def test_lowergamma(): from sympy import meijerg, exp_polar, I, expint assert lowergamma(x, y).diff(y) == y**(x-1)*exp(-y) assert td(lowergamma(randcplx(), y), y) assert lowergamma(x, y).diff(x) == \ gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \ + meijerg([], [1, 1], [0, 0, x], [], y) assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x)) assert not lowergamma(S.Half - 3, x).has(lowergamma) assert not lowergamma(S.Half + 3, x).has(lowergamma) assert lowergamma(S.Half, x, evaluate=False).has(lowergamma) assert tn(lowergamma(S.Half + 3, x, evaluate=False), lowergamma(S.Half + 3, x), x) assert tn(lowergamma(S.Half - 3, x, evaluate=False), lowergamma(S.Half - 3, x), x) assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y) assert tn_branch(-3, lowergamma) assert tn_branch(-4, lowergamma) assert tn_branch(S(1)/3, lowergamma) assert tn_branch(pi, lowergamma) assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x) assert lowergamma(y, exp_polar(5*pi*I)*x) == \ exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I)) assert lowergamma(-2, exp_polar(5*pi*I)*x) == \ lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I assert lowergamma(x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x) k = Symbol('k', integer=True) assert lowergamma(k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k) k = Symbol('k', integer=True, positive=False) assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
def test_uppergamma(): from sympy import meijerg, exp_polar, I, expint assert uppergamma(4, 0) == 6 assert uppergamma(x, y).diff(y) == -y**(x-1)*exp(-y) assert td(uppergamma(randcplx(), y), y) assert uppergamma(x, y).diff(x) == \ uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y) assert td(uppergamma(x, randcplx()), x) assert uppergamma(S.Half, x) == sqrt(pi)*(1 - erf(sqrt(x))) assert not uppergamma(S.Half - 3, x).has(uppergamma) assert not uppergamma(S.Half + 3, x).has(uppergamma) assert uppergamma(S.Half, x, evaluate=False).has(uppergamma) assert tn(uppergamma(S.Half + 3, x, evaluate=False), uppergamma(S.Half + 3, x), x) assert tn(uppergamma(S.Half - 3, x, evaluate=False), uppergamma(S.Half - 3, x), x) assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y) assert tn_branch(-3, uppergamma) assert tn_branch(-4, uppergamma) assert tn_branch(S(1)/3, uppergamma) assert tn_branch(pi, uppergamma) assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x) assert uppergamma(y, exp_polar(5*pi*I)*x) == \ exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + gamma(y)*(1-exp(4*pi*I*y)) assert uppergamma(-2, exp_polar(5*pi*I)*x) == \ uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I assert uppergamma(-2, x) == expint(3, x)/x**2 assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
def test_si(): assert Si(I*x) == I*Shi(x) assert Shi(I*x) == I*Si(x) assert Si(-I*x) == -I*Shi(x) assert Shi(-I*x) == -I*Si(x) assert Si(-x) == -Si(x) assert Shi(-x) == -Shi(x) assert Si(exp_polar(2*pi*I)*x) == Si(x) assert Si(exp_polar(-2*pi*I)*x) == Si(x) assert Shi(exp_polar(2*pi*I)*x) == Shi(x) assert Shi(exp_polar(-2*pi*I)*x) == Shi(x) assert mytd(Si(x), sin(x)/x, x) assert mytd(Shi(x), sinh(x)/x, x) assert mytn(Si(x), Si(x).rewrite(Ei), -I*(-Ei(x*exp_polar(-I*pi/2))/2 \ + Ei(x*exp_polar(I*pi/2))/2 - I*pi) + pi/2, x) assert mytn(Si(x), Si(x).rewrite(expint), -I*(-expint(1, x*exp_polar(-I*pi/2))/2 + \ expint(1, x*exp_polar(I*pi/2))/2) + pi/2, x) assert mytn(Shi(x), Shi(x).rewrite(Ei), Ei(x)/2 - Ei(x*exp_polar(I*pi))/2 + I*pi/2, x) assert mytn(Shi(x), Shi(x).rewrite(expint), expint(1, x)/2 - expint(1, x*exp_polar(I*pi))/2 - I*pi/2, x) assert tn_arg(Si) assert tn_arg(Shi) from sympy import O assert Si(x).nseries(x, n=8) == x - x**3/18 + x**5/600 - x**7/35280 + O(x**9) assert Shi(x).nseries(x, n=8) == x + x**3/18 + x**5/600 + x**7/35280 + O(x**9) assert Si(sin(x)).nseries(x, n=5) == x - 2*x**3/9 + 17*x**5/450 + O(x**6) assert Si(x).nseries(x, 1, n=3) == \ Si(1) + x*sin(1) + x**2/2*cos(1) - x**2/2*sin(1) + O(x**3)
def test_ei(): pos = Symbol('p', positive=True) neg = Symbol('n', negative=True) assert Ei(-pos) == Ei(polar_lift(-1)*pos) - I*pi assert Ei(neg) == Ei(polar_lift(neg)) - I*pi assert tn_branch(Ei) assert mytd(Ei(x), exp(x)/x, x) assert mytn(Ei(x), Ei(x).rewrite(uppergamma), -uppergamma(0, x*polar_lift(-1)) - I*pi, x) assert mytn(Ei(x), Ei(x).rewrite(expint), -expint(1, x*polar_lift(-1)) - I*pi, x) assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x) assert Ei(x*exp_polar(2*I*pi)) == Ei(x) + 2*I*pi assert Ei(x*exp_polar(-2*I*pi)) == Ei(x) - 2*I*pi assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x) assert mytn(Ei(x*polar_lift(I)), Ei(x*polar_lift(I)).rewrite(Si), Ci(x) + I*Si(x) + I*pi/2, x) assert Ei(log(x)).rewrite(li) == li(x) assert Ei(2*log(x)).rewrite(li) == li(x**2) assert gruntz(Ei(x+exp(-x))*exp(-x)*x, x, oo) == 1 assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \ x**3/18 + x**4/96 + x**5/600 + O(x**6)
def test_lowergamma(): from sympy import meijerg, exp_polar, I, expint assert lowergamma(x, 0) == 0 assert lowergamma(x, y).diff(y) == y**(x - 1)*exp(-y) assert td(lowergamma(randcplx(), y), y) assert td(lowergamma(x, randcplx()), x) assert lowergamma(x, y).diff(x) == \ gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \ - meijerg([], [1, 1], [0, 0, x], [], y) assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x)) assert not lowergamma(S.Half - 3, x).has(lowergamma) assert not lowergamma(S.Half + 3, x).has(lowergamma) assert lowergamma(S.Half, x, evaluate=False).has(lowergamma) assert tn(lowergamma(S.Half + 3, x, evaluate=False), lowergamma(S.Half + 3, x), x) assert tn(lowergamma(S.Half - 3, x, evaluate=False), lowergamma(S.Half - 3, x), x) assert tn_branch(-3, lowergamma) assert tn_branch(-4, lowergamma) assert tn_branch(S(1)/3, lowergamma) assert tn_branch(pi, lowergamma) assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x) assert lowergamma(y, exp_polar(5*pi*I)*x) == \ exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I)) assert lowergamma(-2, exp_polar(5*pi*I)*x) == \ lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y)) assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0)) assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo)) assert lowergamma( x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x) k = Symbol('k', integer=True) assert lowergamma( k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k) k = Symbol('k', integer=True, positive=False) assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y) assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y) assert lowergamma(70, 6) == factorial(69) - 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320 * exp(-6) assert (lowergamma(S(77) / 2, 6) - lowergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16 assert (lowergamma(-S(77) / 2, 6) - lowergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
def test_si(): assert Si(I*x) == I*Shi(x) assert Shi(I*x) == I*Si(x) assert Si(-I*x) == -I*Shi(x) assert Shi(-I*x) == -I*Si(x) assert Si(-x) == -Si(x) assert Shi(-x) == -Shi(x) assert Si(exp_polar(2*pi*I)*x) == Si(x) assert Si(exp_polar(-2*pi*I)*x) == Si(x) assert Shi(exp_polar(2*pi*I)*x) == Shi(x) assert Shi(exp_polar(-2*pi*I)*x) == Shi(x) assert Si(oo) == pi/2 assert Si(-oo) == -pi/2 assert Shi(oo) == oo assert Shi(-oo) == -oo assert mytd(Si(x), sin(x)/x, x) assert mytd(Shi(x), sinh(x)/x, x) assert mytn(Si(x), Si(x).rewrite(Ei), -I*(-Ei(x*exp_polar(-I*pi/2))/2 + Ei(x*exp_polar(I*pi/2))/2 - I*pi) + pi/2, x) assert mytn(Si(x), Si(x).rewrite(expint), -I*(-expint(1, x*exp_polar(-I*pi/2))/2 + expint(1, x*exp_polar(I*pi/2))/2) + pi/2, x) assert mytn(Shi(x), Shi(x).rewrite(Ei), Ei(x)/2 - Ei(x*exp_polar(I*pi))/2 + I*pi/2, x) assert mytn(Shi(x), Shi(x).rewrite(expint), expint(1, x)/2 - expint(1, x*exp_polar(I*pi))/2 - I*pi/2, x) assert tn_arg(Si) assert tn_arg(Shi) assert Si(x).nseries(x, n=8) == \ x - x**3/18 + x**5/600 - x**7/35280 + O(x**9) assert Shi(x).nseries(x, n=8) == \ x + x**3/18 + x**5/600 + x**7/35280 + O(x**9) assert Si(sin(x)).nseries(x, n=5) == x - 2*x**3/9 + 17*x**5/450 + O(x**6) assert Si(x).nseries(x, 1, n=3) == \ Si(1) + (x - 1)*sin(1) + (x - 1)**2*(-sin(1)/2 + cos(1)/2) + O((x - 1)**3, (x, 1)) t = Symbol('t', Dummy=True) assert Si(x).rewrite(sinc) == Integral(sinc(t), (t, 0, x))
def test_expint(): """ Test various exponential integrals. """ from sympy import expint, unpolarify, Symbol, Ci, Si, Shi, Chi, sin, cos, sinh, cosh, Ei assert simplify( unpolarify( integrate(exp(-z * x) / x ** y, (x, 1, oo), meijerg=True, conds="none").rewrite(expint).expand(func=True) ) ) == expint(y, z) assert integrate(exp(-z * x) / x, (x, 1, oo), meijerg=True, conds="none").rewrite(expint).expand() == expint(1, z) assert integrate(exp(-z * x) / x ** 2, (x, 1, oo), meijerg=True, conds="none").rewrite(expint).expand() == expint( 2, z ).rewrite(Ei).rewrite(expint) assert ( integrate(exp(-z * x) / x ** 3, (x, 1, oo), meijerg=True, conds="none").rewrite(expint).expand() == expint(3, z).rewrite(Ei).rewrite(expint).expand() ) t = Symbol("t", positive=True) assert integrate(-cos(x) / x, (x, t, oo), meijerg=True).expand() == Ci(t) assert integrate(-sin(x) / x, (x, t, oo), meijerg=True).expand() == Si(t) - pi / 2 assert integrate(sin(x) / x, (x, 0, z), meijerg=True) == Si(z) assert integrate(sinh(x) / x, (x, 0, z), meijerg=True) == Shi(z) assert integrate(exp(-x) / x, x, meijerg=True).expand().rewrite(expint) == I * pi - expint(1, x) assert integrate(exp(-x) / x ** 2, x, meijerg=True).rewrite(expint).expand() == expint(1, x) - exp(-x) / x - I * pi u = Symbol("u", polar=True) assert integrate(cos(u) / u, u, meijerg=True).expand().as_independent(u)[1] == Ci(u) assert integrate(cosh(u) / u, u, meijerg=True).expand().as_independent(u)[1] == Chi(u) assert integrate(expint(1, x), x, meijerg=True).rewrite(expint).expand() == x * expint(1, x) - exp(-x) assert ( integrate(expint(2, x), x, meijerg=True).rewrite(expint).expand() == -x ** 2 * expint(1, x) / 2 + x * exp(-x) / 2 - exp(-x) / 2 ) assert simplify(unpolarify(integrate(expint(y, x), x, meijerg=True).rewrite(expint).expand(func=True))) == -expint( y + 1, x ) assert integrate(Si(x), x, meijerg=True) == x * Si(x) + cos(x) assert integrate(Ci(u), u, meijerg=True).expand() == u * Ci(u) - sin(u) assert integrate(Shi(x), x, meijerg=True) == x * Shi(x) - cosh(x) assert integrate(Chi(u), u, meijerg=True).expand() == u * Chi(u) - sinh(u) assert integrate(Si(x) * exp(-x), (x, 0, oo), meijerg=True) == pi / 4 assert integrate(expint(1, x) * sin(x), (x, 0, oo), meijerg=True) == log(2) / 2
