi + 1/(6*pi) + log(2)/pi) + O(x**4*log(x)) assert bessely(0, x/(1 - x)).series(x, n=3) == const\ + 2*x/pi + x**2*(-log(x)/(2*pi) + (2 - 2*S.EulerGamma)/(4*pi)\ + log(2)/(2*pi) + 1/pi) + O(x**3*log(x)) assert bessely(0, log(1 + x)).series(x, n=3) == const\ - x/pi + x**2*(-log(x)/(2*pi) + (2 - 2*S.EulerGamma)/(4*pi)\ + log(2)/(2*pi) + 5/(12*pi)) + O(x**3*log(x)) assert bessely(1, sin(x)).series(x, n=4) == -1/(pi*(-x**3/12 + x/2)) - \ (1 - 2*S.EulerGamma)*(-x**3/12 + x/2)/pi + x*(log(x)/pi - log(2)/pi) + \ x**3*(-7*log(x)/(24*pi) - 1/(6*pi) + (S(5)/2 - 2*S.EulerGamma)/(16*pi) + 7*log(2)/(24*pi)) + O(x**4*log(x)) assert bessely(1, 2*sqrt(x)).series(x, n=3) == -1/(pi*sqrt(x)) + \ sqrt(x)*(log(x)/pi - (1 - 2*S.EulerGamma)/pi) + x**(S(3)/2)*(-log(x)/(2*pi) + \ (S(5)/2 - 2*S.EulerGamma)/(2*pi)) + x**(S(5)/2)*(log(x)/(12*pi) - \ (S(10)/3 - 2*S.EulerGamma)/(12*pi)) + O(x**3*log(x)) assert bessely(-2, sin(x)).series(x, n=4) == bessely(2, sin(x)).series(x, n=4) def test_besseli_series(): assert besseli(0, x).series(x) == 1 + x**2/4 + x**4/64 + O(x**6) assert besseli(0, x**(1.1)).series(x) == 1 + x**4.4/64 + x**2.2/4 + O(x**6) assert besseli(0, x**2 + x).series(x) == 1 + x**2/4 + x**3/2 + 17*x**4/64 + \ x**5/16 + O(x**6) assert besseli(0, sqrt(x) + x).series(x, n=4) == 1 + x/4 + 17*x**2/64 + \ 217*x**3/2304 + x**(S(3)/2)/2 + x**(S(5)/2)/16 + 25*x**(S(7)/2)/384 + O(x**4) assert besseli(0, x/(1 - x)).series(x) == 1 + x**2/4 + x**3/2 + 49*x**4/64 + \ 17*x**5/16 + O(x**6) assert besseli(0, log(1 + x)).series(x) == 1 + x**2/4 - x**3/4 + 47*x**4/192 - \ 23*x**5/96 + O(x**6) assert besseli(1, sin(x)).series(x) == x/2 - x**3/48 - 47*x**5/1920 + O(x**6) assert besseli(1, 2*sqrt(x)).series(x) == sqrt(x) + x**(S(3)/2)/2 + x**(S(5)/2)/12 + \ x**(S(7)/2)/144 + x**(S(9)/2)/2880 + x**(S(11)/2)/86400 + O(x**6) assert besseli(-2, sin(x)).series(x, n=4) == besseli(2, sin(x)).series(x, n=4) def test_besselk_series(): const = log(2) - S.EulerGamma - log(x) assert besselk(0, x).series(x, n=4) == const + \ x**2*(-log(x)/4 - S.EulerGamma/4 + log(2)/4 + S(1)/4) + O(x**4*log(x)) assert besselk(1, x).series(x, n=4) == 1/x + x*(log(x)/2 - log(2)/2 - \ S(1)/4 + S.EulerGamma/2) + x**3*(log(x)/16 - S(5)/64 - log(2)/16 + \ S.EulerGamma/16) + O(x**4*log(x)) assert besselk(2, x).series(x, n=4) == 2/x**2 - S(1)/2 + x**2*(-log(x)/8 - \ S.EulerGamma/8 + log(2)/8 + S(3)/32) + O(x**4*log(x)) assert besselk(0, x**(1.1)).series(x, n=4) == log(2) - S.EulerGamma - \ 1.1*log(x) + x**2.2*(-0.275*log(x) - S.EulerGamma/4 + \ log(2)/4 + S(1)/4) + O(x**4*log(x)) assert besselk(0, x**2 + x).series(x, n=4) == const + \ (2 - 2*S.EulerGamma)*(x**3/4 + x**2/8) - x + x**2*(-log(x)/4 + \ log(2)/4 + S(1)/2) + x**3*(-log(x)/2 - S(7)/12 + log(2)/2) + O(x**4*log(x)) assert besselk(0, x/(1 - x)).series(x, n=3) == const - x + x**2*(-log(x)/4 - \ S(1)/4 - S.EulerGamma/4 + log(2)/4) + O(x**3*log(x)) assert besselk(0, log(1 + x)).series(x, n=3) == const + x/2 + \ x**2*(-log(x)/4 - S.EulerGamma/4 + S(1)/24 + log(2)/4) + O(x**3*log(x)) assert besselk(1, 2*sqrt(x)).series(x, n=3) == 1/(2*sqrt(x)) + \ sqrt(x)*(log(x)/2 - S(1)/2 + S.EulerGamma) + x**(S(3)/2)*(log(x)/4 - S(5)/8 + \ S.EulerGamma/2) + x**(S(5)/2)*(log(x)/24 - S(5)/36 + S.EulerGamma/12) + O(x**3*log(x)) assert besselk(-2, sin(x)).series(x, n=4) == besselk(2, sin(x)).series(x, n=4) 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 def test_rewrite(): assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S.Half, z) assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S.Half, z) assert besseli(n, z).rewrite(besselj) == \ exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z) assert