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) == 1 assert f(2.1, 0) == 0 assert f(-3, 0) == 0 assert f(-10.2, 0) == zoo assert f(1 + 3 * I, 0) == 0 assert f(-3 + I, 0) == zoo assert f(-2 * I, 0) == nan assert f(n, 0) not in (0, 1) assert f(m, 0) not in (0, 1) assert f(k, 0) == 0 assert bessely(0, 0) == -oo assert besselk(0, 0) == +oo for f in [bessely, besselk]: assert f(1 + I, 0) == zoo assert f(I, 0) == nan for x in (oo, -oo, I * oo, -I * oo): assert besselk(m, x) == 0 for f in [besselj, bessely]: assert f(m, +oo) == 0 assert f(m, -oo) == 0 for f in [besseli, besselk]: assert f(m, +I * oo) == 0 assert f(m, -I * oo) == 0 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_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_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_bessel_eval(): from diofant import I, Symbol n, m, k = Symbol('n', integer=True), Symbol('m'), Symbol('k', integer=True, zero=False) for f in [besselj, besseli]: assert f(0, 0) == S.One assert f(2.1, 0) == S.Zero assert f(-3, 0) == S.Zero assert f(-10.2, 0) == S.ComplexInfinity assert f(1 + 3 * I, 0) == S.Zero assert f(-3 + I, 0) == S.ComplexInfinity assert f(-2 * I, 0) == 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) == S.Zero assert bessely(0, 0) == S.NegativeInfinity assert besselk(0, 0) == S.Infinity for f in [bessely, besselk]: assert f(1 + I, 0) == S.ComplexInfinity assert f(I, 0) == S.NaN for f in [besselj, bessely]: assert f(m, S.Infinity) == S.Zero assert f(m, S.NegativeInfinity) == S.Zero for f in [besseli, besselk]: assert f(m, I * S.Infinity) == S.Zero assert f(m, I * S.NegativeInfinity) == 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_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) == 1 assert f(2.1, 0) == 0 assert f(-3, 0) == 0 assert f(-10.2, 0) == zoo assert f(1 + 3*I, 0) == 0 assert f(-3 + I, 0) == zoo assert f(-2*I, 0) == nan assert f(n, 0) not in (0, 1) assert f(m, 0) not in (0, 1) assert f(k, 0) == 0 assert bessely(0, 0) == -oo assert besselk(0, 0) == +oo for f in [bessely, besselk]: assert f(1 + I, 0) == zoo assert f(I, 0) == nan for f in [besselj, bessely]: assert f(m, +oo) == 0 assert f(m, -oo) == 0 for f in [besseli, besselk]: assert f(m, +I*oo) == 0 assert f(m, -I*oo) == 0 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_rewrite(): from diofant import polar_lift, exp, I assert besselj( n, z).rewrite(jn) == sqrt(2 * z / pi) * jn(n - Rational(1, 2), z) assert bessely( n, z).rewrite(yn) == sqrt(2 * z / pi) * yn(n - Rational(1, 2), 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)
def test_mellin_transform_bessel(): from diofant import Max MT = mellin_transform # 8.4.19 assert MT(besselj(a, 2*sqrt(x)), x, s) == \ (gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True) assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (2**a*gamma(-2*s + Rational(1, 2))*gamma(a/2 + s + Rational(1, 2))/( gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), ( -re(a)/2 - Rational(1, 2), Rational(1, 4)), True) assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (2**a*gamma(a/2 + s)*gamma(-2*s + Rational(1, 2))/( gamma(-a/2 - s + Rational(1, 2))*gamma(a - 2*s + 1)), ( -re(a)/2, Rational(1, 4)), True) assert MT(besselj(a, sqrt(x))**2, x, s) == \ (gamma(a + s)*gamma(Rational(1, 2) - s) / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)), (-re(a), Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \ (gamma(s)*gamma(Rational(1, 2) - s) / (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)), (0, Rational(1, 2)), True) # NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as # I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large) assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (gamma(1 - s)*gamma(a + s - Rational(1, 2)) / (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + Rational(1, 2))), (Rational(1, 2) - re(a), Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \ (4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s) / (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2) * gamma( 1 - s + (a + b)/2)), (-(re(a) + re(b))/2, Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \ ((Max(re(a), -re(a)), Rational(1, 2)), True) # Section 8.4.20 assert MT(bessely(a, 2*sqrt(x)), x, s) == \ (-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi, (Max(-re(a)/2, re(a)/2), Rational(3, 4)), True) assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \ (-4**s*sin(pi*(a/2 - s))*gamma(Rational(1, 2) - 2*s) * gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s) / (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)), (Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True) assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \ (-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(Rational(1, 2) - 2*s) / (sqrt(pi)*gamma(Rational(1, 2) - s - a/2)*gamma(Rational(1, 2) - s + a/2)), (Max(-re(a)/2, re(a)/2), Rational(1, 4)), True) assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \ (-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(Rational(1, 2) - s) / (pi**Rational(3, 2)*gamma(1 + a - s)), (Max(-re(a), 0), Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \ (-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s) * gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) / (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)), (Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), Rational(1, 2)), True) # NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x)) # are a mess (no matter what way you look at it ...) assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \ ((Max(-re(a), 0, re(a)), Rational(1, 2)), True) # Section 8.4.22 # TODO we can't do any of these (delicate cancellation) # Section 8.4.23 assert MT(besselk(a, 2*sqrt(x)), x, s) == \ (gamma( s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True) assert MT( besselj(a, 2 * sqrt(2 * sqrt(x))) * besselk(a, 2 * sqrt(2 * sqrt(x))), x, s) == (4**(-s) * gamma(2 * s) * gamma(a / 2 + s) / (2 * gamma(a / 2 - s + 1)), (Max(0, -re(a) / 2), oo), True) # TODO bessely(a, x)*besselk(a, x) is a mess assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \ (gamma(s)*gamma( a + s)*gamma(-s + Rational(1, 2))/(2*sqrt(pi)*gamma(a - s + 1)), (Max(-re(a), 0), Rational(1, 2)), True) assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \ (2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s) * gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1) * gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2, re(a)/2 - re(b)/2), Rational(1, 2)), True) # TODO products of besselk are a mess mt = MT(exp(-x / 2) * besselk(a, x / 2), x, s) mt0 = combsimp((trigsimp(combsimp(mt[0].expand(func=True))))) assert mt0 == 2 * pi**Rational(3, 2) * cos( pi * s) * gamma(-s + Rational(1, 2)) / ( (cos(2 * pi * a) - cos(2 * pi * s)) * gamma(-a - s + 1) * gamma(a - s + 1)) assert mt[1:] == ((Max(-re(a), re(a)), oo), True)
def test_pole(): pytest.raises(PoleError, lambda: sin(1 / x).series(x, 0, 5)) pytest.raises(PoleError, lambda: sin(1 + 1 / x).series(x, 0, 5)) pytest.raises(PoleError, lambda: (x * sin(1 / x)).series(x, 0, 5)) pytest.raises(PoleError, lambda: besselk(0, x).series(x, 0, 2))
def test_pole(): pytest.raises(PoleError, lambda: sin(1/x).series(x, 0, 5)) pytest.raises(PoleError, lambda: sin(1 + 1/x).series(x, 0, 5)) pytest.raises(PoleError, lambda: (x*sin(1/x)).series(x, 0, 5)) pytest.raises(PoleError, lambda: besselk(0, x).series(x, 0, 2))
def test_expand(): assert expand_func(besselj(Rational(1, 2), z).rewrite(jn)) == \ sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z)) assert expand_func(bessely(Rational(1, 2), z).rewrite(yn)) == \ -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z)) assert expand_func(besselj(I, z)) == besselj(I, z) # Test simplify helper assert simplify(besselj(Rational(1, 2), z)) == sqrt(2) * sin(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(Rational(1, 2), 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(Rational(1, 2), 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(Rational(1, 2), 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(Rational(1, 2), 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)) 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 = Rational(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) 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 + Rational(1, 2), z).rewrite(jn)) == \ (sqrt(2)*sqrt(z)*exp(-I*pi*(n + Rational(1, 2))/2) * exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)) assert expand_func(besselj(n + Rational(1, 2), z).rewrite(jn)) == \ sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi) r = Symbol('r', extended_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 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 for besselx in [bessely, besselk]: assert besselx(i, r).is_extended_real is None
def test_sympyissue_14874(): assert limit(besselk(0, x), x, oo) == 0
def test_expand(): assert expand_func(besselj(Rational(1, 2), z).rewrite(jn)) == \ sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z)) assert expand_func(bessely(Rational(1, 2), z).rewrite(yn)) == \ -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z)) assert expand_func(besselj(I, z)) == besselj(I, z) # Test simplify helper assert simplify(besselj(Rational(1, 2), z)) == sqrt(2)*sin(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(Rational(1, 2), 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(Rational(1, 2), 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(Rational(1, 2), 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(Rational(1, 2), 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)) 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 = Rational(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) 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 + Rational(1, 2), z).rewrite(jn)) == \ (sqrt(2)*sqrt(z)*exp(-I*pi*(n + Rational(1, 2))/2) * exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)) assert expand_func(besselj(n + Rational(1, 2), z).rewrite(jn)) == \ sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi) r = Symbol('r', extended_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 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 for besselx in [bessely, besselk]: assert besselx(i, r).is_extended_real is None