def test_uppergamma(): from sympy import meijerg, exp_polar, I, expint assert uppergamma(4, 0) == 6 assert uppergamma(x, y).diff(y) == -y**(x - 1)*exp(-y) assert td(uppergamma(randcplx(), y), y) assert uppergamma(x, y).diff(x) == \ uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y) assert td(uppergamma(x, randcplx()), x) assert uppergamma(S.Half, x) == sqrt(pi)*erfc(sqrt(x)) assert not uppergamma(S.Half - 3, x).has(uppergamma) assert not uppergamma(S.Half + 3, x).has(uppergamma) assert uppergamma(S.Half, x, evaluate=False).has(uppergamma) assert tn(uppergamma(S.Half + 3, x, evaluate=False), uppergamma(S.Half + 3, x), x) assert tn(uppergamma(S.Half - 3, x, evaluate=False), uppergamma(S.Half - 3, x), x) assert tn_branch(-3, uppergamma) assert tn_branch(-4, uppergamma) assert tn_branch(S(1)/3, uppergamma) assert tn_branch(pi, uppergamma) assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x) assert uppergamma(y, exp_polar(5*pi*I)*x) == \ exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + \ gamma(y)*(1 - exp(4*pi*I*y)) assert uppergamma(-2, exp_polar(5*pi*I)*x) == \ uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I assert uppergamma(-2, x) == expint(3, x)/x**2 assert conjugate(uppergamma(x, y)) == uppergamma(conjugate(x), conjugate(y)) assert conjugate(uppergamma(x, 0)) == gamma(conjugate(x)) assert conjugate(uppergamma(x, -oo)) == conjugate(uppergamma(x, -oo)) assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y) assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y) assert uppergamma(70, 6) == 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320*exp(-6) assert (uppergamma(S(77) / 2, 6) - uppergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16 assert (uppergamma(-S(77) / 2, 6) - uppergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
def test_erf(): assert erf(nan) == nan assert erf(oo) == 1 assert erf(-oo) == -1 assert erf(0) == 0 assert erf(I*oo) == oo*I assert erf(-I*oo) == -oo*I assert erf(-2) == -erf(2) assert erf(-x*y) == -erf(x*y) assert erf(-x - y) == -erf(x + y) assert erf(erfinv(x)) == x assert erf(erfcinv(x)) == 1 - x assert erf(erf2inv(0, x)) == x assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x assert erf(I).is_real is False assert erf(0).is_real is True assert conjugate(erf(z)) == erf(conjugate(z)) assert erf(x).as_leading_term(x) == 2*x/sqrt(pi) assert erf(1/x).as_leading_term(x) == erf(1/x) assert erf(z).rewrite('uppergamma') == sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z assert erf(z).rewrite('erfc') == S.One - erfc(z) assert erf(z).rewrite('erfi') == -I*erfi(I*z) assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erf(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erf(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi) assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi) assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \ 2/sqrt(pi) assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi) assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1 assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1 assert erf(x).as_real_imag() == \ ((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
def test_ei(): pos = Symbol("p", positive=True) neg = Symbol("n", negative=True) assert Ei(-pos) == Ei(polar_lift(-1) * pos) - I * pi assert Ei(neg) == Ei(polar_lift(neg)) - I * pi assert tn_branch(Ei) assert mytd(Ei(x), exp(x) / x, x) assert mytn(Ei(x), Ei(x).rewrite(uppergamma), -uppergamma(0, x * polar_lift(-1)) - I * pi, x) assert mytn(Ei(x), Ei(x).rewrite(expint), -expint(1, x * polar_lift(-1)) - I * pi, x) assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x) assert Ei(x * exp_polar(2 * I * pi)) == Ei(x) + 2 * I * pi assert Ei(x * exp_polar(-2 * I * pi)) == Ei(x) - 2 * I * pi assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x) assert mytn(Ei(x * polar_lift(I)), Ei(x * polar_lift(I)).rewrite(Si), Ci(x) + I * Si(x) + I * pi / 2, x)
def test_function__eval_nseries(): n = Symbol("n") assert sin(x)._eval_nseries(x, 2, None) == x + O(x ** 2) assert sin(x + 1)._eval_nseries(x, 2, None) == x * cos(1) + sin(1) + O(x ** 2) assert sin(pi * (1 - x))._eval_nseries(x, 2, None) == pi * x + O(x ** 2) assert acos(1 - x ** 2)._eval_nseries(x, 2, None) == sqrt(2) * x + O(x ** 2) assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == polygamma(n, 1) + polygamma(n + 1, 1) * x + O(x ** 2) raises(PoleError, lambda: sin(1 / x)._eval_nseries(x, 2, None)) raises(PoleError, lambda: acos(1 - x)._eval_nseries(x, 2, None)) raises(PoleError, lambda: acos(1 + x)._eval_nseries(x, 2, None)) assert loggamma(1 / x)._eval_nseries(x, 0, None) == log(x) / 2 - log(x) / x - 1 / x + O(1, x) assert loggamma(log(1 / x)).nseries(x, n=1, logx=y) == loggamma(-y) # issue 6725: assert expint(S(3) / 2, -x)._eval_nseries(x, 5, None) == 2 - 2 * sqrt(pi) * sqrt( -x ) - 2 * x - x ** 2 / 3 - x ** 3 / 15 - x ** 4 / 84 + O(x ** 5) assert sin(sqrt(x))._eval_nseries(x, 3, None) == sqrt(x) - x ** (S(3) / 2) / 6 + x ** (S(5) / 2) / 120 + O(x ** 3)
def test_erfi(): assert erfi(nan) == nan assert erfi(oo) == S.Infinity assert erfi(-oo) == S.NegativeInfinity assert erfi(0) == S.Zero assert erfi(I*oo) == I assert erfi(-I*oo) == -I assert erfi(-x) == -erfi(x) assert erfi(I*erfinv(x)) == I*x assert erfi(I*erfcinv(x)) == I*(1 - x) assert erfi(I*erf2inv(0, x)) == I*x assert erfi(I).is_real is False assert erfi(0).is_real is True assert conjugate(erfi(z)) == erfi(conjugate(z)) assert erfi(z).rewrite('erf') == -I*erf(I*z) assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi) assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], -z**2)/sqrt(pi) assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(S.Pi) - S.One)) assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi) assert expand_func(erfi(I*z)) == I*erf(z) assert erfi(x).as_real_imag() == \ ((erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_erfc(): assert erfc(nan) == nan assert erfc(oo) == 0 assert erfc(-oo) == 2 assert erfc(0) == 1 assert erfc(I*oo) == -oo*I assert erfc(-I*oo) == oo*I assert erfc(-x) == S(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_real is False assert erfc(0).is_real is True assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) == S.One assert erfc(1/x).as_leading_term(x) == erfc(1/x) assert erfc(z).rewrite('erf') == 1 - erf(z) assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z) assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi) assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi) assert expand_func(erf(x) + erfc(x)) == S.One assert erfc(x).as_real_imag() == \ ((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
def eval(cls, a, z): from sympy import unpolarify, I, expint if z.is_Number: if z is S.NaN: return S.NaN elif z is S.Infinity: return S.Zero elif z is S.Zero: # TODO: Holds only for Re(a) > 0: return gamma(a) # We extract branching information here. C/f lowergamma. nx, n = z.extract_branch_factor() if a.is_integer and (a > 0) == True: nx = unpolarify(z) if z != nx: return uppergamma(a, nx) elif a.is_integer and (a <= 0) == True: if n != 0: return -2*pi*I*n*(-1)**(-a)/factorial(-a) + uppergamma(a, nx) elif n != 0: return gamma(a)*(1 - exp(2*pi*I*n*a)) + exp(2*pi*I*n*a)*uppergamma(a, nx) # Special values. if a.is_Number: if a is S.One: return exp(-z) elif a is S.Half: return sqrt(pi)*erfc(sqrt(z)) elif a.is_Integer or (2*a).is_Integer: b = a - 1 if b.is_positive: if a.is_integer: return exp(-z) * factorial(b) * Add(*[z**k / factorial(k) for k in range(a)]) else: return gamma(a) * erfc(sqrt(z)) + (-1)**(a - S(3)/2) * exp(-z) * sqrt(z) * Add(*[gamma(-S.Half - k) * (-z)**k / gamma(1-a) for k in range(a - S.Half)]) elif b.is_Integer: return expint(-b, z)*unpolarify(z)**(b + 1) if not a.is_Integer: return (-1)**(S.Half - a) * pi*erfc(sqrt(z))/gamma(1-a) - z**a * exp(-z) * Add(*[z**k * gamma(a) / gamma(a+k+1) for k in range(S.Half - a)])
def test_erfc(): assert erfc(nan) == nan assert erfc(oo) == 0 assert erfc(-oo) == 2 assert erfc(0) == 1 assert erfc(I*oo) == -oo*I assert erfc(-I*oo) == oo*I assert erfc(-x) == S(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_real is False assert erfc(0).is_real is True assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) == S.One assert erfc(1/x).as_leading_term(x) == erfc(1/x) assert erfc(z).rewrite('erf') == 1 - erf(z) assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z) assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi) assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*erf(sqrt(z**2))/z assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi) assert erfc(x).as_real_imag() == \ ((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
def test_li(): z = Symbol("z") zr = Symbol("z", real=True) zp = Symbol("z", positive=True) zn = Symbol("z", negative=True) assert li(0) == 0 assert li(1) == -oo assert li(oo) == oo assert isinstance(li(z), li) assert diff(li(z), z) == 1 / log(z) assert conjugate(li(z)) == li(conjugate(z)) assert conjugate(li(-zr)) == li(-zr) assert conjugate(li(-zp)) == conjugate(li(-zp)) assert conjugate(li(zn)) == conjugate(li(zn)) assert li(z).rewrite(Li) == Li(z) + li(2) assert li(z).rewrite(Ei) == Ei(log(z)) assert li(z).rewrite(uppergamma) == (-log(1 / log(z)) / 2 - log(-log(z)) + log(log(z)) / 2 - expint(1, -log(z))) assert li(z).rewrite(Si) == (-log(I * log(z)) - log(1 / log(z)) / 2 + log(log(z)) / 2 + Ci(I * log(z)) + Shi(log(z))) assert li(z).rewrite(Ci) == (-log(I * log(z)) - log(1 / log(z)) / 2 + log(log(z)) / 2 + Ci(I * log(z)) + Shi(log(z))) assert li(z).rewrite(Shi) == (-log(1 / log(z)) / 2 + log(log(z)) / 2 + Chi(log(z)) - Shi(log(z))) assert li(z).rewrite(Chi) == (-log(1 / log(z)) / 2 + log(log(z)) / 2 + Chi(log(z)) - Shi(log(z))) assert li(z).rewrite(hyper) == (log(z) * hyper( (1, 1), (2, 2), log(z)) - log(1 / log(z)) / 2 + log(log(z)) / 2 + EulerGamma) assert li(z).rewrite(meijerg) == (-log(1 / log(z)) / 2 - log(-log(z)) + log(log(z)) / 2 - meijerg( ((), (1, )), ((0, 0), ()), -log(z))) assert gruntz(1 / li(z), z, oo) == 0