besselj(n, z).rewrite(besseli) == \ exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z) nu = randcplx() assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z) assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z) assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z) assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z) assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z) assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z) # check that a rewrite was triggered, when the order is set to a generic # symbol 'nu' assert yn(nu, z) != yn(nu, z).rewrite(jn) assert hn1(nu, z) != hn1(nu, z).rewrite(jn) assert hn2(nu, z) != hn2(nu, z).rewrite(jn) assert jn(nu, z) != jn(nu, z).rewrite(yn) assert hn1(nu, z) != hn1(nu, z).rewrite(yn) assert hn2(nu, z) != hn2(nu, z).rewrite(yn) # rewriting spherical bessel functions (SBFs) w.r.t. besselj, bessely is # not allowed if a generic symbol 'nu' is used as the order of the SBFs # to avoid inconsistencies (the order of bessel[jy] is allowed to be # complex-valued, whereas SBFs are defined only for integer orders) order = nu for f in (besselj, bessely): assert hn1(order, z) == hn1(order, z).rewrite(f) assert hn2(order, z) == hn2(order, z).rewrite(f) assert jn(order, z).rewrite(besselj) == sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(order + S.Half, z)/2 assert jn(order, z).rewrite(bessely) == (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-order - S.Half, z)/2 # for integral orders rewriting SBFs w.r.t bessel[jy] is allowed N = Symbol('n', integer=True) ri = randint(-11, 10) for order in (ri, N): for f in (besselj, bessely): assert yn(order, z) != yn(order, z).rewrite(f) assert jn(order, z) != jn(order, z).rewrite(f) assert hn1(order, z) != hn1(order, z).rewrite(f) assert hn2(order, z) != hn2(order, z).rewrite(f) for func, refunc in product((yn, jn, hn1, hn2), (jn, yn, besselj, bessely)): assert tn(func(ri, z), func(ri, z).rewrite(refunc), z) def test_expand(): assert expand_func(besselj(S.Half, z).rewrite(jn)) == \ sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z)) assert expand_func(bessely(S.Half, z).rewrite(yn)) == \ -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z)) # XXX: teach sin/cos to work around arguments like # x*exp_polar(I*pi*n/2). Then change besselsimp -> expand_func assert besselsimp(besselj(S.Half, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z)) assert besselsimp(besselj(Rational(-1, 2), z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z)) assert besselsimp(besselj(Rational(5, 2), z)) == \ -sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2)) assert besselsimp(besselj(Rational(-5, 2), z)) == \ -sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2)) assert besselsimp(bessely(S.Half, z)) == \ -(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z)) assert besselsimp(bessely(Rational(-1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z)) assert besselsimp(bessely(Rational(5, 2), z)) == \ sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2)) assert besselsimp(bessely(Rational(-5, 2), z)) == \ -sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2)) assert besselsimp(besseli(S.Half, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z)) assert besselsimp(besseli(Rational(-1, 2), z)) == \ sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z)) assert besselsimp(besseli(Rational(5, 2), z)) == \ sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**Rational(5, 2)) assert besselsimp(besseli(Rational(-5, 2), z)) == \ sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**Rational(5, 2)) assert besselsimp(besselk(S.Half, z)) == \ besselsimp(besselk(Rational(-1, 