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_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_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_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_to_hyper(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper() q = 3 * hyper([], [], 2 * x) assert p == q p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand() q = 2 * x**3 + 6 * x**2 + 6 * x + 2 assert p == q p = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1]).to_hyper() q = -x**2 * hyper((2, 2, 1), (3, 2), -x) / 2 + x assert p == q p = HolonomicFunction(2 * x * Dx + Dx**2, x, 0, [0, 2 / sqrt(pi)]).to_hyper() q = 2 * x * hyper((S.Half, ), (Rational(3, 2), ), -x**2) / sqrt(pi) assert p == q p = hyperexpand( HolonomicFunction(2 * x * Dx + Dx**2, x, 0, [1, -2 / sqrt(pi)]).to_hyper()) q = erfc(x) assert p.rewrite(erfc) == q p = hyperexpand( HolonomicFunction((x**2 - 1) + x * Dx + x**2 * Dx**2, x, 0, [0, S.Half]).to_hyper()) q = besselj(1, x) assert p == q p = hyperexpand( HolonomicFunction(x * Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper()) q = besselj(0, x) assert p == q
def test_inverse_laplace_transform(): from sympy.core.exprtools import factor_terms from sympy.functions.special.delta_functions import DiracDelta from sympy.simplify.simplify import simplify ILT = inverse_laplace_transform a, b, c, = symbols('a b c', positive=True) t = symbols('t') def simp_hyp(expr): return factor_terms(expand_mul(expr)).rewrite(sin) assert ILT(1, s, t) == DiracDelta(t) assert ILT(1/s, s, t) == Heaviside(t) assert ILT(a/(a + s), s, t) == a*exp(-a*t)*Heaviside(t) assert ILT(s/(a + s), s, t) == -a*exp(-a*t)*Heaviside(t) + DiracDelta(t) assert ILT((a + s)**(-2), s, t) == t*exp(-a*t)*Heaviside(t) assert ILT((a + s)**(-5), s, t) == t**4*exp(-a*t)*Heaviside(t)/24 assert ILT(a/(a**2 + s**2), s, t) == sin(a*t)*Heaviside(t) assert ILT(s/(s**2 + a**2), s, t) == cos(a*t)*Heaviside(t) assert ILT(b/(b**2 + (a + s)**2), s, t) == exp(-a*t)*sin(b*t)*Heaviside(t) assert ILT(b*s/(b**2 + (a + s)**2), s, t) +\ (a*sin(b*t) - b*cos(b*t))*exp(-a*t)*Heaviside(t) == 0 assert ILT(exp(-a*s)/s, s, t) == Heaviside(-a + t) assert ILT(exp(-a*s)/(b + s), s, t) == exp(b*(a - t))*Heaviside(-a + t) assert ILT((b + s)/(a**2 + (b + s)**2), s, t) == \ exp(-b*t)*cos(a*t)*Heaviside(t) assert ILT(exp(-a*s)/s**b, s, t) == \ (-a + t)**(b - 1)*Heaviside(-a + t)/gamma(b) assert ILT(exp(-a*s)/sqrt(s**2 + 1), s, t) == \ Heaviside(-a + t)*besselj(0, a - t) assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t)) assert ILT(1/(s**2*(s**2 + 1)), s, t) == (t - sin(t))*Heaviside(t) assert ILT(s**2/(s**2 + 1), s, t) == -sin(t)*Heaviside(t) + DiracDelta(t) assert ILT(1 - 1/(s**2 + 1), s, t) == -sin(t)*Heaviside(t) + DiracDelta(t) assert ILT(1/s**2, s, t) == t*Heaviside(t) assert ILT(1/s**5, s, t) == t**4*Heaviside(t)/24 assert simp_hyp(ILT(a/(s**2 - a**2), s, t)) == sinh(a*t)*Heaviside(t) assert simp_hyp(ILT(s/(s**2 - a**2), s, t)) == cosh(a*t)*Heaviside(t) # TODO sinh/cosh shifted come out a mess. also delayed trig is a mess # TODO should this simplify further? assert ILT(exp(-a*s)/s**b, s, t) == \ (t - a)**(b - 1)*Heaviside(t - a)/gamma(b) assert ILT(exp(-a*s)/sqrt(1 + s**2), s, t) == \ Heaviside(t - a)*besselj(0, a - t) # note: besselj(0, x) is even # XXX ILT turns these branch factor into trig functions ... assert simplify(ILT(a**b*(s + sqrt(s**2 - a**2))**(-b)/sqrt(s**2 - a**2), s, t).rewrite(exp)) == \ Heaviside(t)*besseli(b, a*t) assert ILT(a**b*(s + sqrt(s**2 + a**2))**(-b)/sqrt(s**2 + a**2), s, t).rewrite(exp) == \ Heaviside(t)*besselj(b, a*t) assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t)) # TODO can we make erf(t) work? assert ILT(1/(s**2*(s**2 + 1)),s,t) == (t - sin(t))*Heaviside(t) assert ILT( (s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t) ==\ Matrix([[exp(t)*Heaviside(t), 0], [0, exp(2*t)*Heaviside(t)]])
def test_requires_partial(): x, y, z, t, nu = symbols('x y z t nu') n = symbols('n', integer=True) f = x * y assert requires_partial(Derivative(f, x)) is True assert requires_partial(Derivative(f, y)) is True ## integrating out one of the variables assert requires_partial( Derivative(Integral(exp(-x * y), (x, 0, oo)), y, evaluate=False)) is False ## bessel function with smooth parameter f = besselj(nu, x) assert requires_partial(Derivative(f, x)) is True assert requires_partial(Derivative(f, nu)) is True ## bessel function with integer parameter f = besselj(n, x) assert requires_partial(Derivative(f, x)) is False # this is not really valid (differentiating with respect to an integer) # but there's no reason to use the partial derivative symbol there. make # sure we don't throw an exception here, though assert requires_partial(Derivative(f, n)) is False ## bell polynomial f = bell(n, x) assert requires_partial(Derivative(f, x)) is False # again, invalid assert requires_partial(Derivative(f, n)) is False ## legendre polynomial f = legendre(0, x) assert requires_partial(Derivative(f, x)) is False f = legendre(n, x) assert requires_partial(Derivative(f, x)) is False # again, invalid assert requires_partial(Derivative(f, n)) is False f = x**n assert requires_partial(Derivative(f, x)) is False assert requires_partial( Derivative( Integral((x * y)**n * exp(-x * y), (x, 0, oo)), y, evaluate=False)) is False # parametric equation f = (exp(t), cos(t)) g = sum(f) assert requires_partial(Derivative(g, t)) is False # function of unspecified variables f = symbols('f', cls=Function) assert requires_partial(Derivative(f, x)) is False assert requires_partial(Derivative(f, x, y)) is True