def eval(cls, a, z): from sympy import unpolarify, I, factorial, exp, expint if z.is_Number: if z is S.NaN: return S.NaN elif z is S.Infinity: return S.Zero elif z is S.Zero: # TODO: Holds only for Re(a) > 0: return gamma(a) # We extract branching information here. C/f lowergamma. nx, n = z.extract_branch_factor() if a.is_integer and (a > 0) == True: nx = unpolarify(z) if z != nx: return uppergamma(a, nx) elif a.is_integer and (a <= 0) == True: if n != 0: return -2 * pi * I * n * (-1)**( -a) / factorial(-a) + uppergamma(a, nx) elif n != 0: return gamma(a) * (1 - exp(2 * pi * I * n * a)) + exp( 2 * pi * I * n * a) * uppergamma(a, nx) # Special values. if a.is_Number: # TODO this should be non-recursive if a is S.One: return C.exp(-z) elif a is S.Half: return sqrt(pi) * (1 - erf(sqrt(z))) # TODO could use erfc... elif a.is_Integer or (2 * a).is_Integer: b = a - 1 if b.is_positive: return b * cls(b, z) + z**b * C.exp(-z) elif b.is_Integer: return expint(-b, z) * unpolarify(z)**(b + 1) if not a.is_Integer: return (cls(a + 1, z) - z**a * C.exp(-z)) / a
def test_ei(): pos = Symbol('p', positive=True) neg = Symbol('n', negative=True) assert Ei(-pos) == Ei(polar_lift(-1) * pos) - I * pi assert Ei(neg) == Ei(polar_lift(neg)) - I * pi assert tn_branch(Ei) assert mytd(Ei(x), exp(x) / x, x) assert mytn(Ei(x), Ei(x).rewrite(uppergamma), -uppergamma(0, x * polar_lift(-1)) - I * pi, x) assert mytn(Ei(x), Ei(x).rewrite(expint), -expint(1, x * polar_lift(-1)) - I * pi, x) assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x) assert Ei(x * exp_polar(2 * I * pi)) == Ei(x) + 2 * I * pi assert Ei(x * exp_polar(-2 * I * pi)) == Ei(x) - 2 * I * pi assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x) assert mytn(Ei(x * polar_lift(I)), Ei(x * polar_lift(I)).rewrite(Si), Ci(x) + I * Si(x) + I * pi / 2, x)
def test_function__eval_nseries(): n = Symbol('n') assert sin(x)._eval_nseries(x, 2, None) == x + O(x**2) assert sin(x + 1)._eval_nseries(x, 2, None) == x*cos(1) + sin(1) + O(x**2) assert sin(pi*(1 - x))._eval_nseries(x, 2, None) == pi*x + O(x**2) assert acos(1 - x**2)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x**2) + O(x**2) assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == \ polygamma(n, 1) + polygamma(n + 1, 1)*x + O(x**2) raises(PoleError, lambda: sin(1/x)._eval_nseries(x, 2, None)) assert acos(1 - x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x) + O(x) assert acos(1 + x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(-x) + O(x) # XXX: wrong, branch cuts assert loggamma(1/x)._eval_nseries(x, 0, None) == \ log(x)/2 - log(x)/x - 1/x + O(1, x) assert loggamma(log(1/x)).nseries(x, n=1, logx=y) == loggamma(-y) # issue 6725: assert expint(Rational(3, 2), -x)._eval_nseries(x, 5, None) == \ 2 - 2*sqrt(pi)*sqrt(-x) - 2*x - x**2/3 - x**3/15 - x**4/84 + O(x**5) assert sin(sqrt(x))._eval_nseries(x, 3, None) == \ sqrt(x) - x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + O(x**3)
def eval(cls, a, z): from sympy import unpolarify, I, factorial, exp, expint if z.is_Number: if z is S.NaN: return S.NaN elif z is S.Infinity: return S.Zero elif z is S.Zero: # TODO: Holds only for Re(a) > 0: return gamma(a) # We extract branching information here. C/f lowergamma. nx, n = z.extract_branch_factor() if a.is_integer and (a > 0) == True: nx = unpolarify(z) if z != nx: return uppergamma(a, nx) elif a.is_integer and (a <= 0) == True: if n != 0: return -2 * pi * I * n * (-1) ** (-a) / factorial(-a) + uppergamma(a, nx) elif n != 0: return gamma(a) * (1 - exp(2 * pi * I * n * a)) + exp(2 * pi * I * n * a) * uppergamma(a, nx) # Special values. if a.is_Number: # TODO this should be non-recursive if a is S.One: return C.exp(-z) elif a is S.Half: return sqrt(pi) * (1 - erf(sqrt(z))) # TODO could use erfc... elif a.is_Integer or (2 * a).is_Integer: b = a - 1 if b.is_positive: return b * cls(b, z) + z ** b * C.exp(-z) elif b.is_Integer: return expint(-b, z) * unpolarify(z) ** (b + 1) if not a.is_Integer: return (cls(a + 1, z) - z ** a * C.exp(-z)) / a
def test_expint(): from sympy import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei aneg = Symbol("a", negative=True) u = Symbol("u", polar=True) assert mellin_transform(E1(x), x, s) == (gamma(s) / s, (0, oo), True) assert inverse_mellin_transform(gamma(s) / s, s, x, (0, oo)).rewrite(expint).expand() == E1(x) assert mellin_transform(expint(a, x), x, s) == (gamma(s) / (a + s - 1), (Max(1 - re(a), 0), oo), True) # XXX IMT has hickups with complicated strips ... assert simplify( unpolarify( inverse_mellin_transform(gamma(s) / (aneg + s - 1), s, x, (1 - aneg, oo)).rewrite(expint).expand(func=True) ) ) == expint(aneg, x) assert mellin_transform(Si(x), x, s) == ( -2 ** s * sqrt(pi) * gamma(s / 2 + S(1) / 2) / (2 * s * gamma(-s / 2 + 1)), (-1, 0), True, ) assert inverse_mellin_transform( -2 ** s * sqrt(pi) * gamma((s + 1) / 2) / (2 * s * gamma(-s / 2 + 1)), s, x, (-1, 0) ) == Si(x) assert mellin_transform(Ci(sqrt(x)), x, s) == ( -2 ** (2 * s - 1) * sqrt(pi) * gamma(s) / (s * gamma(-s + S(1) / 2)), (0, 1), True, ) assert inverse_mellin_transform( -4 ** s * sqrt(pi) * gamma(s) / (2 * s * gamma(-s + S(1) / 2)), s, u, (0, 1) ).expand() == Ci(sqrt(u)) # TODO LT of Si, Shi, Chi is a mess ... assert laplace_transform(Ci(x), x, s) == (-log(1 + s ** 2) / 2 / s, 0, True) assert laplace_transform(expint(a, x), x, s) == (lerchphi(s * polar_lift(-1), 1, a), 0, S(0) < re(a)) assert laplace_transform(expint(1, x), x, s) == (log(s + 1) / s, 0, True) assert laplace_transform(expint(2, x), x, s) == ((s - log(s + 1)) / s ** 2, 0, True) assert inverse_laplace_transform(-log(1 + s ** 2) / 2 / s, s, u).expand() == Heaviside(u) * Ci(u) assert inverse_laplace_transform(log(s + 1) / s, s, x).rewrite(expint) == Heaviside(x) * E1(x) assert ( inverse_laplace_transform((s - log(s + 1)) / s ** 2, s, x).rewrite(expint).expand() == (expint(2, x) * Heaviside(x)).rewrite(Ei).rewrite(expint).expand() )
def test_li(): z = Symbol("z") zr = Symbol("z", real=True) zp = Symbol("z", positive=True) zn = Symbol("z", negative=True) assert li(0) == 0 assert li(1) == -oo assert li(oo) == oo assert isinstance(li(z), li) assert diff(li(z), z) == 1/log(z) assert conjugate(li(z)) == li(conjugate(z)) assert conjugate(li(-zr)) == li(-zr) assert conjugate(li(-zp)) == conjugate(li(-zp)) assert conjugate(li(zn)) == conjugate(li(zn)) assert li(z).rewrite(Li) == Li(z) + li(2) assert li(z).rewrite(Ei) == Ei(log(z)) assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 - expint(1, -log(z))) assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))) assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))) assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))) assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))) assert li(z).rewrite(hyper) ==(log(z)*hyper((1, 1), (2, 2), log(z)) - log(1/log(z))/2 + log(log(z))/2 + EulerGamma) assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 - meijerg(((), (1,)), ((0, 0), ()), -log(z))) assert gruntz(1/li(z), z, oo) == 0
def test_expint(): from sympy import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei aneg = Symbol('a', negative=True) u = Symbol('u', polar=True) assert mellin_transform(E1(x), x, s) == (gamma(s) / s, (0, oo), True) assert inverse_mellin_transform(gamma(s) / s, s, x, (0, oo)).rewrite(expint).expand() == E1(x) assert mellin_transform(expint(a, x), x, s) == \ (gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True) # XXX IMT has hickups with complicated strips ... assert simplify(unpolarify( inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x, (1 - aneg, oo)).rewrite(expint).expand(func=True))) == \ expint(aneg, x) assert mellin_transform(Si(x), x, s) == \ (-2**s*sqrt(pi)*gamma(s/2 + S(1)/2)/( 2*s*gamma(-s/2 + 1)), (-1, 0), True) assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2) /(2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \ == Si(x) assert mellin_transform(Ci(sqrt(x)), x, s) == \ (-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + S(1)/2)), (0, 1), True) assert inverse_mellin_transform( -4**s * sqrt(pi) * gamma(s) / (2 * s * gamma(-s + S(1) / 2)), s, u, (0, 1)).expand() == Ci(sqrt(u)) # TODO LT of Si, Shi, Chi is a mess ... assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2) / 2 / s, 0, True) assert laplace_transform(expint(a, x), x, s) == \ (lerchphi(s*polar_lift(-1), 1, a), 0, S(0) < re(a)) assert laplace_transform(expint(1, x), x, s) == (log(s + 1) / s, 0, True) assert laplace_transform(expint(2, x), x, s) == \ ((s - log(s + 1))/s**2, 0, True) assert inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() == \ Heaviside(u)*Ci(u) assert inverse_laplace_transform(log(s + 1)/s, s, x).rewrite(expint) == \ Heaviside(x)*E1(x) assert inverse_laplace_transform((s - log(s + 1))/s**2, s, x).rewrite(expint).expand() == \ (expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
def test_ei(): assert Ei(0) is S.NegativeInfinity assert Ei(oo) is S.Infinity assert Ei(-oo) is S.Zero assert tn_branch(Ei) assert mytd(Ei(x), exp(x) / x, x) assert mytn(Ei(x), Ei(x).rewrite(uppergamma), -uppergamma(0, x * polar_lift(-1)) - I * pi, x) assert mytn(Ei(x), Ei(x).rewrite(expint), -expint(1, x * polar_lift(-1)) - I * pi, x) assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x) assert Ei(x * exp_polar(2 * I * pi)) == Ei(x) + 2 * I * pi assert Ei(x * exp_polar(-2 * I * pi)) == Ei(x) - 2 * I * pi assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x) assert mytn( Ei(x * polar_lift(I)), Ei(x * polar_lift(I)).rewrite(Si), Ci(x) + I * Si(x) + I * pi / 2, x, ) assert Ei(log(x)).rewrite(li) == li(x) assert Ei(2 * log(x)).rewrite(li) == li(x**2) assert gruntz(Ei(x + exp(-x)) * exp(-x) * x, x, oo) == 1 assert Ei(x).series(x) == EulerGamma + log( x) + x + x**2 / 4 + x**3 / 18 + x**4 / 96 + x**5 / 600 + O(x**6) assert Ei(x).series(x, 1, 3) == Ei(1) + E * (x - 1) + O((x - 1)**3, (x, 1)) assert str(Ei(cos(2)).evalf(n=10)) == "-0.6760647401" raises(ArgumentIndexError, lambda: Ei(x).fdiff(2))