2), z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z)) assert besselsimp(besselk(Rational(5, 2), z)) == \ besselsimp(besselk(Rational(-5, 2), z)) == \ sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**Rational(5, 2)) n = Symbol('n', integer=True, positive=True) assert expand_func(besseli(n + 2, z)) == \ besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z assert expand_func(besselj(n + 2, z)) == \ -besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z assert expand_func(besselk(n + 2, z)) == \ besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z assert expand_func(bessely(n + 2, z)) == \ -bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z assert expand_func(besseli(n + S.Half, z).rewrite(jn)) == \ (sqrt(2)*sqrt(z)*exp(-I*pi*(n + S.Half)/2) * exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)) assert expand_func(besselj(n + S.Half, z).rewrite(jn)) == \ sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi) r = Symbol('r', real=True) p = Symbol('p', positive=True) i = Symbol('i', integer=True) for besselx in [besselj, bessely, besseli, besselk]: assert besselx(i, p).is_extended_real is True assert besselx(i, x).is_extended_real is None assert besselx(x, z).is_extended_real is None for besselx in [besselj, besseli]: assert besselx(i, r).is_extended_real is True for besselx in [bessely, besselk]: assert besselx(i, r).is_extended_real is None for besselx in [besselj, bessely, besseli, besselk]: assert expand_func(besselx(oo, x)) == besselx(oo, x, evaluate=False) assert expand_func(besselx(-oo, x)) == besselx(-oo, x, evaluate=False) # Quite varying time, but often really slow @slow def test_slow_expand(): def check(eq, ans): return tn(eq, ans) and eq == ans rn = randcplx(a=1, b=0, d=0, c=2) for besselx in [besselj, bessely, besseli, besselk]: ri = S(2*randint(-11, 10) + 1) / 2 # half integer in [-21/2, 21/2] assert tn(besselsimp(besselx(ri, z)), besselx(ri, z)) assert check(expand_func(besseli(rn, x)), besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x) assert check(expand_func(besseli(-rn, x)), besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x) assert check(expand_func(besselj(rn, x)), -besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x) assert check(expand_func(besselj(-rn, x)), -besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x) assert check(expand_func(besselk(rn, x)), besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x) assert check(expand_func(besselk(-rn, x)), besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x) assert check(expand_func(bessely(rn, x)), -bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x) assert check(expand_func(bessely(-rn, x)), -bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x) def mjn(n, z): return expand_func(jn(n, z)) def myn(n, z): return expand_func(yn(n, z)) def test_jn(): z = symbols("z") assert jn(0, 0) == 1 assert jn(1, 0) == 0 assert jn(-1, 0) == S.ComplexInfinity assert jn(z, 0) == jn(z, 0, evaluate=False) assert jn(0, oo) == 0 assert jn(0, -oo) == 0 assert mjn(0, z) == sin(z)/z assert mjn(1, z) == sin(z)/z**2 - cos(z)/z assert mjn(2, z) == (3/z**3 - 1/z)*sin(z) - (3/z**2) * cos(z) assert mjn(3, z) == (15/z**4 - 6/z**2)*sin(z) + (1/z - 15/z**3)*cos(z) assert mjn(4, z) == (1/z + 105/z**5 - 45/z**3)*sin(z) + \ (-105/z**4 + 10/z**2)*cos(z) assert mjn(5, z) == (945/z**6 - 420/z**4 + 15/z**2)*sin(z) + \ (-1/z - 945/z**5 + 105/z**3)*cos(z) assert mjn(6, z) == (-1/z + 10395/z**7 - 4725/z**5 + 210/z**3)*sin(z) + \ (-10395/z**6 + 1260/z**4 - 21/z**2)*cos(z) assert expand_func(jn(n, z)) == jn(n, z) # SBFs not defined for complex-valued orders assert jn(2+3j, 