def test_requires_partial(): x, y, z, t, nu = symbols('x y z t nu') n = symbols('n', integer=True) f = x * y assert requires_partial(Derivative(f, x)) is True assert requires_partial(Derivative(f, y)) is True ## integrating out one of the variables assert requires_partial(Derivative(Integral(exp(-x * y), (x, 0, oo)), y, evaluate=False)) is False ## bessel function with smooth parameter f = besselj(nu, x) assert requires_partial(Derivative(f, x)) is True assert requires_partial(Derivative(f, nu)) is True ## bessel function with integer parameter f = besselj(n, x) assert requires_partial(Derivative(f, x)) is False # this is not really valid (differentiating with respect to an integer) # but there's no reason to use the partial derivative symbol there. make # sure we don't throw an exception here, though assert requires_partial(Derivative(f, n)) is False ## bell polynomial f = bell(n, x) assert requires_partial(Derivative(f, x)) is False # again, invalid assert requires_partial(Derivative(f, n)) is False ## legendre polynomial f = legendre(0, x) assert requires_partial(Derivative(f, x)) is False f = legendre(n, x) assert requires_partial(Derivative(f, x)) is False # again, invalid assert requires_partial(Derivative(f, n)) is False f = x ** n assert requires_partial(Derivative(f, x)) is False assert requires_partial(Derivative(Integral((x*y) ** n * exp(-x * y), (x, 0, oo)), y, evaluate=False)) is False # parametric equation f = (exp(t), cos(t)) g = sum(f) assert requires_partial(Derivative(g, t)) is False f = symbols('f', cls=Function) assert requires_partial(Derivative(f(x), x)) is False assert requires_partial(Derivative(f(x), y)) is False assert requires_partial(Derivative(f(x, y), x)) is True assert requires_partial(Derivative(f(x, y), y)) is True assert requires_partial(Derivative(f(x, y), z)) is True assert requires_partial(Derivative(f(x, y), x, y)) is True
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_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_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_latex_bessel(): from sympy.functions.special.bessel import besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn from sympy.abc import z assert latex(besselj(n, z ** 2) ** k) == r"J^{k}_{n}\left(z^{2}\right)" assert latex(bessely(n, z)) == r"Y_{n}\left(z\right)" assert latex(besseli(n, z)) == r"I_{n}\left(z\right)" assert latex(besselk(n, z)) == r"K_{n}\left(z\right)" assert latex(hankel1(n, z ** 2) ** 2) == r"\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}" assert latex(hankel2(n, z)) == r"H^{(2)}_{n}\left(z\right)" assert latex(jn(n, z)) == r"j_{n}\left(z\right)" assert latex(yn(n, z)) == r"y_{n}\left(z\right)"
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 test_latex_bessel(): from sympy.functions.special.bessel import (besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn) from sympy.abc import z assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)' assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)' assert latex(besseli(n, z)) == r'I_{n}\left(z\right)' assert latex(besselk(n, z)) == r'K_{n}\left(z\right)' assert latex(hankel1(n, z**2)**2) == \ r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}' assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)' assert latex(jn(n, z)) == r'j_{n}\left(z\right)' assert latex(yn(n, z)) == r'y_{n}\left(z\right)'
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_messy(): from sympy.functions.elementary.complexes import re from sympy.functions.elementary.hyperbolic import (acosh, acoth) from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import (asin, atan) from sympy.functions.special.bessel import besselj from sympy.functions.special.error_functions import (Chi, E1, Shi, Si) from sympy.integrals.transforms import (fourier_transform, laplace_transform) assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi / 2) / s, 0, True) assert laplace_transform(Shi(x), x, s) == (acoth(s) / s, -oo, s**2 > 1) # where should the logs be simplified? assert laplace_transform(Chi(x), x, s) == ((log(s**(-2)) - log(1 - 1 / s**2)) / (2 * s), -oo, s**2 > 1) # TODO maybe simplify the inequalities? when the simplification # allows for generators instead of symbols this will work assert laplace_transform(besselj(a, x), x, s)[1:] == \ (0, (re(a) > -2) & (re(a) > -1)) # NOTE s < 0 can be done, but argument reduction is not good enough yet ans = fourier_transform(besselj(1, x) / x, x, s, noconds=False) assert tuple([ans[0].factor(deep=True).expand(), ans[1]]) == \ (Piecewise((0, (s > 1/(2*pi)) | (s < -1/(2*pi))), (2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0) # TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons) # - folding could be better assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \ log(1 + sqrt(2)) assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \ log(S.Half + sqrt(2)/2) assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \ Piecewise((-acosh(1/x), abs(x**(-2)) > 1), (I*asin(1/x), True))