def test_ei(): assert tn_branch(Ei) assert mytd(Ei(x), exp(x)/x, x) assert mytn(Ei(x), Ei(x).rewrite(uppergamma), -uppergamma(0, x*polar_lift(-1)) - I*pi, x) assert mytn(Ei(x), Ei(x).rewrite(expint), -expint(1, x*polar_lift(-1)) - I*pi, x) assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x) assert Ei(x*exp_polar(2*I*pi)) == Ei(x) + 2*I*pi assert Ei(x*exp_polar(-2*I*pi)) == Ei(x) - 2*I*pi assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x) assert mytn(Ei(x*polar_lift(I)), Ei(x*polar_lift(I)).rewrite(Si), Ci(x) + I*Si(x) + I*pi/2, x) assert Ei(log(x)).rewrite(li) == li(x) assert Ei(2*log(x)).rewrite(li) == li(x**2) assert gruntz(Ei(x+exp(-x))*exp(-x)*x, x, oo) == 1 assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \ x**3/18 + x**4/96 + x**5/600 + O(x**6) assert str(Ei(cos(2)).evalf(n=10)) == '-0.6760647401'
def test_issue_14692(): b = Symbol('b', negative=True) assert laplace_transform(1/(I*x - b), x, s) == \ (-I*exp(I*b*s)*expint(1, b*s*exp_polar(I*pi/2)), 0, True)
def test_latex_functions(): assert latex(exp(x)) == "e^{x}" assert latex(exp(1) + exp(2)) == "e + e^{2}" f = Function('f') assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}' beta = Function('beta') assert latex(beta(x)) == r"\beta{\left (x \right )}" assert latex(sin(x)) == r"\sin{\left (x \right )}" assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" assert latex(sin(2*x**2), fold_func_brackets=True) == \ r"\sin {2 x^{2}}" assert latex(sin(x**2), fold_func_brackets=True) == \ r"\sin {x^{2}}" assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="full") == \ r"\arcsin^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="power") == \ r"\sin^{-1}{\left (x \right )}^{2}" assert latex(asin(x**2), inv_trig_style="power", fold_func_brackets=True) == \ r"\sin^{-1} {x^{2}}" assert latex(factorial(k)) == r"k!" assert latex(factorial(-k)) == r"\left(- k\right)!" assert latex(subfactorial(k)) == r"!k" assert latex(subfactorial(-k)) == r"!\left(- k\right)" assert latex(factorial2(k)) == r"k!!" assert latex(factorial2(-k)) == r"\left(- k\right)!!" assert latex(binomial(2, k)) == r"{\binom{2}{k}}" assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}" assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}" assert latex(floor(x)) == r"\lfloor{x}\rfloor" assert latex(ceiling(x)) == r"\lceil{x}\rceil" assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)" assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}" assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)" assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}" assert latex(Abs(x)) == r"\lvert{x}\rvert" assert latex(re(x)) == r"\Re{x}" assert latex(re(x + y)) == r"\Re{x} + \Re{y}" assert latex(im(x)) == r"\Im{x}" assert latex(conjugate(x)) == r"\overline{x}" assert latex(gamma(x)) == r"\Gamma\left(x\right)" assert latex(Order(x)) == r"\mathcal{O}\left(x\right)" assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)' assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)' assert latex(cot(x)) == r'\cot{\left (x \right )}' assert latex(coth(x)) == r'\coth{\left (x \right )}' assert latex(re(x)) == r'\Re{x}' assert latex(im(x)) == r'\Im{x}' assert latex(root(x, y)) == r'x^{\frac{1}{y}}' assert latex(arg(x)) == r'\arg{\left (x \right )}' assert latex(zeta(x)) == r'\zeta\left(x\right)' assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)" assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)" assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)" assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" assert latex(polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)" assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}' assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}' assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)' assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}' assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}' assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}' assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}' assert latex(jacobi(n, a, b, x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)' assert latex(jacobi( n, a, b, x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}' assert latex(gegenbauer(n, a, x)) == r'C_{n}^{\left(a\right)}\left(x\right)' assert latex(gegenbauer( n, a, x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)' assert latex(chebyshevt(n, x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}' assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)' assert latex(chebyshevu(n, x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}' assert latex(legendre(n, x)) == r'P_{n}\left(x\right)' assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}' assert latex(assoc_legendre(n, a, x)) == r'P_{n}^{\left(a\right)}\left(x\right)' assert latex(assoc_legendre( n, a, x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)' assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}' assert latex(assoc_laguerre(n, a, x)) == r'L_{n}^{\left(a\right)}\left(x\right)' assert latex(assoc_laguerre( n, a, x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(hermite(n, x)) == r'H_{n}\left(x\right)' assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}' # Test latex printing of function names with "_" assert latex( polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}" assert latex(polar_lift(0)** 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
def _eval_rewrite_as_expint(self, s, x): from sympy import expint return expint(1 - s, x) * x ** s
def test_expint(): assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma), y**(x - 1) * uppergamma(1 - x, y), x) assert mytd(expint(x, y), -y**(x - 1) * meijerg([], [1, 1], [0, 0, 1 - x], [], y), x) assert mytd(expint(x, y), -expint(x - 1, y), y) assert mytn(expint(1, x), expint(1, x).rewrite(Ei), -Ei(x * polar_lift(-1)) + I * pi, x) assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \ + 24*exp(-x)/x**4 + 24*exp(-x)/x**5 assert expint(Rational(-3, 2), x) == \ exp(-x)/x + 3*exp(-x)/(2*x**2) + 3*sqrt(pi)*erfc(sqrt(x))/(4*x**S('5/2')) assert tn_branch(expint, 1) assert tn_branch(expint, 2) assert tn_branch(expint, 3) assert tn_branch(expint, 1.7) assert tn_branch(expint, pi) assert expint(y, x*exp_polar(2*I*pi)) == \ x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(y, x*exp_polar(-2*I*pi)) == \ x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(2, x * exp_polar(2 * I * pi)) == 2 * I * pi * x + expint(2, x) assert expint(2, x * exp_polar(-2 * I * pi)) == -2 * I * pi * x + expint(2, x) assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x) assert expint(x, y).rewrite(Ei) == expint(x, y) assert expint(x, y).rewrite(Ci) == expint(x, y) assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x) assert mytn(E1(polar_lift(I) * x), E1(polar_lift(I) * x).rewrite(Si), -Ci(x) + I * Si(x) - I * pi / 2, x) assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint), -x * E1(x) + exp(-x), x) assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint), x**2 * E1(x) / 2 + (1 - x) * exp(-x) / 2, x) assert expint(Rational(3, 2), z).nseries(z) == \ 2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \ 2*sqrt(pi)*sqrt(z) + O(z**6) assert E1(z).series(z) == -EulerGamma - log(z) + z - \ z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6) assert expint(4, z).series(z) == Rational(1, 3) - z/2 + z**2/2 + \ z**3*(log(z)/6 - Rational(11, 36) + EulerGamma/6 - I*pi/6) - z**4/24 + \ z**5/240 + O(z**6) assert expint(z, y).series(z, 0, 2) == exp(-y) / y - z * meijerg( ((), (1, 1)), ((0, 0, 1), ()), y) / y + O(z**2) raises(ArgumentIndexError, lambda: expint(x, y).fdiff(3)) neg = Symbol('neg', negative=True) assert Ei(neg).rewrite(Si) == Shi(neg) + Chi(neg) - I * pi
def test_expint(): assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma), y**(x - 1) * uppergamma(1 - x, y), x) assert mytd(expint(x, y), -y**(x - 1) * meijerg([], [1, 1], [0, 0, 1 - x], [], y), x) assert mytd(expint(x, y), -expint(x - 1, y), y) assert mytn(expint(1, x), expint(1, x).rewrite(Ei), -Ei(x * polar_lift(-1)) + I * pi, x) assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \ + 24*exp(-x)/x**4 + 24*exp(-x)/x**5 assert expint(-S(3)/2, x) == \ exp(-x)/x + 3*exp(-x)/(2*x**2) - 3*sqrt(pi)*erf(sqrt(x))/(4*x**S('5/2')) \ + 3*sqrt(pi)/(4*x**S('5/2')) assert tn_branch(expint, 1) assert tn_branch(expint, 2) assert tn_branch(expint, 3) assert tn_branch(expint, 1.7) assert tn_branch(expint, pi) assert expint(y, x*exp_polar(2*I*pi)) == \ x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(y, x*exp_polar(-2*I*pi)) == \ x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(2, x * exp_polar(2 * I * pi)) == 2 * I * pi * x + expint(2, x) assert expint(2, x * exp_polar(-2 * I * pi)) == -2 * I * pi * x + expint(2, x) assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x) assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x) assert mytn(E1(polar_lift(I) * x), E1(polar_lift(I) * x).rewrite(Si), -Ci(x) + I * Si(x) - I * pi / 2, x) assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint), -x * E1(x) + exp(-x), x) assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint), x**2 * E1(x) / 2 + (1 - x) * exp(-x) / 2, x)