5.2+0.3j).evalf() == jn(2+3j, 5.2+0.3j) assert eq([jn(2, 5.2+0.3j).evalf(10)], [0.09941975672 - 0.05452508024*I]) def test_yn(): z = symbols("z") assert myn(0, z) == -cos(z)/z assert myn(1, z) == -cos(z)/z**2 - sin(z)/z assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z)) assert expand_func(yn(n, z)) == yn(n, z) # SBFs not defined for complex-valued orders assert yn(2+3j, 5.2+0.3j).evalf() == yn(2+3j, 5.2+0.3j) assert eq([yn(2, 5.2+0.3j).evalf(10)], [0.185250342 + 0.01489557397*I]) def test_sympify_yn(): assert S(15) in myn(3, pi).atoms() assert myn(3, pi) == 15/pi**4 - 6/pi**2 def eq(a, b, tol=1e-6): for u, v in zip(a, b): if not (abs(u - v) < tol): return False return True def test_jn_zeros(): assert eq(jn_zeros(0, 4), [3.141592, 6.283185, 9.424777, 12.566370]) assert eq(jn_zeros(1, 4), [4.493409, 7.725251, 10.904121, 14.066193]) assert eq(jn_zeros(2, 4), [5.763459, 9.095011, 12.322940, 15.514603]) assert eq(jn_zeros(3, 4), [6.987932, 10.417118, 13.698023, 16.923621]) assert eq(jn_zeros(4, 4), [8.182561, 11.704907, 15.039664, 18.301255]) def test_bessel_eval(): n, m, k = Symbol('n', integer=True), Symbol('m'), Symbol('k', integer=True, zero=False) for f in [besselj, besseli]: assert f(0, 0) is S.One assert f(2.1, 0) is S.Zero assert f(-3, 0) is S.Zero assert f(-10.2, 0) is S.ComplexInfinity assert f(1 + 3*I, 0) is S.Zero assert f(-3 + I, 0) is S.ComplexInfinity assert f(-2*I, 0) is S.NaN assert f(n, 0) != S.One and f(n, 0) != S.Zero assert f(m, 0) != S.One and f(m, 0) != S.Zero assert f(k, 0) is S.Zero assert bessely(0, 0) is S.NegativeInfinity assert besselk(0, 0) is S.Infinity for f in [bessely, besselk]: assert f(1 + I, 0) is S.ComplexInfinity assert f(I, 0) is S.NaN for f in [besselj, bessely]: assert f(m, S.Infinity) is S.Zero assert f(m, S.NegativeInfinity) is S.Zero for f in [besseli, besselk]: assert f(m, I*S.Infinity) is S.Zero assert f(m, I*S.NegativeInfinity) is S.Zero for f in [besseli, besselk]: assert f(-4, z) == f(4, z) assert f(-3, z) == f(3, z) assert f(-n, z) == f(n, z) assert f(-m, z) != f(m, z) for f in [besselj, bessely]: assert f(-4, z) == f(4, z) assert f(-3, z) == -f(3, z) assert f(-n, z) == (-1)**n*f(n, z) assert f(-m, z) != (-1)**m*f(m, z) for f in [besselj, besseli]: assert f(m, -z) == (-z)**m*z**(-m)*f(m, z) assert besseli(2, -z) == besseli(2, z) assert besseli(3, -z) == -besseli(3, z) assert besselj(0, -z) == besselj(0, z) assert besselj(1, -z) == -besselj(1, z) assert besseli(0, I*z) == besselj(0, z) assert besseli(1, I*z) == I*besselj(1, z) assert besselj(3, I*z) == -I*besseli(3, z) def test_bessel_nan(): # FIXME: could have these return NaN; for now just fix infinite recursion for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, yn, jn]: assert f(1, S.NaN) == f(1, S.NaN, evaluate=False) def test_meromorphic(): assert besselj(2, x).is_meromorphic(x, 1) == True assert besselj(2, x).is_meromorphic(x, 0) == True assert besselj(2, x).is_meromorphic(x, oo) == False assert besselj(S(2)/3, x).is_meromorphic(x, 1) == True assert besselj(S(2)/3, x).is_meromorphic(x, 0) == False assert besselj(S(2)/3, x).is_meromorphic(x, oo) == False assert besselj(x, 2*x).is_meromorphic(x, 2) == False assert besselk(0, x).is_meromorphic(x, 1) == True assert besselk(2, x).is_meromorphic(x, 0) == True assert besseli(0, x).is_meromorphic(x, 1) == True assert besseli(2, x).is_meromorphic(x, 0) == True assert bessely(0, x).is_meromorphic(x, 1) == True assert bessely(0, x).is_meromorphic(x, 0) == False assert bessely(2, x).is_meromorphic(x, 0) == True assert hankel1(3, x**2 + 2*x).is_meromorphic(x, 1) == True assert hankel1(0, x).is_meromorphic(x, 0) == False assert hankel2(11, 