def test_rewrite(): assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - Rational(1, 2), z) assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - Rational(1, 2), 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) assert besselj(2, z).rewrite(bessely) == besselj(2, z) assert bessely(2, z).rewrite(besselj) == bessely(2, z) assert bessely(2, z).rewrite(besseli) == bessely(2, z) assert besselk(2, z).rewrite(besseli) == besselk(2, z) assert besselk(2, z).rewrite(besselj) == besselk(2, z) assert besselk(2, z).rewrite(bessely) == besselk(2, 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)
def test_mellin_transform_bessel(): MT = mellin_transform # 8.4.19 assert MT(besselj(a, 2*sqrt(x)), x, s) == \ (gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True) assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (2**a*gamma(-2*s + Rational(1, 2))*gamma(a/2 + s + Rational(1, 2))/( gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), ( -re(a)/2 - Rational(1, 2), Rational(1, 4)), True) assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (2**a*gamma(a/2 + s)*gamma(-2*s + Rational(1, 2))/( gamma(-a/2 - s + Rational(1, 2))*gamma(a - 2*s + 1)), ( -re(a)/2, Rational(1, 4)), True) assert MT(besselj(a, sqrt(x))**2, x, s) == \ (gamma(a + s)*gamma(Rational(1, 2) - s) / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)), (-re(a), Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \ (gamma(s)*gamma(Rational(1, 2) - s) / (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)), (0, Rational(1, 2)), True) # NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as # I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large) assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (gamma(1 - s)*gamma(a + s - Rational(1, 2)) / (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + Rational(1, 2))), (Rational(1, 2) - re(a), Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \ (4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s) / (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2) * gamma( 1 - s + (a + b)/2)), (-(re(a) + re(b))/2, Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \ ((Max(re(a), -re(a)), Rational(1, 2)), True) # Section 8.4.20 assert MT(bessely(a, 2*sqrt(x)), x, s) == \ (-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi, (Max(-re(a)/2, re(a)/2), Rational(3, 4)), True) assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \ (-4**s*sin(pi*(a/2 - s))*gamma(Rational(1, 2) - 2*s) * gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s) / (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)), (Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True) assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \ (-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(Rational(1, 2) - 2*s) / (sqrt(pi)*gamma(Rational(1, 2) - s - a/2)*gamma(Rational(1, 2) - s + a/2)), (Max(-re(a)/2, re(a)/2), Rational(1, 4)), True) assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \ (-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(Rational(1, 2) - s) / (pi**Rational(3, 2)*gamma(1 + a - s)), (Max(-re(a), 0), Rational(1, 2)), True) assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \ (-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s) * gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) / (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)), (Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), Rational(1, 2)), True) # NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x)) # are a mess (no matter what way you look at it ...) assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \ ((Max(-re(a), 0, re(a)), Rational(1, 2)), True) # Section 8.4.22 # TODO we can't do any of these (delicate cancellation) # Section 8.4.23 assert MT(besselk(a, 2*sqrt(x)), x, s) == \ (gamma( s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True) assert MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk( a, 2*sqrt(2*sqrt(x))), x, s) == (4**(-s)*gamma(2*s) * gamma(a/2 + s)/(2*gamma(a/2 - s + 1)), (Max(0, -re(a)/2), oo), True) # TODO bessely(a, x)*besselk(a, x) is a mess assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \ (gamma(s)*gamma( a + s)*gamma(-s + Rational(1, 2))/(2*sqrt(pi)*gamma(a - s + 1)), (Max(-re(a), 0), Rational(1, 2)), True) assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \ (2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s) * gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1) * gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2, re(a)/2 - re(b)/2), Rational(1, 2)), True) # TODO products of besselk are a mess mt = MT(exp(-x/2)*besselk(a, x/2), x, s) mt0 = combsimp((trigsimp(combsimp(mt[0].expand(func=True))))) assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(-s + Rational(1, 2))/( (cos(2*pi*a) - cos(2*pi*s))*gamma(-a - s + 1)*gamma(a - s + 1)) assert mt[1:] == ((Max(-re(a), re(a)), oo), True)