def test_expint(): """ Test various exponential integrals. """ from sympy import (expint, unpolarify, Symbol, Ci, Si, Shi, Chi, sin, cos, sinh, cosh, Ei) assert simplify( unpolarify( integrate(exp(-z * x) / x**y, (x, 1, oo), meijerg=True, conds='none').rewrite(expint).expand( func=True))) == expint(y, z) assert integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True, conds='none').rewrite(expint).expand() == \ expint(1, z) assert integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True, conds='none').rewrite(expint).expand() == \ expint(2, z).rewrite(Ei).rewrite(expint) assert integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True, conds='none').rewrite(expint).expand() == \ expint(3, z).rewrite(Ei).rewrite(expint).expand() t = Symbol('t', positive=True) assert integrate(-cos(x) / x, (x, t, oo), meijerg=True).expand() == Ci(t) assert integrate(-sin(x)/x, (x, t, oo), meijerg=True).expand() == \ Si(t) - pi/2 assert integrate(sin(x) / x, (x, 0, z), meijerg=True) == Si(z) assert integrate(sinh(x) / x, (x, 0, z), meijerg=True) == Shi(z) assert integrate(exp(-x)/x, x, meijerg=True).expand().rewrite(expint) == \ I*pi - expint(1, x) assert integrate(exp(-x)/x**2, x, meijerg=True).rewrite(expint).expand() \ == expint(1, x) - exp(-x)/x - I*pi u = Symbol('u', polar=True) assert integrate(cos(u)/u, u, meijerg=True).expand().as_independent(u)[1] \ == Ci(u) assert integrate(cosh(u)/u, u, meijerg=True).expand().as_independent(u)[1] \ == Chi(u) assert integrate( expint(1, x), x, meijerg=True).rewrite(expint).expand() == x * expint(1, x) - exp(-x) assert integrate(expint(2, x), x, meijerg=True ).rewrite(expint).expand() == \ -x**2*expint(1, x)/2 + x*exp(-x)/2 - exp(-x)/2 assert simplify(unpolarify(integrate(expint(y, x), x, meijerg=True).rewrite(expint).expand(func=True))) == \ -expint(y + 1, x) assert integrate(Si(x), x, meijerg=True) == x * Si(x) + cos(x) assert integrate(Ci(u), u, meijerg=True).expand() == u * Ci(u) - sin(u) assert integrate(Shi(x), x, meijerg=True) == x * Shi(x) - cosh(x) assert integrate(Chi(u), u, meijerg=True).expand() == u * Chi(u) - sinh(u) assert integrate(Si(x) * exp(-x), (x, 0, oo), meijerg=True) == pi / 4 assert integrate(expint(1, x) * sin(x), (x, 0, oo), meijerg=True) == log(2) / 2
def test_latex_functions(): assert latex(exp(x)) == "e^{x}" assert latex(exp(1) + exp(2)) == "e + e^{2}" f = Function('f') assert latex(f(x)) == r'f{\left (x \right )}' assert latex(f) == r'f' g = Function('g') assert latex(g(x, y)) == r'g{\left (x,y \right )}' assert latex(g) == r'g' h = Function('h') assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}' assert latex(h) == r'h' Li = Function('Li') assert latex(Li) == r'\operatorname{Li}' assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}' beta = Function('beta') # not to be confused with the beta function assert latex(beta(x)) == r"\beta{\left (x \right )}" assert latex(beta) == r"\beta" assert latex(sin(x)) == r"\sin{\left (x \right )}" assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" assert latex(sin(2*x**2), fold_func_brackets=True) == \ r"\sin {2 x^{2}}" assert latex(sin(x**2), fold_func_brackets=True) == \ r"\sin {x^{2}}" assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="full") == \ r"\arcsin^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="power") == \ r"\sin^{-1}{\left (x \right )}^{2}" assert latex(asin(x**2), inv_trig_style="power", fold_func_brackets=True) == \ r"\sin^{-1} {x^{2}}" assert latex(factorial(k)) == r"k!" assert latex(factorial(-k)) == r"\left(- k\right)!" assert latex(subfactorial(k)) == r"!k" assert latex(subfactorial(-k)) == r"!\left(- k\right)" assert latex(factorial2(k)) == r"k!!" assert latex(factorial2(-k)) == r"\left(- k\right)!!" assert latex(binomial(2, k)) == r"{\binom{2}{k}}" assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}" assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}" assert latex(floor(x)) == r"\lfloor{x}\rfloor" assert latex(ceiling(x)) == r"\lceil{x}\rceil" assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)" assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}" assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)" assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}" assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert" assert latex(re(x)) == r"\Re{x}" assert latex(re(x + y)) == r"\Re{x} + \Re{y}" assert latex(im(x)) == r"\Im{x}" assert latex(conjugate(x)) == r"\overline{x}" assert latex(gamma(x)) == r"\Gamma{\left(x \right)}" w = Wild('w') assert latex(gamma(w)) == r"\Gamma{\left(w \right)}" assert latex(Order(x)) == r"\mathcal{O}\left(x\right)" assert latex(Order(x, x)) == r"\mathcal{O}\left(x\right)" assert latex(Order(x, x, 0)) == r"\mathcal{O}\left(x\right)" assert latex(Order(x, x, oo)) == r"\mathcal{O}\left(x; x\rightarrow\infty\right)" assert latex( Order(x, x, y) ) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)" assert latex( Order(x, x, y, 0) ) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)" assert latex( Order(x, x, y, oo) ) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow\infty\right)" assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)' assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)' assert latex(cot(x)) == r'\cot{\left (x \right )}' assert latex(coth(x)) == r'\coth{\left (x \right )}' assert latex(re(x)) == r'\Re{x}' assert latex(im(x)) == r'\Im{x}' assert latex(root(x, y)) == r'x^{\frac{1}{y}}' assert latex(arg(x)) == r'\arg{\left (x \right )}' assert latex(zeta(x)) == r'\zeta\left(x\right)' assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)" assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)" assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)" assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" assert latex(polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)" assert latex(elliptic_k(z)) == r"K\left(z\right)" assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)" assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)" assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)" assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)" assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)" assert latex(elliptic_e(z)) == r"E\left(z\right)" assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)" assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)" assert latex(elliptic_pi(x, y, z)**2) == \ r"\Pi^{2}\left(x; y\middle| z\right)" assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)" assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)" assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}' assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}' assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)' assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}' assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}' assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}' assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}' assert latex(Chi(x)) == r'\operatorname{Chi}{\left (x \right )}' assert latex(jacobi(n, a, b, x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)' assert latex(jacobi( n, a, b, x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}' assert latex(gegenbauer(n, a, x)) == r'C_{n}^{\left(a\right)}\left(x\right)' assert latex(gegenbauer( n, a, x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)' assert latex(chebyshevt(n, x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}' assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)' assert latex(chebyshevu(n, x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}' assert latex(legendre(n, x)) == r'P_{n}\left(x\right)' assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}' assert latex(assoc_legendre(n, a, x)) == r'P_{n}^{\left(a\right)}\left(x\right)' assert latex(assoc_legendre( n, a, x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)' assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}' assert latex(assoc_laguerre(n, a, x)) == r'L_{n}^{\left(a\right)}\left(x\right)' assert latex(assoc_laguerre( n, a, x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(hermite(n, x)) == r'H_{n}\left(x\right)' assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}' theta = Symbol("theta", real=True) phi = Symbol("phi", real=True) assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)' assert latex( Ynm(n, m, theta, phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}' assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)' assert latex( Znm(n, m, theta, phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}' # Test latex printing of function names with "_" assert latex( polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}" assert latex(polar_lift(0)** 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}" assert latex(totient(n)) == r'\phi\left( n \right)' # some unknown function name should get rendered with \operatorname fjlkd = Function('fjlkd') assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}' # even when it is referred to without an argument assert latex(fjlkd) == r'\operatorname{fjlkd}'