4).is_meromorphic(x, 5) == True assert hn1(6, 7*x**3 + 4).is_meromorphic(x, 7) == True assert hn2(3, 2*x).is_meromorphic(x, 9) == True assert jn(5, 2*x + 7).is_meromorphic(x, 4) == True assert yn(8, x**2 + 11).is_meromorphic(x, 6) == True def test_conjugate(): n = Symbol('n') z = Symbol('z', extended_real=False) x = Symbol('x', extended_real=True) y = Symbol('y', positive=True) t = Symbol('t', negative=True) for f in [besseli, besselj, besselk, bessely, hankel1, hankel2]: assert f(n, -1).conjugate() != f(conjugate(n), -1) assert f(n, x).conjugate() != f(conjugate(n), x) assert f(n, t).conjugate() != f(conjugate(n), t) rz = randcplx(b=0.5) for f in [besseli, besselj, besselk, bessely]: assert f(n, 1 + I).conjugate() == f(conjugate(n), 1 - I) assert f(n, 0).conjugate() == f(conjugate(n), 0) assert f(n, 1).conjugate() == f(conjugate(n), 1) assert f(n, z).conjugate() == f(conjugate(n), conjugate(z)) assert f(n, y).conjugate() == f(conjugate(n), y) assert tn(f(n, rz).conjugate(), f(conjugate(n), conjugate(rz))) assert hankel1(n, 1 + I).conjugate() == hankel2(conjugate(n), 1 - I) assert hankel1(n, 0).conjugate() == hankel2(conjugate(n), 0) assert hankel1(n, 1).conjugate() == hankel2(conjugate(n), 1) assert hankel1(n, y).conjugate() == hankel2(conjugate(n), y) assert hankel1(n, z).conjugate() == hankel2(conjugate(n), conjugate(z)) assert tn(hankel1(n, rz).conjugate(), hankel2(conjugate(n), conjugate(rz))) assert hankel2(n, 1 + I).conjugate() == hankel1(conjugate(n), 1 - I) assert hankel2(n, 0).conjugate() == hankel1(conjugate(n), 0) assert hankel2(n, 1).conjugate() == hankel1(conjugate(n), 1) assert hankel2(n, y).conjugate() == hankel1(conjugate(n), y) assert hankel2(n, z).conjugate() == hankel1(conjugate(n), conjugate(z)) assert tn(hankel2(n, rz).conjugate(), hankel1(conjugate(n), conjugate(rz))) def test_branching(): assert besselj(polar_lift(k), x) == besselj(k, x) assert besseli(polar_lift(k), x) == besseli(k, x) n = Symbol('n', integer=True) assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x) assert besselj(n, polar_lift(x)) == besselj(n, x) assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x) assert besseli(n, polar_lift(x)) == besseli(n, x) def tn(func, s): from sympy.core.random import uniform c = uniform(1, 5) expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi)) eps = 1e-15 expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I) return abs(expr.n() - expr2.n()).n() < 1e-10 nu = Symbol('nu') assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x) assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x) assert tn(besselj, 2) assert tn(besselj, pi) assert tn(besselj, I) assert tn(besseli, 2) assert tn(besseli, pi) assert tn(besseli, I) def test_airy_base(): z = Symbol('z') x = Symbol('x', real=True) y = Symbol('y', real=True) assert conjugate(airyai(z)) == airyai(conjugate(z)) assert airyai(x).is_extended_real assert airyai(x+I*y).as_real_imag() == ( airyai(x - I*y)/2 + airyai(x + I*y)/2, I*(airyai(x - I*y) - airyai(x + I*y))/2) def test_airyai(): z = Symbol('z', 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**3/9)/(3*gamma(Rational(1, 3))) + 3**Rational(1, 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*(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_airybi(): z = Symbol('z', 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) is oo assert airybi(-oo) == 0 assert diff(airybi(z), z) == airybiprime(z) assert series(airybi(z), z, 0, 3) == ( 3**Rational(1, 3)*gamma(Rational(1, 3))/(2*pi) + 3**Rational(2, 3)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3)) assert airybi(z).rewrite(hyper) == ( 3**Rational(1, 