def test_inversion(): from sympy.functions.elementary.piecewise import piecewise_fold from sympy.functions.special.bessel import besselj from sympy.functions.special.delta_functions import Heaviside def inv(f): return piecewise_fold(meijerint_inversion(f, s, t)) assert inv(1 / (s**2 + 1)) == sin(t) * Heaviside(t) assert inv(s / (s**2 + 1)) == cos(t) * Heaviside(t) assert inv(exp(-s) / s) == Heaviside(t - 1) assert inv(1 / sqrt(1 + s**2)) == besselj(0, t) * Heaviside(t) # Test some antcedents checking. assert meijerint_inversion(sqrt(s) / sqrt(1 + s**2), s, t) is None assert inv(exp(s**2)) is None assert meijerint_inversion(exp(-s**2), s, t) is None
def test_pmint_besselj(): f = besselj(nu + 1, x) / besselj(nu, x) g = nu * log(x) - log(besselj(nu, x)) assert heurisch(f, x) == g f = (nu * besselj(nu, x) - x * besselj(nu + 1, x)) / x g = besselj(nu, x) assert heurisch(f, x) == g f = jn(nu + 1, x) / jn(nu, x) g = nu * log(x) - log(jn(nu, x)) assert heurisch(f, x) == g
def test_pmint_bessel_products(): # Note: Derivatives of Bessel functions have many forms. # Recurrence relations are needed for comparisons. if ON_TRAVIS: skip("Too slow for travis.") f = x * besselj(nu, x) * bessely(nu, 2 * x) g = -2 * x * besselj(nu, x) * bessely(nu - 1, 2 * x) / 3 + x * besselj( nu - 1, x) * bessely(nu, 2 * x) / 3 assert heurisch(f, x) == g f = x * besselj(nu, x) * besselk(nu, 2 * x) g = -2 * x * besselj(nu, x) * besselk(nu - 1, 2 * x) / 5 - x * besselj( nu - 1, x) * besselk(nu, 2 * x) / 5 assert heurisch(f, x) == g
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_to_meijerg(): x = symbols('x') assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x) assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x) assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x) assert hyperexpand(expr_to_holonomic( log(x)).to_meijerg()).simplify() == log(x) assert expr_to_holonomic(4 * x**2 / 3 + 7).to_meijerg() == 4 * x**2 / 3 + 7 assert hyperexpand( expr_to_holonomic(besselj(2, x), lenics=3).to_meijerg()) == besselj(2, x) p = hyper((Rational(-1, 2), -3), (), x) assert from_hyper(p).to_meijerg() == hyperexpand(p) p = hyper((S.One, S(3)), (S(2), ), x) assert (hyperexpand(from_hyper(p).to_meijerg()) - hyperexpand(p)).expand() == 0 p = from_hyper(hyper((-2, -3), (S.Half, ), x)) s = hyperexpand(hyper((-2, -3), (S.Half, ), x)) C_0 = Symbol('C_0') C_1 = Symbol('C_1') D_0 = Symbol('D_0') assert (hyperexpand(p.to_meijerg()).subs({ C_0: 1, D_0: 0 }) - s).simplify() == 0 p.y0 = {0: [1], S.Half: [0]} assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0 p = expr_to_holonomic(besselj(S.Half, x), initcond=False) assert ( p.to_expr() - (D_0 * sin(x) + C_0 * cos(x) + C_1 * sin(x)) / sqrt(x)).simplify() == 0 p = expr_to_holonomic( besselj(S.Half, x), y0={Rational(-1, 2): [sqrt(2) / sqrt(pi), sqrt(2) / sqrt(pi)]}) assert (p.to_expr() - besselj(S.Half, x) - besselj(Rational(-1, 2), x)).simplify() == 0
def test_expr_to_holonomic(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = expr_to_holonomic((sin(x) / x)**2) q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \ [1, 0, Rational(-2, 3)]) assert p == q p = expr_to_holonomic(1 / (1 + x**2)**2) q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, [1]) assert p == q p = expr_to_holonomic(exp(x) * sin(x) + x * log(1 + x)) q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \ - 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \ (-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \ 7*x**2/2 + x + Rational(5, 2))*Dx**4, x, 0, [0, 1, 4, -1]) assert p == q p = expr_to_holonomic(x * exp(x) + cos(x) + 1) q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \ 0, [2, 1, 1, 3]) assert p == q assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10) p = expr_to_holonomic(log(1 + x)**2 + 1) q = HolonomicFunction( Dx + (3 * x + 3) * Dx**2 + (x**2 + 2 * x + 1) * Dx**3, x, 0, [1, 0, 2]) assert p == q p = expr_to_holonomic(erf(x)**2 + x) q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \ (x**2+ Rational(1, 4))*Dx**4, x, 0, [0, 1, 8/pi, 0]) assert p == q p = expr_to_holonomic(cosh(x) * x) q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0, [0, 1]) assert p == q p = expr_to_holonomic(besselj(2, x)) q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0]) assert p == q p = expr_to_holonomic(besselj(0, x) + exp(x)) q = HolonomicFunction((-x**2 - x/2 + S.Half) + (x**2 - x/2 - Rational(3, 2))*Dx + (-x**2 + x/2 + 1)*Dx**2 +\ (x**2 + x/2)*Dx**3, x, 0, [2, 1, S.Half]) assert p == q p = expr_to_holonomic(sin(x)**2 / x) q = HolonomicFunction(4 + 4 * x * Dx + 3 * Dx**2 + x * Dx**3, x, 0, [0, 1, 0]) assert p == q p = expr_to_holonomic(sin(x)**2 / x, x0=2) q = HolonomicFunction((4) + (4 * x) * Dx + (3) * Dx**2 + (x) * Dx**3, x, 2, [ sin(2)**2 / 2, sin(2) * cos(2) - sin(2)**2 / 4, -3 * sin(2)**2 / 4 + cos(2)**2 - sin(2) * cos(2) ]) assert p == q p = expr_to_holonomic(log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2) q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \ [-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0]) assert p == q p = p.to_expr() q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2 assert p == q p = expr_to_holonomic(x**S.Half, x0=1) q = HolonomicFunction(x * Dx - S.Half, x, 1, [1]) assert p == q p = expr_to_holonomic(sqrt(1 + x**2)) q = HolonomicFunction((-x) + (x**2 + 1) * Dx, x, 0, [1]) assert p == q assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\ (sqrt(x) + sqrt(2*x))).simplify() == 0 assert expr_to_holonomic(3 * x + 2 * sqrt(x)).to_expr() == 3 * x + 2 * sqrt(x) p = expr_to_holonomic((x**4 + x**3 + 5 * x**2 + 3 * x + 2) / x**2, lenics=3) q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \ 2*x)*Dx, x, 0, {-2: [2, 3, 5]}) assert p == q p = expr_to_holonomic(1 / (x - 1)**2, lenics=3, x0=1) q = HolonomicFunction((2) + (x - 1) * Dx, x, 1, {-2: [1, 0, 0]}) assert p == q a = symbols("a") p = expr_to_holonomic(sqrt(a * x), x=x) assert p.to_expr() == sqrt(a) * sqrt(x)
def test_linear_subs(): from sympy.functions.special.bessel import besselj assert integrate(sin(x - 1), x, meijerg=True) == -cos(1 - x) assert integrate(besselj(1, x - 1), x, meijerg=True) == -besselj(0, 1 - x)
def test_bessel(): from sympy.functions.special.bessel import (besseli, besselj) assert simplify(integrate(besselj(a, z)*besselj(b, z)/z, (z, 0, oo), meijerg=True, conds='none')) == \ 2*sin(pi*(a/2 - b/2))/(pi*(a - b)*(a + b)) assert simplify( integrate(besselj(a, z) * besselj(a, z) / z, (z, 0, oo), meijerg=True, conds='none')) == 1 / (2 * a) # TODO more orthogonality integrals assert simplify(integrate(sin(z*x)*(x**2 - 1)**(-(y + S.Half)), (x, 1, oo), meijerg=True, conds='none') *2/((z/2)**y*sqrt(pi)*gamma(S.Half - y))) == \ besselj(y, z) # Werner Rosenheinrich # SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS assert integrate(x * besselj(0, x), x, meijerg=True) == x * besselj(1, x) assert integrate(x * besseli(0, x), x, meijerg=True) == x * besseli(1, x) # TODO can do higher powers, but come out as high order ... should they be # reduced to order 0, 1? assert integrate(besselj(1, x), x, meijerg=True) == -besselj(0, x) assert integrate(besselj(1, x)**2/x, x, meijerg=True) == \ -(besselj(0, x)**2 + besselj(1, x)**2)/2 # TODO more besseli when tables are extended or recursive mellin works assert integrate(besselj(0, x)**2/x**2, x, meijerg=True) == \ -2*x*besselj(0, x)**2 - 2*x*besselj(1, x)**2 \ + 2*besselj(0, x)*besselj(1, x) - besselj(0, x)**2/x assert integrate(besselj(0, x)*besselj(1, x), x, meijerg=True) == \ -besselj(0, x)**2/2 assert integrate(x**2*besselj(0, x)*besselj(1, x), x, meijerg=True) == \ x**2*besselj(1, x)**2/2 assert integrate(besselj(0, x)*besselj(1, x)/x, x, meijerg=True) == \ (x*besselj(0, x)**2 + x*besselj(1, x)**2 - besselj(0, x)*besselj(1, x)) # TODO how does besselj(0, a*x)*besselj(0, b*x) work? # TODO how does besselj(0, x)**2*besselj(1, x)**2 work? # TODO sin(x)*besselj(0, x) etc come out a mess # TODO can x*log(x)*besselj(0, x) be done? # TODO how does besselj(1, x)*besselj(0, x+a) work? # TODO more indefinite integrals when struve functions etc are implemented # test a substitution assert integrate(besselj(1, x**2)*x, x, meijerg=True) == \ -besselj(0, x**2)/2
def test_mellin_transform_bessel(): from sympy.functions.elementary.miscellaneous 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 + S.Half)*gamma(a/2 + s + S.Half)/( gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), ( -re(a)/2 - S.Half, Rational(1, 4)), True) assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \ (2**a*gamma(a/2 + s)*gamma(-2*s + S.Half)/( gamma(-a/2 - s + S.Half)*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(S.Half - s) / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)), (-re(a), S.Half), True) assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \ (gamma(s)*gamma(S.Half - s) / (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)), (0, S.Half), 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 - S.Half) / (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + S.Half)), (S.Half - re(a), S.Half), 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, S.Half), True) assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \ ((Max(re(a), -re(a)), S.Half), 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(S.Half - 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(S.Half - 2*s) / (sqrt(pi)*gamma(S.Half - s - a/2)*gamma(S.Half - 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(S.Half - s) / (pi**S('3/2')*gamma(1 + a - s)), (Max(-re(a), 0), S.Half), 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), S.Half), 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)), S.Half), 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 + S.Half)/(2*sqrt(pi)*gamma(a - s + 1)), (Max(-re(a), 0), S.Half), 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), S.Half), True) # TODO products of besselk are a mess mt = MT(exp(-x/2)*besselk(a, x/2), x, s) mt0 = gammasimp(trigsimp(gammasimp(mt[0].expand(func=True)))) assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(S.Half - s)/( (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_sympy__functions__special__bessel__besselj(): from sympy.functions.special.bessel import besselj assert _test_args(besselj(x, 1))