def test_lowergamma(): from sympy import meijerg, expint assert lowergamma(x, 0) == 0 assert lowergamma(x, y).diff(y) == y**(x - 1) * exp(-y) assert td(lowergamma(randcplx(), y), y) assert td(lowergamma(x, randcplx()), x) assert lowergamma(x, y).diff(x) == \ gamma(x)*digamma(x) - uppergamma(x, y)*log(y) \ - meijerg([], [1, 1], [0, 0, x], [], y) assert lowergamma(S.Half, x) == sqrt(pi) * erf(sqrt(x)) assert not lowergamma(S.Half - 3, x).has(lowergamma) assert not lowergamma(S.Half + 3, x).has(lowergamma) assert lowergamma(S.Half, x, evaluate=False).has(lowergamma) assert tn(lowergamma(S.Half + 3, x, evaluate=False), lowergamma(S.Half + 3, x), x) assert tn(lowergamma(S.Half - 3, x, evaluate=False), lowergamma(S.Half - 3, x), x) assert tn_branch(-3, lowergamma) assert tn_branch(-4, lowergamma) assert tn_branch(Rational(1, 3), lowergamma) assert tn_branch(pi, lowergamma) assert lowergamma(3, exp_polar(4 * pi * I) * x) == lowergamma(3, x) assert lowergamma(y, exp_polar(5*pi*I)*x) == \ exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I)) assert lowergamma(-2, exp_polar(5*pi*I)*x) == \ lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y)) assert conjugate(lowergamma(x, 0)) == 0 assert unchanged(conjugate, lowergamma(x, -oo)) assert lowergamma(0, x)._eval_is_meromorphic(x, 0) == False assert lowergamma(S(1) / 3, x)._eval_is_meromorphic(x, 0) == False assert lowergamma(1, x, evaluate=False)._eval_is_meromorphic(x, 0) == True assert lowergamma(x, x)._eval_is_meromorphic(x, 0) == False assert lowergamma(x + 1, x)._eval_is_meromorphic(x, 0) == False assert lowergamma(1 / x, x)._eval_is_meromorphic(x, 0) == False assert lowergamma(0, x + 1)._eval_is_meromorphic(x, 0) == False assert lowergamma(S(1) / 3, x + 1)._eval_is_meromorphic(x, 0) == True assert lowergamma(1, x + 1, evaluate=False)._eval_is_meromorphic(x, 0) == True assert lowergamma(x, x + 1)._eval_is_meromorphic(x, 0) == True assert lowergamma(x + 1, x + 1)._eval_is_meromorphic(x, 0) == True assert lowergamma(1 / x, x + 1)._eval_is_meromorphic(x, 0) == False assert lowergamma(0, 1 / x)._eval_is_meromorphic(x, 0) == False assert lowergamma(S(1) / 3, 1 / x)._eval_is_meromorphic(x, 0) == False assert lowergamma(1, 1 / x, evaluate=False)._eval_is_meromorphic(x, 0) == False assert lowergamma(x, 1 / x)._eval_is_meromorphic(x, 0) == False assert lowergamma(x + 1, 1 / x)._eval_is_meromorphic(x, 0) == False assert lowergamma(1 / x, 1 / x)._eval_is_meromorphic(x, 0) == False assert lowergamma(x, 2).series(x, oo, 3) == \ 2**x*(1 + 2/(x + 1))*exp(-2)/x + O(exp(x*log(2))/x**3, (x, oo)) assert lowergamma( x, y).rewrite(expint) == -y**x * expint(-x + 1, y) + gamma(x) k = Symbol('k', integer=True) assert lowergamma( k, y).rewrite(expint) == -y**k * expint(-k + 1, y) + gamma(k) k = Symbol('k', integer=True, positive=False) assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y) assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y) assert lowergamma(70, 6) == factorial( 69 ) - 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320 * exp( -6) assert (lowergamma(S(77) / 2, 6) - lowergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16 assert (lowergamma(-S(77) / 2, 6) - lowergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
def test_latex_functions(): assert latex(exp(x)) == "e^{x}" assert latex(exp(1) + exp(2)) == "e + e^{2}" f = Function("f") assert latex(f(x)) == "\\operatorname{f}{\\left (x \\right )}" beta = Function("beta") assert latex(beta(x)) == r"\beta{\left (x \right )}" assert latex(sin(x)) == r"\sin{\left (x \right )}" assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" assert latex(sin(2 * x ** 2), fold_func_brackets=True) == r"\sin {2 x^{2}}" assert latex(sin(x ** 2), fold_func_brackets=True) == r"\sin {x^{2}}" assert latex(asin(x) ** 2) == r"\operatorname{asin}^{2}{\left (x \right )}" assert latex(asin(x) ** 2, inv_trig_style="full") == r"\arcsin^{2}{\left (x \right )}" assert latex(asin(x) ** 2, inv_trig_style="power") == r"\sin^{-1}{\left (x \right )}^{2}" assert latex(asin(x ** 2), inv_trig_style="power", fold_func_brackets=True) == r"\sin^{-1} {x^{2}}" assert latex(factorial(k)) == r"k!" assert latex(factorial(-k)) == r"\left(- k\right)!" assert latex(subfactorial(k)) == r"!k" assert latex(subfactorial(-k)) == r"!\left(- k\right)" assert latex(factorial2(k)) == r"k!!" assert latex(factorial2(-k)) == r"\left(- k\right)!!" assert latex(binomial(2, k)) == r"{\binom{2}{k}}" assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}" assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}" assert latex(floor(x)) == r"\lfloor{x}\rfloor" assert latex(ceiling(x)) == r"\lceil{x}\rceil" assert latex(Min(x, 2, x ** 3)) == r"\min\left(2, x, x^{3}\right)" assert latex(Min(x, y) ** 2) == r"\min\left(x, y\right)^{2}" assert latex(Max(x, 2, x ** 3)) == r"\max\left(2, x, x^{3}\right)" assert latex(Max(x, y) ** 2) == r"\max\left(x, y\right)^{2}" assert latex(Abs(x)) == r"\lvert{x}\rvert" assert latex(re(x)) == r"\Re{x}" assert latex(re(x + y)) == r"\Re{x} + \Re{y}" assert latex(im(x)) == r"\Im{x}" assert latex(conjugate(x)) == r"\overline{x}" assert latex(gamma(x)) == r"\Gamma\left(x\right)" assert latex(Order(x)) == r"\mathcal{O}\left(x\right)" assert latex(lowergamma(x, y)) == r"\gamma\left(x, y\right)" assert latex(uppergamma(x, y)) == r"\Gamma\left(x, y\right)" assert latex(cot(x)) == r"\cot{\left (x \right )}" assert latex(coth(x)) == r"\coth{\left (x \right )}" assert latex(re(x)) == r"\Re{x}" assert latex(im(x)) == r"\Im{x}" assert latex(root(x, y)) == r"x^{\frac{1}{y}}" assert latex(arg(x)) == r"\arg{\left (x \right )}" assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x) ** 2) == r"\zeta^{2}\left(x\right)" assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" assert latex(zeta(x, y) ** 2) == r"\zeta^{2}\left(x, y\right)" assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" assert latex(dirichlet_eta(x) ** 2) == r"\eta^{2}\left(x\right)" assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" assert latex(polylog(x, y) ** 2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" assert latex(lerchphi(x, y, n) ** 2) == r"\Phi^{2}\left(x, y, n\right)" assert latex(Ei(x)) == r"\operatorname{Ei}{\left (x \right )}" assert latex(Ei(x) ** 2) == r"\operatorname{Ei}^{2}{\left (x \right )}" assert latex(expint(x, y) ** 2) == r"\operatorname{E}_{x}^{2}\left(y\right)" assert latex(Shi(x) ** 2) == r"\operatorname{Shi}^{2}{\left (x \right )}" assert latex(Si(x) ** 2) == r"\operatorname{Si}^{2}{\left (x \right )}" assert latex(Ci(x) ** 2) == r"\operatorname{Ci}^{2}{\left (x \right )}" assert latex(Chi(x) ** 2) == r"\operatorname{Chi}^{2}{\left (x \right )}" assert latex(jacobi(n, a, b, x)) == r"P_{n}^{\left(a,b\right)}\left(x\right)" assert latex(jacobi(n, a, b, x) ** 2) == r"\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}" assert latex(gegenbauer(n, a, x)) == r"C_{n}^{\left(a\right)}\left(x\right)" assert latex(gegenbauer(n, a, x) ** 2) == r"\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}" assert latex(chebyshevt(n, x)) == r"T_{n}\left(x\right)" assert latex(chebyshevt(n, x) ** 2) == r"\left(T_{n}\left(x\right)\right)^{2}" assert latex(chebyshevu(n, x)) == r"U_{n}\left(x\right)" assert latex(chebyshevu(n, x) ** 2) == r"\left(U_{n}\left(x\right)\right)^{2}" assert latex(legendre(n, x)) == r"P_{n}\left(x\right)" assert latex(legendre(n, x) ** 2) == r"\left(P_{n}\left(x\right)\right)^{2}" assert latex(assoc_legendre(n, a, x)) == r"P_{n}^{\left(a\right)}\left(x\right)" assert latex(assoc_legendre(n, a, x) ** 2) == r"\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}" assert latex(laguerre(n, x)) == r"L_{n}\left(x\right)" assert latex(laguerre(n, x) ** 2) == r"\left(L_{n}\left(x\right)\right)^{2}" assert latex(assoc_laguerre(n, a, x)) == r"L_{n}^{\left(a\right)}\left(x\right)" assert latex(assoc_laguerre(n, a, x) ** 2) == r"\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}" assert latex(hermite(n, x)) == r"H_{n}\left(x\right)" assert latex(hermite(n, x) ** 2) == r"\left(H_{n}\left(x\right)\right)^{2}" # Test latex printing of function names with "_" assert latex(polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}" assert latex(polar_lift(0) ** 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
def test_latex_functions(): assert latex(exp(x)) == "e^{x}" assert latex(exp(1) + exp(2)) == "e + e^{2}" f = Function('f') assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}' beta = Function('beta') assert latex(beta(x)) == r"\beta{\left (x \right )}" assert latex(sin(x)) == r"\sin{\left (x \right )}" assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" assert latex(sin(2*x**2), fold_func_brackets=True) == \ r"\sin {2 x^{2}}" assert latex(sin(x**2), fold_func_brackets=True) == \ r"\sin {x^{2}}" assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="full") == \ r"\arcsin^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="power") == \ r"\sin^{-1}{\left (x \right )}^{2}" assert latex(asin(x**2), inv_trig_style="power", fold_func_brackets=True) == \ r"\sin^{-1} {x^{2}}" assert latex(factorial(k)) == r"k!" assert latex(factorial(-k)) == r"\left(- k\right)!" assert latex(factorial2(k)) == r"k!!" assert latex(factorial2(-k)) == r"\left(- k\right)!!" assert latex(binomial(2, k)) == r"{\binom{2}{k}}" assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}" assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}" assert latex(floor(x)) == r"\lfloor{x}\rfloor" assert latex(ceiling(x)) == r"\lceil{x}\rceil" assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)" assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)" assert latex(Abs(x)) == r"\lvert{x}\rvert" assert latex(re(x)) == r"\Re{x}" assert latex(re(x + y)) == r"\Re {\left (x + y \right )}" assert latex(im(x)) == r"\Im{x}" assert latex(conjugate(x)) == r"\overline{x}" assert latex(gamma(x)) == r"\Gamma\left(x\right)" assert latex(Order(x)) == r"\mathcal{O}\left(x\right)" assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)' assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)' assert latex(cot(x)) == r'\cot{\left (x \right )}' assert latex(coth(x)) == r'\coth{\left (x \right )}' assert latex(re(x)) == r'\Re{x}' assert latex(im(x)) == r'\Im{x}' assert latex(root(x, y)) == r'x^{\frac{1}{y}}' assert latex(arg(x)) == r'\arg{\left (x \right )}' assert latex(zeta(x)) == r'\zeta\left(x\right)' assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)" assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)" assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)" assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" assert latex(polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)" assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}' assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}' assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)' assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}' assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}' assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}' assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