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 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 airybi(z).rewrite(besseli) == ( sqrt(3)*(z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/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 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 expand_func(airybi(2*(3*z**5)**Rational(1, 3))) == ( sqrt(3)*(1 - (z**5)**Rational(1, 3)/z**Rational(5, 3))*airyai(2*3**Rational(1, 3)*z**Rational(5, 3))/2 + (1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airybi(2*3**Rational(1, 3)*z**Rational(5, 3))/2) def test_airyaiprime(): z = Symbol('z', 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))) + 3**Rational(1, 3)*z**2/(6*gamma(Rational(2, 3))) + O(z**3)) assert airyaiprime(z).rewrite(hyper) == ( 3**Rational(1, 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 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 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 expand_func(airyaiprime(2*(3*z**5)**Rational(1, 3))) == ( sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/6 + (z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2) def test_airybiprime(): z = Symbol('z', real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybiprime(z), airybiprime) assert airybiprime(0) == 3**Rational(1, 6)/gamma(Rational(1, 3)) assert airybiprime(oo) is oo assert airybiprime(-oo) == 0 assert diff(airybiprime(z), z) == z*airybi(z) assert series(airybiprime(z), z, 0, 3) == ( 3**Rational(1, 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))) + 3**Rational(1, 6)*hyper((), (Rational(1, 3),), z**3/9)/gamma(Rational(1, 3))) assert isinstance(airybiprime(z).rewrite(besselj), airybiprime) 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 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 expand_func(airybiprime(2*(3*z**5)**Rational(1, 3))) == ( sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2 + (z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2) def test_marcumq(): m = Symbol('m') a = Symbol('a') b = Symbol('b') assert marcumq(0, 0, 0) == 0 assert marcumq(m, 0, b) == uppergamma(m, b**2/2)/gamma(m) assert marcumq(2, 0, 5) == 27*exp(Rational(-25, 2))/2 assert marcumq(0, a, 0) == 1 - exp(-a**2/2) assert marcumq(0, pi, 0) == 1 - exp(-pi**2/2) assert marcumq(1, a, a) == S.Half + exp(-a**2)*besseli(0, a**2)/2 assert marcumq(2, a, a) == S.Half + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) assert diff(marcumq(1, a, 3), a) == a*(-marcumq(1, a, 3) + marcumq(2, a, 3)) assert diff(marcumq(2, 3, b), b) == -b**2*exp(-b**2/2 - Rational(9, 2))*besseli(1, 3*b)/3 x = Symbol('x') assert marcumq(2, 3, 4).rewrite(Integral, x=x) == \ Integral(x**2*exp(-x**2/2 - Rational(9, 2))*besseli(1, 3*x), (x, 4, oo))/3 assert eq([marcumq(5, -2, 3).rewrite(Integral).evalf(10)], [0.7905769565]) k = Symbol('k') assert marcumq(-3, -5, -7).rewrite(Sum, k=k) == \ exp(-37)*Sum((Rational(5, 7))**k*besseli(k, 35), (k, 4, oo)) assert eq([marcumq(1, 3, 1).rewrite(Sum).evalf(10)], [0.9891705502]) assert marcumq(1, a, a, evaluate=False).rewrite(besseli) == S.Half + exp(-a**2)*besseli(0, a**2)/2 assert marcumq(2, a, a, evaluate=False).rewrite(besseli) == S.Half + exp(-a**2)*besseli(0, a**2)/2 + \ exp(-a**2)*besseli(1, a**2) assert marcumq(3, a, a).rewrite(besseli) == (besseli(1, a**2) + besseli(2, a**2))*exp(-a**2) + \ S.Half + exp(-a**2)*besseli(0, a**2)/2 assert marcumq(5, 8, 8).rewrite(besseli) == exp(-64)*besseli(0, 64)/2 + \ (besseli(4, 64) + besseli(3, 64) + besseli(2, 64) + besseli(1, 64))*exp(-64) + S.Half assert marcumq(m, a, a).rewrite(besseli) == marcumq(m, a, a) x = Symbol('x', integer=True) assert marcumq(x, a, a).rewrite(besseli) == marcumq(x, a, a)