def test_inverse_mellin_transform(): from sympy.core.function import expand from sympy.functions.elementary.miscellaneous import (Max, Min) from sympy.functions.elementary.trigonometric import cot from sympy.simplify.powsimp import powsimp from sympy.simplify.simplify import simplify IMT = inverse_mellin_transform assert IMT(gamma(s), s, x, (0, oo)) == exp(-x) assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x) assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \ (x**2 + 1)*Heaviside(1 - x)/(4*x) # test passing "None" assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \ -x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x) assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \ -x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x) # test expansion of sums assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x # test factorisation of polys r = symbols('r', real=True) assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo) ).subs(x, r).rewrite(sin).simplify() \ == sin(r)*Heaviside(1 - exp(-r)) # test multiplicative substitution _a, _b = symbols('a b', positive=True) assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a) assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b) def simp_pows(expr): return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp) # Now test the inverses of all direct transforms tested above # Section 8.4.2 nu = symbols('nu', real=True) assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1) assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x) assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \ == (1 - x)**(beta - 1)*Heaviside(1 - x) assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s), s, x, (-oo, None))) \ == (x - 1)**(beta - 1)*Heaviside(x - 1) assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \ == (1/(x + 1))**rho assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c) *gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi, s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \ == (x**c - d**c)/(x - d) assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s) *gamma(-c/2 - s)/gamma(1 - c - s), s, x, (0, -re(c)/2))) == \ (1 + sqrt(x + 1))**c assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s) /gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \ b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) + b**2 + x)/(b**2 + x) assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s) / gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \ b**c*(sqrt(1 + x/b**2) + 1)**c # Section 8.4.5 assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x) assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \ log(x)**3*Heaviside(x - 1) assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1) assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1) assert IMT(pi/(s*sin(2*pi*s)), s, x, (Rational(-1, 2), 0)) == log(sqrt(x) + 1) assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x) # TODO def mysimp(expr): from sympy.core.function import expand from sympy.simplify.powsimp import powsimp from sympy.simplify.simplify import logcombine return expand( powsimp(logcombine(expr, force=True), force=True, deep=True), force=True).replace(exp_polar, exp) assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [ log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1), log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x + 1)*Heaviside(-x + 1)] # test passing cot assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [ log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1), -log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x + 1)*Heaviside(-x + 1), ] # 8.4.14 assert IMT(-gamma(s + S.Half)/(sqrt(pi)*s), s, x, (Rational(-1, 2), 0)) == \ erf(sqrt(x)) # 8.4.19 assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \ == besselj(a, 2*sqrt(x)) assert simplify(IMT(2**a*gamma(S.Half - 2*s)*gamma(s + (a + 1)/2) / (gamma(1 - s - a/2)*gamma(1 - 2*s + a)), s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \ sin(sqrt(x))*besselj(a, sqrt(x)) assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S.Half - 2*s) / (gamma(S.Half - s - a/2)*gamma(1 - 2*s + a)), s, x, (-re(a)/2, Rational(1, 4)))) == \ cos(sqrt(x))*besselj(a, sqrt(x)) # TODO this comes out as an amazing mess, but simplifies nicely assert simplify(IMT(gamma(a + s)*gamma(S.Half - s) / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)), s, x, (-re(a), S.Half))) == \ besselj(a, sqrt(x))**2 assert simplify(IMT(gamma(s)*gamma(S.Half - s) / (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)), s, x, (0, S.Half))) == \ besselj(-a, sqrt(x))*besselj(a, sqrt(x)) assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s) / (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1) *gamma(a/2 + b/2 - s + 1)), s, x, (-(re(a) + re(b))/2, S.Half))) == \ besselj(a, sqrt(x))*besselj(b, sqrt(x)) # Section 8.4.20 # TODO this can be further simplified! assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) * 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)), s, x, (Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S.Half))) == \ besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) - besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b) # TODO more # for coverage assert IMT(pi/cos(pi*s), s, x, (0, S.Half)) == sqrt(x)/(x + 1)