def test_erf(): assert erf(nan) == nan assert erf(oo) == 1 assert erf(-oo) == -1 assert erf(0) == 0 assert erf(I * oo) == oo * I assert erf(-I * oo) == -oo * I assert erf(-2) == -erf(2) assert erf(-x * y) == -erf(x * y) assert erf(-x - y) == -erf(x + y) assert erf(erfinv(x)) == x assert erf(erfcinv(x)) == 1 - x assert erf(erf2inv(0, x)) == x assert erf(erf2inv(0, x, evaluate=False)) == x # To cover code in erf assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x assert erf(I).is_real is False assert erf(0).is_real is True assert conjugate(erf(z)) == erf(conjugate(z)) assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi) assert erf(1 / x).as_leading_term(x) == erf(1 / x) assert erf(z).rewrite('uppergamma') == sqrt(z** 2) * (1 - erfc(sqrt(z**2))) / z assert erf(z).rewrite('erfc') == S.One - erfc(z) assert erf(z).rewrite('erfi') == -I * erfi(I * z) assert erf(z).rewrite('fresnels') == (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erf(z).rewrite('fresnelc') == (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erf(z).rewrite('hyper') == 2 * z * hyper([S.Half], [3 * S.Half], -z**2) / sqrt(pi) assert erf(z).rewrite('meijerg') == z * meijerg([S.Half], [], [0], [-S.Half], z**2) / sqrt(pi) assert erf(z).rewrite( 'expint') == sqrt(z**2) / z - z * expint(S.Half, z**2) / sqrt(S.Pi) assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \ 2/sqrt(pi) assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi) assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1 assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x, oo) == -1 assert erf(x).as_real_imag() == \ (erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2, -I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2) assert erf(x).as_real_imag(deep=False) == \ (erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2, -I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2) assert erf(w).as_real_imag() == (erf(w), 0) assert erf(w).as_real_imag(deep=False) == (erf(w), 0) # issue 13575 assert erf(I).as_real_imag() == (0, -I * erf(I)) raises(ArgumentIndexError, lambda: erf(x).fdiff(2)) assert erf(x).inverse() == erfinv
def test_latex_functions(): assert latex(exp(x)) == "e^{x}" assert latex(exp(1) + exp(2)) == "e + e^{2}" f = Function('f') assert latex(f(x)) == r'f{\left (x \right )}' assert latex(f) == r'f' g = Function('g') assert latex(g(x, y)) == r'g{\left (x,y \right )}' assert latex(g) == r'g' h = Function('h') assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}' assert latex(h) == r'h' Li = Function('Li') assert latex(Li) == r'\operatorname{Li}' assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}' beta = Function('beta') # not to be confused with the beta function assert latex(beta(x)) == r"\beta{\left (x \right )}" assert latex(beta) == r"\beta" assert latex(sin(x)) == r"\sin{\left (x \right )}" assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" assert latex(sin(2*x**2), fold_func_brackets=True) == \ r"\sin {2 x^{2}}" assert latex(sin(x**2), fold_func_brackets=True) == \ r"\sin {x^{2}}" assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="full") == \ r"\arcsin^{2}{\left (x \right )}" assert latex(asin(x)**2, inv_trig_style="power") == \ r"\sin^{-1}{\left (x \right )}^{2}" assert latex(asin(x**2), inv_trig_style="power", fold_func_brackets=True) == \ r"\sin^{-1} {x^{2}}" assert latex(factorial(k)) == r"k!" assert latex(factorial(-k)) == r"\left(- k\right)!" assert latex(subfactorial(k)) == r"!k" assert latex(subfactorial(-k)) == r"!\left(- k\right)" assert latex(factorial2(k)) == r"k!!" assert latex(factorial2(-k)) == r"\left(- k\right)!!" assert latex(binomial(2, k)) == r"{\binom{2}{k}}" assert latex( FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}" assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}" assert latex(floor(x)) == r"\lfloor{x}\rfloor" assert latex(ceiling(x)) == r"\lceil{x}\rceil" assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)" assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}" assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)" assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}" assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert" assert latex(re(x)) == r"\Re{x}" assert latex(re(x + y)) == r"\Re{x} + \Re{y}" assert latex(im(x)) == r"\Im{x}" assert latex(conjugate(x)) == r"\overline{x}" assert latex(gamma(x)) == r"\Gamma\left(x\right)" assert latex(Order(x)) == r"\mathcal{O}\left(x\right)" assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)' assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)' assert latex(cot(x)) == r'\cot{\left (x \right )}' assert latex(coth(x)) == r'\coth{\left (x \right )}' assert latex(re(x)) == r'\Re{x}' assert latex(im(x)) == r'\Im{x}' assert latex(root(x, y)) == r'x^{\frac{1}{y}}' assert latex(arg(x)) == r'\arg{\left (x \right )}' assert latex(zeta(x)) == r'\zeta\left(x\right)' assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)" assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)" assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)" assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" assert latex( polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)" assert latex(elliptic_k(z)) == r"K\left(z\right)" assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)" assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)" assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)" assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)" assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)" assert latex(elliptic_e(z)) == r"E\left(z\right)" assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)" assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)" assert latex(elliptic_pi(x, y, z)**2) == \ r"\Pi^{2}\left(x; y\middle| z\right)" assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)" assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)" assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}' assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}' assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)' assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}' assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}' assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}' assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}', latex(Chi(x)**2) assert latex( jacobi(n, a, b, x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)' assert latex(jacobi(n, a, b, x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}' assert latex( gegenbauer(n, a, x)) == r'C_{n}^{\left(a\right)}\left(x\right)' assert latex(gegenbauer(n, a, x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)' assert latex( chebyshevt(n, x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}' assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)' assert latex( chebyshevu(n, x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}' assert latex(legendre(n, x)) == r'P_{n}\left(x\right)' assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}' assert latex( assoc_legendre(n, a, x)) == r'P_{n}^{\left(a\right)}\left(x\right)' assert latex(assoc_legendre(n, a, x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)' assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}' assert latex( assoc_laguerre(n, a, x)) == r'L_{n}^{\left(a\right)}\left(x\right)' assert latex(assoc_laguerre(n, a, x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}' assert latex(hermite(n, x)) == r'H_{n}\left(x\right)' assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}' theta = Symbol("theta", real=True) phi = Symbol("phi", real=True) assert latex(Ynm(n,m,theta,phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)' assert latex(Ynm(n, m, theta, phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}' assert latex(Znm(n,m,theta,phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)' assert latex(Znm(n, m, theta, phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}' # Test latex printing of function names with "_" assert latex( polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}" assert latex(polar_lift( 0)**3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}" assert latex(totient(n)) == r'\phi\left( n \right)' # some unknown function name should get rendered with \operatorname fjlkd = Function('fjlkd') assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}' # even when it is referred to without an argument assert latex(fjlkd) == r'\operatorname{fjlkd}'
def test_expint(): assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma), y**(x - 1)*uppergamma(1 - x, y), x) assert mytd( expint(x, y), -y**(x - 1)*meijerg([], [1, 1], [0, 0, 1 - x], [], y), x) assert mytd(expint(x, y), -expint(x - 1, y), y) assert mytn(expint(1, x), expint(1, x).rewrite(Ei), -Ei(x*polar_lift(-1)) + I*pi, x) assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \ + 24*exp(-x)/x**4 + 24*exp(-x)/x**5 assert expint(-S(3)/2, x) == \ exp(-x)/x + 3*exp(-x)/(2*x**2) + 3*sqrt(pi)*erfc(sqrt(x))/(4*x**S('5/2')) assert tn_branch(expint, 1) assert tn_branch(expint, 2) assert tn_branch(expint, 3) assert tn_branch(expint, 1.7) assert tn_branch(expint, pi) assert expint(y, x*exp_polar(2*I*pi)) == \ x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(y, x*exp_polar(-2*I*pi)) == \ x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(2, x*exp_polar(2*I*pi)) == 2*I*pi*x + expint(2, x) assert expint(2, x*exp_polar(-2*I*pi)) == -2*I*pi*x + expint(2, x) assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x) assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x) assert mytn(E1(polar_lift(I)*x), E1(polar_lift(I)*x).rewrite(Si), -Ci(x) + I*Si(x) - I*pi/2, x) assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint), -x*E1(x) + exp(-x), x) assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint), x**2*E1(x)/2 + (1 - x)*exp(-x)/2, x) assert expint(S(3)/2, z).nseries(z) == \ 2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \ 2*sqrt(pi)*sqrt(z) + O(z**6) assert E1(z).series(z) == -EulerGamma - log(z) + z - \ z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6) assert expint(4, z).series(z) == S(1)/3 - z/2 + z**2/2 + \ z**3*(log(z)/6 - S(11)/36 + EulerGamma/6) - z**4/24 + \ z**5/240 + O(z**6)