def test_issue_21701(): assert limit((besselj(z, x)/x**z).subs(z, 7), x, 0) == S(1)/645120
def test_laplace_transform(): from sympy import lowergamma from sympy.functions.special.delta_functions import DiracDelta from sympy.functions.special.error_functions import (fresnelc, fresnels) LT = laplace_transform a, b, c, = symbols('a, b, c', positive=True) t, w, x = symbols('t, w, x') f = Function("f") g = Function("g") # Test rule-base evaluation according to # http://eqworld.ipmnet.ru/en/auxiliary/inttrans/ # Power-law functions (laplace2.pdf) assert LT(a*t+t**2+t**(S(5)/2), t, s) ==\ (a/s**2 + 2/s**3 + 15*sqrt(pi)/(8*s**(S(7)/2)), 0, True) assert LT(b/(t+a), t, s) == (-b*exp(-a*s)*Ei(-a*s), 0, True) assert LT(1/sqrt(t+a), t, s) ==\ (sqrt(pi)*sqrt(1/s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True) assert LT(sqrt(t)/(t+a), t, s) ==\ (-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s), 0, True) assert LT((t+a)**(-S(3)/2), t, s) ==\ (-2*sqrt(pi)*sqrt(s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + 2/sqrt(a), 0, True) assert LT(t**(S(1)/2)*(t+a)**(-1), t, s) ==\ (-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s), 0, True) assert LT(1/(a*sqrt(t) + t**(3/2)), t, s) ==\ (pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True) assert LT((t+a)**b, t, s) ==\ (s**(-b - 1)*exp(-a*s)*lowergamma(b + 1, a*s), 0, True) assert LT(t**5/(t+a), t, s) == (120*a**5*lowergamma(-5, a*s), 0, True) # Exponential functions (laplace3.pdf) assert LT(exp(t), t, s) == (1/(s - 1), 1, True) assert LT(exp(2*t), t, s) == (1/(s - 2), 2, True) assert LT(exp(a*t), t, s) == (1/(s - a), a, True) assert LT(exp(a*(t-b)), t, s) == (exp(-a*b)/(-a + s), a, True) assert LT(t*exp(-a*(t)), t, s) == ((a + s)**(-2), -a, True) assert LT(t*exp(-a*(t-b)), t, s) == (exp(a*b)/(a + s)**2, -a, True) assert LT(b*t*exp(-a*t), t, s) == (b/(a + s)**2, -a, True) assert LT(t**(S(7)/4)*exp(-8*t)/gamma(S(11)/4), t, s) ==\ ((s + 8)**(-S(11)/4), -8, True) assert LT(t**(S(3)/2)*exp(-8*t), t, s) ==\ (3*sqrt(pi)/(4*(s + 8)**(S(5)/2)), -8, True) assert LT(t**a*exp(-a*t), t, s) == ((a+s)**(-a-1)*gamma(a+1), -a, True) assert LT(b*exp(-a*t**2), t, s) ==\ (sqrt(pi)*b*exp(s**2/(4*a))*erfc(s/(2*sqrt(a)))/(2*sqrt(a)), 0, True) assert LT(exp(-2*t**2), t, s) ==\ (sqrt(2)*sqrt(pi)*exp(s**2/8)*erfc(sqrt(2)*s/4)/4, 0, True) assert LT(b*exp(2*t**2), t, s) == b*LaplaceTransform(exp(2*t**2), t, s) assert LT(t*exp(-a*t**2), t, s) ==\ (1/(2*a) - s*erfc(s/(2*sqrt(a)))/(4*sqrt(pi)*a**(S(3)/2)), 0, True) assert LT(exp(-a/t), t, s) ==\ (2*sqrt(a)*sqrt(1/s)*besselk(1, 2*sqrt(a)*sqrt(s)), 0, True) assert LT(sqrt(t)*exp(-a/t), t, s) ==\ (sqrt(pi)*(2*sqrt(a)*sqrt(s) + 1)*sqrt(s**(-3))*exp(-2*sqrt(a)*\ sqrt(s))/2, 0, True) assert LT(exp(-a/t)/sqrt(t), t, s) ==\ (sqrt(pi)*sqrt(1/s)*exp(-2*sqrt(a)*sqrt(s)), 0, True) assert LT( exp(-a/t)/(t*sqrt(t)), t, s) ==\ (sqrt(pi)*sqrt(1/a)*exp(-2*sqrt(a)*sqrt(s)), 0, True) assert LT(exp(-2*sqrt(a*t)), t, s) ==\ ( 1/s -sqrt(pi)*sqrt(a) * exp(a/s)*erfc(sqrt(a)*sqrt(1/s))/\ s**(S(3)/2), 0, True) assert LT(exp(-2*sqrt(a*t))/sqrt(t), t, s) == (exp(a/s)*erfc(sqrt(a)*\ sqrt(1/s))*(sqrt(pi)*sqrt(1/s)), 0, True) assert LT(t**4*exp(-2/t), t, s) ==\ (8*sqrt(2)*(1/s)**(S(5)/2)*besselk(5, 2*sqrt(2)*sqrt(s)), 0, True) # Hyperbolic functions (laplace4.pdf) assert LT(sinh(a*t), t, s) == (a/(-a**2 + s**2), a, True) assert LT(b*sinh(a*t)**2, t, s) == (2*a**2*b/(-4*a**2*s**2 + s**3), 2*a, True) # The following line confirms that issue #21202 is solved assert LT(cosh(2*t), t, s) == (s/(-4 + s**2), 2, True) assert LT(cosh(a*t), t, s) == (s/(-a**2 + s**2), a, True) assert LT(cosh(a*t)**2, t, s) == ((-2*a**2 + s**2)/(-4*a**2*s**2 + s**3), 2*a, True) assert LT(sinh(x + 3), x, s) == ( (-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 0, Abs(s) > 1) # The following line replaces the old test test_issue_7173() assert LT(sinh(a*t)*cosh(a*t), t, s) == (a/(-4*a**2 + s**2), 2*a, True) assert LT(sinh(a*t)/t, t, s) == (log((a + s)/(-a + s))/2, a, True) assert LT(t**(-S(3)/2)*sinh(a*t), t, s) ==\ (-sqrt(pi)*(sqrt(-a + s) - sqrt(a + s)), a, True) assert LT(sinh(2*sqrt(a*t)), t, s) ==\ (sqrt(pi)*sqrt(a)*exp(a/s)/s**(S(3)/2), 0, True) assert LT(sqrt(t)*sinh(2*sqrt(a*t)), t, s) ==\ (-sqrt(a)/s**2 + sqrt(pi)*(a + s/2)*exp(a/s)*erf(sqrt(a)*\ sqrt(1/s))/s**(S(5)/2), 0, True) assert LT(sinh(2*sqrt(a*t))/sqrt(t), t, s) ==\ (sqrt(pi)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/sqrt(s), 0, True) assert LT(sinh(sqrt(a*t))**2/sqrt(t), t, s) ==\ (sqrt(pi)*(exp(a/s) - 1)/(2*sqrt(s)), 0, True) assert LT(t**(S(3)/7)*cosh(a*t), t, s) ==\ (((a + s)**(-S(10)/7) + (-a+s)**(-S(10)/7))*gamma(S(10)/7)/2, a, True) assert LT(cosh(2*sqrt(a*t)), t, s) ==\ (sqrt(pi)*sqrt(a)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/s**(S(3)/2) + 1/s, 0, True) assert LT(sqrt(t)*cosh(2*sqrt(a*t)), t, s) ==\ (sqrt(pi)*(a + s/2)*exp(a/s)/s**(S(5)/2), 0, True) assert LT(cosh(2*sqrt(a*t))/sqrt(t), t, s) ==\ (sqrt(pi)*exp(a/s)/sqrt(s), 0, True) assert LT(cosh(sqrt(a*t))**2/sqrt(t), t, s) ==\ (sqrt(pi)*(exp(a/s) + 1)/(2*sqrt(s)), 0, True) # logarithmic functions (laplace5.pdf) assert LT(log(t), t, s) == (-log(s+S.EulerGamma)/s, 0, True) assert LT(log(t/a), t, s) == (-log(a*s + S.EulerGamma)/s, 0, True) assert LT(log(1+a*t), t, s) == (-exp(s/a)*Ei(-s/a)/s, 0, True) assert LT(log(t+a), t, s) == ((log(a) - exp(s/a)*Ei(-s/a)/s)/s, 0, True) assert LT(log(t)/sqrt(t), t, s) ==\ (sqrt(pi)*(-log(s) - 2*log(2) - S.EulerGamma)/sqrt(s), 0, True) assert LT(t**(S(5)/2)*log(t), t, s) ==\ (15*sqrt(pi)*(-log(s)-2*log(2)-S.EulerGamma+S(46)/15)/(8*s**(S(7)/2)), 0, True) assert (LT(t**3*log(t), t, s, noconds=True)-6*(-log(s) - S.EulerGamma\ + S(11)/6)/s**4).simplify() == S.Zero assert LT(log(t)**2, t, s) ==\ (((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True) assert LT(exp(-a*t)*log(t), t, s) ==\ ((-log(a + s) - S.EulerGamma)/(a + s), -a, True) # Trigonometric functions (laplace6.pdf) assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True) assert LT(Abs(sin(a*t)), t, s) ==\ (a*coth(pi*s/(2*a))/(a**2 + s**2), 0, True) assert LT(sin(a*t)/t, t, s) == (atan(a/s), 0, True) assert LT(sin(a*t)**2/t, t, s) == (log(4*a**2/s**2 + 1)/4, 0, True) assert LT(sin(a*t)**2/t**2, t, s) ==\ (a*atan(2*a/s) - s*log(4*a**2/s**2 + 1)/4, 0, True) assert LT(sin(2*sqrt(a*t)), t, s) ==\ (sqrt(pi)*sqrt(a)*exp(-a/s)/s**(S(3)/2), 0, True) assert LT(sin(2*sqrt(a*t))/t, t, s) == (pi*erf(sqrt(a)*sqrt(1/s)), 0, True) assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True) assert LT(cos(a*t)**2, t, s) ==\ ((2*a**2 + s**2)/(s*(4*a**2 + s**2)), 0, True) assert LT(sqrt(t)*cos(2*sqrt(a*t)), t, s) ==\ (sqrt(pi)*(-2*a + s)*exp(-a/s)/(2*s**(S(5)/2)), 0, True) assert LT(cos(2*sqrt(a*t))/sqrt(t), t, s) ==\ (sqrt(pi)*sqrt(1/s)*exp(-a/s), 0, True) assert LT(sin(a*t)*sin(b*t), t, s) ==\ (2*a*b*s/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True) assert LT(cos(a*t)*sin(b*t), t, s) ==\ (b*(-a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True) assert LT(cos(a*t)*cos(b*t), t, s) ==\ (s*(a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True) assert LT(c*exp(-b*t)*sin(a*t), t, s) == (a*c/(a**2 + (b + s)**2), -b, True) assert LT(c*exp(-b*t)*cos(a*t), t, s) == ((b + s)*c/(a**2 + (b + s)**2), -b, True) assert LT(cos(x + 3), x, s) == ((s*cos(3) - sin(3))/(s**2 + 1), 0, True) # Error functions (laplace7.pdf) assert LT(erf(a*t), t, s) == (exp(s**2/(4*a**2))*erfc(s/(2*a))/s, 0, True) assert LT(erf(sqrt(a*t)), t, s) == (sqrt(a)/(s*sqrt(a + s)), 0, True) assert LT(exp(a*t)*erf(sqrt(a*t)), t, s) ==\ (sqrt(a)/(sqrt(s)*(-a + s)), a, True) assert LT(erf(sqrt(a/t)/2), t, s) == ((1-exp(-sqrt(a)*sqrt(s)))/s, 0, True) assert LT(erfc(sqrt(a*t)), t, s) ==\ ((-sqrt(a) + sqrt(a + s))/(s*sqrt(a + s)), 0, True) assert LT(exp(a*t)*erfc(sqrt(a*t)), t, s) ==\ (1/(sqrt(a)*sqrt(s) + s), 0, True) assert LT(erfc(sqrt(a/t)/2), t, s) == (exp(-sqrt(a)*sqrt(s))/s, 0, True) # Bessel functions (laplace8.pdf) assert LT(besselj(0, a*t), t, s) == (1/sqrt(a**2 + s**2), 0, True) assert LT(besselj(1, a*t), t, s) ==\ (a/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))), 0, True) assert LT(besselj(2, a*t), t, s) ==\ (a**2/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))**2), 0, True) assert LT(t*besselj(0, a*t), t, s) ==\ (s/(a**2 + s**2)**(S(3)/2), 0, True) assert LT(t*besselj(1, a*t), t, s) ==\ (a/(a**2 + s**2)**(S(3)/2), 0, True) assert LT(t**2*besselj(2, a*t), t, s) ==\ (3*a**2/(a**2 + s**2)**(S(5)/2), 0, True) assert LT(besselj(0, 2*sqrt(a*t)), t, s) == (exp(-a/s)/s, 0, True) assert LT(t**(S(3)/2)*besselj(3, 2*sqrt(a*t)), t, s) ==\ (a**(S(3)/2)*exp(-a/s)/s**4, 0, True) assert LT(besselj(0, a*sqrt(t**2+b*t)), t, s) ==\ (exp(b*s - b*sqrt(a**2 + s**2))/sqrt(a**2 + s**2), 0, True) assert LT(besseli(0, a*t), t, s) == (1/sqrt(-a**2 + s**2), a, True) assert LT(besseli(1, a*t), t, s) ==\ (a/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))), a, True) assert LT(besseli(2, a*t), t, s) ==\ (a**2/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))**2), a, True) assert LT(t*besseli(0, a*t), t, s) == (s/(-a**2 + s**2)**(S(3)/2), a, True) assert LT(t*besseli(1, a*t), t, s) == (a/(-a**2 + s**2)**(S(3)/2), a, True) assert LT(t**2*besseli(2, a*t), t, s) ==\ (3*a**2/(-a**2 + s**2)**(S(5)/2), a, True) assert LT(t**(S(3)/2)*besseli(3, 2*sqrt(a*t)), t, s) ==\ (a**(S(3)/2)*exp(a/s)/s**4, 0, True) assert LT(bessely(0, a*t), t, s) ==\ (-2*asinh(s/a)/(pi*sqrt(a**2 + s**2)), 0, True) assert LT(besselk(0, a*t), t, s) ==\ (log(s + sqrt(-a**2 + s**2))/sqrt(-a**2 + s**2), a, True) assert LT(sin(a*t)**8, t, s) ==\ (40320*a**8/(s*(147456*a**8 + 52480*a**6*s**2 + 4368*a**4*s**4 +\ 120*a**2*s**6 + s**8)), 0, True) # Test general rules and unevaluated forms # These all also test whether issue #7219 is solved. assert LT(Heaviside(t-1)*cos(t-1), t, s) == (s*exp(-s)/(s**2 + 1), 0, True) assert LT(a*f(t), t, w) == a*LaplaceTransform(f(t), t, w) assert LT(a*Heaviside(t+1)*f(t+1), t, s) ==\ a*LaplaceTransform(f(t + 1)*Heaviside(t + 1), t, s) assert LT(a*Heaviside(t-1)*f(t-1), t, s) ==\ a*LaplaceTransform(f(t), t, s)*exp(-s) assert LT(b*f(t/a), t, s) == a*b*LaplaceTransform(f(t), t, a*s) assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -f(x), True) assert LT(exp(-a*t)*f(t), t, s) == LaplaceTransform(f(t), t, a + s) assert LT(exp(-a*t)*erfc(sqrt(b/t)/2), t, s) ==\ (exp(-sqrt(b)*sqrt(a + s))/(a + s), -a, True) assert LT(sinh(a*t)*f(t), t, s) ==\ LaplaceTransform(f(t), t, -a+s)/2 - LaplaceTransform(f(t), t, a+s)/2 assert LT(sinh(a*t)*t, t, s) ==\ (-1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True) assert LT(cosh(a*t)*f(t), t, s) ==\ LaplaceTransform(f(t), t, -a+s)/2 + LaplaceTransform(f(t), t, a+s)/2 assert LT(cosh(a*t)*t, t, s) ==\ (1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True) assert LT(sin(a*t)*f(t), t, s) ==\ I*(-LaplaceTransform(f(t), t, -I*a + s) +\ LaplaceTransform(f(t), t, I*a + s))/2 assert LT(sin(a*t)*t, t, s) ==\ (2*a*s/(a**4 + 2*a**2*s**2 + s**4), 0, True) assert LT(cos(a*t)*f(t), t, s) ==\ LaplaceTransform(f(t), t, -I*a + s)/2 +\ LaplaceTransform(f(t), t, I*a + s)/2 assert LT(cos(a*t)*t, t, s) ==\ ((-a**2 + s**2)/(a**4 + 2*a**2*s**2 + s**4), 0, True) # The following two lines test whether issues #5813 and #7176 are solved. assert LT(diff(f(t), (t, 1)), t, s) == s*LaplaceTransform(f(t), t, s)\ - f(0) assert LT(diff(f(t), (t, 3)), t, s) == s**3*LaplaceTransform(f(t), t, s)\ - s**2*f(0) - s*Subs(Derivative(f(t), t), t, 0)\ - Subs(Derivative(f(t), (t, 2)), t, 0) assert LT(a*f(b*t)+g(c*t), t, s) == a*LaplaceTransform(f(t), t, s/b)/b +\ LaplaceTransform(g(t), t, s/c)/c assert inverse_laplace_transform( f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0) assert LT(f(t)*g(t), t, s) == LaplaceTransform(f(t)*g(t), t, s) # additional basic tests from wikipedia assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \ ((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True) assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \ == exp(-b)/(s**2 - 1) # DiracDelta function: standard cases assert LT(DiracDelta(t), t, s) == (1, 0, True) assert LT(DiracDelta(a*t), t, s) == (1/a, 0, True) assert LT(DiracDelta(t/42), t, s) == (42, 0, True) assert LT(DiracDelta(t+42), t, s) == (0, 0, True) assert LT(DiracDelta(t)+DiracDelta(t-42), t, s) == \ (1 + exp(-42*s), 0, True) assert LT(DiracDelta(t)-a*exp(-a*t), t, s) == (s/(a + s), 0, True) assert LT(exp(-t)*(DiracDelta(t)+DiracDelta(t-42)), t, s) == \ (exp(-42*s - 42) + 1, -oo, True) # Collection of cases that cannot be fully evaluated and/or would catch # some common implementation errors assert LT(DiracDelta(t**2), t, s) == LaplaceTransform(DiracDelta(t**2), t, s) assert LT(DiracDelta(t**2 - 1), t, s) == (exp(-s)/2, -oo, True) assert LT(DiracDelta(t*(1 - t)), t, s) == \ LaplaceTransform(DiracDelta(-t**2 + t), t, s) assert LT((DiracDelta(t) + 1)*(DiracDelta(t - 1) + 1), t, s) == \ (LaplaceTransform(DiracDelta(t)*DiracDelta(t - 1), t, s) + \ 1 + exp(-s) + 1/s, 0, True) assert LT(DiracDelta(2*t-2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, True) assert LT(DiracDelta(-2*t+2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, True) # Heaviside tests assert LT(Heaviside(t), t, s) == (1/s, 0, True) assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True) assert LT(Heaviside(t-1), t, s) == (exp(-s)/s, 0, True) assert LT(Heaviside(2*t-4), t, s) == (exp(-2*s)/s, 0, True) assert LT(Heaviside(-2*t+4), t, s) == ((1 - exp(-2*s))/s, 0, True) assert LT(Heaviside(2*t+4), t, s) == (1/s, 0, True) assert LT(Heaviside(-2*t+4), t, s) == ((1 - exp(-2*s))/s, 0, True) # Fresnel functions assert laplace_transform(fresnels(t), t, s) == \ ((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 - cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True) assert laplace_transform(fresnelc(t), t, s) == ( ((2*sin(s**2/(2*pi))*fresnelc(s/pi) - 2*cos(s**2/(2*pi))*fresnels(s/pi) + sqrt(2)*cos(s**2/(2*pi) + pi/4))/(2*s), 0, True)) # Matrix tests Mt = Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]]) Ms = Matrix([[ 1/(s - 1), (s + 1)**(-2)], [(s + 1)**(-2), 1/(s - 1)]]) # The default behaviour for Laplace tranform of a Matrix returns a Matrix # of Tuples and is deprecated: with warns_deprecated_sympy(): Ms_conds = Matrix([[(1/(s - 1), 1, True), ((s + 1)**(-2), -1, True)], [((s + 1)**(-2), -1, True), (1/(s - 1), 1, True)]]) with warns_deprecated_sympy(): assert LT(Mt, t, s) == Ms_conds # The new behavior is to return a tuple of a Matrix and the convergence # conditions for the matrix as a whole: assert LT(Mt, t, s, legacy_matrix=False) == (Ms, 1, True) # With noconds=True the transformed matrix is returned without conditions # either way: assert LT(Mt, t, s, noconds=True) == Ms assert LT(Mt, t, s, legacy_matrix=False, noconds=True) == Ms
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)