def test_si(): assert Si(I * x) == I * Shi(x) assert Shi(I * x) == I * Si(x) assert Si(-I * x) == -I * Shi(x) assert Shi(-I * x) == -I * Si(x) assert Si(-x) == -Si(x) assert Shi(-x) == -Shi(x) assert Si(exp_polar(2 * pi * I) * x) == Si(x) assert Si(exp_polar(-2 * pi * I) * x) == Si(x) assert Shi(exp_polar(2 * pi * I) * x) == Shi(x) assert Shi(exp_polar(-2 * pi * I) * x) == Shi(x) assert Si(oo) == pi / 2 assert Si(-oo) == -pi / 2 assert Shi(oo) is oo assert Shi(-oo) is -oo assert mytd(Si(x), sin(x) / x, x) assert mytd(Shi(x), sinh(x) / x, x) assert mytn( Si(x), Si(x).rewrite(Ei), -I * (-Ei(x * exp_polar(-I * pi / 2)) / 2 + Ei(x * exp_polar(I * pi / 2)) / 2 - I * pi) + pi / 2, x, ) assert mytn( Si(x), Si(x).rewrite(expint), -I * (-expint(1, x * exp_polar(-I * pi / 2)) / 2 + expint(1, x * exp_polar(I * pi / 2)) / 2) + pi / 2, x, ) assert mytn( Shi(x), Shi(x).rewrite(Ei), Ei(x) / 2 - Ei(x * exp_polar(I * pi)) / 2 + I * pi / 2, x, ) assert mytn( Shi(x), Shi(x).rewrite(expint), expint(1, x) / 2 - expint(1, x * exp_polar(I * pi)) / 2 - I * pi / 2, x, ) assert tn_arg(Si) assert tn_arg(Shi) assert Si(x).nseries( x, n=8) == x - x**3 / 18 + x**5 / 600 - x**7 / 35280 + O(x**9) assert Shi(x).nseries( x, n=8) == x + x**3 / 18 + x**5 / 600 + x**7 / 35280 + O(x**9) assert Si(sin(x)).nseries( x, n=5) == x - 2 * x**3 / 9 + 17 * x**5 / 450 + O(x**6) assert Si(x).nseries( x, 1, n=3) == Si(1) + (x - 1) * sin(1) + (x - 1)**2 * (-sin(1) / 2 + cos(1) / 2) + O( (x - 1)**3, (x, 1)) t = Symbol("t", Dummy=True) assert Si(x).rewrite(sinc) == Integral(sinc(t), (t, 0, x))
def test_periodicity(): x = Symbol('x') y = Symbol('y') z = Symbol('z', real=True) assert periodicity(sin(2 * x), x) == pi assert periodicity((-2) * tan(4 * x), x) == pi / 4 assert periodicity(sin(x)**2, x) == 2 * pi assert periodicity(3**tan(3 * x), x) == pi / 3 assert periodicity(tan(x) * cos(x), x) == 2 * pi assert periodicity(sin(x)**(tan(x)), x) == 2 * pi assert periodicity(tan(x) * sec(x), x) == 2 * pi assert periodicity(sin(2 * x) * cos(2 * x) - y, x) == pi / 2 assert periodicity(tan(x) + cot(x), x) == pi assert periodicity(sin(x) - cos(2 * x), x) == 2 * pi assert periodicity(sin(x) - 1, x) == 2 * pi assert periodicity(sin(4 * x) + sin(x) * cos(x), x) == pi assert periodicity(exp(sin(x)), x) == 2 * pi assert periodicity(log(cot(2 * x)) - sin(cos(2 * x)), x) == pi assert periodicity(sin(2 * x) * exp(tan(x) - csc(2 * x)), x) == pi assert periodicity(cos(sec(x) - csc(2 * x)), x) == 2 * pi assert periodicity(tan(sin(2 * x)), x) == pi assert periodicity(2 * tan(x)**2, x) == pi assert periodicity(sin(x % 4), x) == 4 assert periodicity(sin(x) % 4, x) == 2 * pi assert periodicity(tan((3 * x - 2) % 4), x) == Rational(4, 3) assert periodicity((sqrt(2) * (x + 1) + x) % 3, x) == 3 / (sqrt(2) + 1) assert periodicity((x**2 + 1) % x, x) is None assert periodicity(sin(re(x)), x) == 2 * pi assert periodicity(sin(x)**2 + cos(x)**2, x) is S.Zero assert periodicity(tan(x), y) is S.Zero assert periodicity(sin(x) + I * cos(x), x) == 2 * pi assert periodicity(x - sin(2 * y), y) == pi assert periodicity(exp(x), x) is None assert periodicity(exp(I * x), x) == 2 * pi assert periodicity(exp(I * z), z) == 2 * pi assert periodicity(exp(z), z) is None assert periodicity(exp(log(sin(z) + I * cos(2 * z)), evaluate=False), z) == 2 * pi assert periodicity(exp(log(sin(2 * z) + I * cos(z)), evaluate=False), z) == 2 * pi assert periodicity(exp(sin(z)), z) == 2 * pi assert periodicity(exp(2 * I * z), z) == pi assert periodicity(exp(z + I * sin(z)), z) is None assert periodicity(exp(cos(z / 2) + sin(z)), z) == 4 * pi assert periodicity(log(x), x) is None assert periodicity(exp(x)**sin(x), x) is None assert periodicity(sin(x)**y, y) is None assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi assert all( periodicity(Abs(f(x)), x) == pi for f in (cos, sin, sec, csc, tan, cot)) assert periodicity(Abs(sin(tan(x))), x) == pi assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2 * pi assert periodicity(sin(x) > S.Half, x) == 2 * pi assert periodicity(x > 2, x) is None assert periodicity(x**3 - x**2 + 1, x) is None assert periodicity(Abs(x), x) is None assert periodicity(Abs(x**2 - 1), x) is None assert periodicity((x**2 + 4) % 2, x) is None assert periodicity((E**x) % 3, x) is None assert periodicity(sin(expint(1, x)) / expint(1, x), x) is None # returning `None` for any Piecewise p = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True)) assert periodicity(p, x) is None m = MatrixSymbol('m', 3, 3) raises(NotImplementedError, lambda: periodicity(sin(m), m)) raises(NotImplementedError, lambda: periodicity(sin(m[0, 0]), m)) raises(NotImplementedError, lambda: periodicity(sin(m), m[0, 0])) raises(NotImplementedError, lambda: periodicity(sin(m[0, 0]), m[0, 0]))
def test_moment_generating_function(): t = symbols('t', positive=True) # Symbolic tests a, b, c = symbols('a b c') mgf = moment_generating_function(Beta('x', a, b))(t) assert mgf == hyper((a, ), (a + b, ), t) mgf = moment_generating_function(Chi('x', a))(t) assert mgf == sqrt(2)*t*gamma(a/2 + S.Half)*\ hyper((a/2 + S.Half,), (Rational(3, 2),), t**2/2)/gamma(a/2) +\ hyper((a/2,), (S.Half,), t**2/2) mgf = moment_generating_function(ChiSquared('x', a))(t) assert mgf == (1 - 2 * t)**(-a / 2) mgf = moment_generating_function(Erlang('x', a, b))(t) assert mgf == (1 - t / b)**(-a) mgf = moment_generating_function(ExGaussian("x", a, b, c))(t) assert mgf == exp(a * t + b**2 * t**2 / 2) / (1 - t / c) mgf = moment_generating_function(Exponential('x', a))(t) assert mgf == a / (a - t) mgf = moment_generating_function(Gamma('x', a, b))(t) assert mgf == (-b * t + 1)**(-a) mgf = moment_generating_function(Gumbel('x', a, b))(t) assert mgf == exp(b * t) * gamma(-a * t + 1) mgf = moment_generating_function(Gompertz('x', a, b))(t) assert mgf == b * exp(b) * expint(t / a, b) mgf = moment_generating_function(Laplace('x', a, b))(t) assert mgf == exp(a * t) / (-b**2 * t**2 + 1) mgf = moment_generating_function(Logistic('x', a, b))(t) assert mgf == exp(a * t) * beta(-b * t + 1, b * t + 1) mgf = moment_generating_function(Normal('x', a, b))(t) assert mgf == exp(a * t + b**2 * t**2 / 2) mgf = moment_generating_function(Pareto('x', a, b))(t) assert mgf == b * (-a * t)**b * uppergamma(-b, -a * t) mgf = moment_generating_function(QuadraticU('x', a, b))(t) assert str(mgf) == ( "(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - " "3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)") mgf = moment_generating_function(RaisedCosine('x', a, b))(t) assert mgf == pi**2 * exp(a * t) * sinh(b * t) / (b * t * (b**2 * t**2 + pi**2)) mgf = moment_generating_function(Rayleigh('x', a))(t) assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\ *exp(a**2*t**2/2)/2 + 1 mgf = moment_generating_function(Triangular('x', a, b, c))(t) assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + " "2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))") mgf = moment_generating_function(Uniform('x', a, b))(t) assert mgf == (-exp(a * t) + exp(b * t)) / (t * (-a + b)) mgf = moment_generating_function(UniformSum('x', a))(t) assert mgf == ((exp(t) - 1) / t)**a mgf = moment_generating_function(WignerSemicircle('x', a))(t) assert mgf == 2 * besseli(1, a * t) / (a * t) # Numeric tests mgf = moment_generating_function(Beta('x', 1, 1))(t) assert mgf.diff(t).subs(t, 1) == hyper((2, ), (3, ), 1) / 2 mgf = moment_generating_function(Chi('x', 1))(t) assert mgf.diff(t).subs(t, 1) == sqrt(2) * hyper( (1, ), (Rational(3, 2), ), S.Half) / sqrt(pi) + hyper( (Rational(3, 2), ), (Rational(3, 2), ), S.Half) + 2 * sqrt(2) * hyper( (2, ), (Rational(5, 2), ), S.Half) / (3 * sqrt(pi)) mgf = moment_generating_function(ChiSquared('x', 1))(t) assert mgf.diff(t).subs(t, 1) == I mgf = moment_generating_function(Erlang('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(ExGaussian("x", 0, 1, 1))(t) assert mgf.diff(t).subs(t, 2) == -exp(2) mgf = moment_generating_function(Exponential('x', 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(Gamma('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(Gumbel('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == EulerGamma + 1 mgf = moment_generating_function(Gompertz('x', 1, 1))(t) assert mgf.diff(t).subs(t, 1) == -e * meijerg(((), (1, 1)), ((0, 0, 0), ()), 1) mgf = moment_generating_function(Laplace('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(Logistic('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == beta(1, 1) mgf = moment_generating_function(Normal('x', 0, 1))(t) assert mgf.diff(t).subs(t, 1) == exp(S.Half) mgf = moment_generating_function(Pareto('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == expint(1, 0) mgf = moment_generating_function(QuadraticU('x', 1, 2))(t) assert mgf.diff(t).subs(t, 1) == -12 * e - 3 * exp(2) mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t) assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\ (1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2) mgf = moment_generating_function(Rayleigh('x', 1))(t) assert mgf.diff(t).subs(t, 0) == sqrt(2) * sqrt(pi) / 2 mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t) assert mgf.diff(t).subs(t, 1) == -e + exp(3) mgf = moment_generating_function(Uniform('x', 0, 1))(t) assert mgf.diff(t).subs(t, 1) == 1 mgf = moment_generating_function(UniformSum('x', 1))(t) assert mgf.diff(t).subs(t, 1) == 1 mgf = moment_generating_function(WignerSemicircle('x', 1))(t) assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\ besseli(0, 1)
def test_periodicity(): x = Symbol('x') y = Symbol('y') z = Symbol('z', real=True) assert periodicity(sin(2*x), x) == pi assert periodicity((-2)*tan(4*x), x) == pi/4 assert periodicity(sin(x)**2, x) == 2*pi assert periodicity(3**tan(3*x), x) == pi/3 assert periodicity(tan(x)*cos(x), x) == 2*pi assert periodicity(sin(x)**(tan(x)), x) == 2*pi assert periodicity(tan(x)*sec(x), x) == 2*pi assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2 assert periodicity(tan(x) + cot(x), x) == pi assert periodicity(sin(x) - cos(2*x), x) == 2*pi assert periodicity(sin(x) - 1, x) == 2*pi assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi assert periodicity(exp(sin(x)), x) == 2*pi assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi assert periodicity(tan(sin(2*x)), x) == pi assert periodicity(2*tan(x)**2, x) == pi assert periodicity(sin(x%4), x) == 4 assert periodicity(sin(x)%4, x) == 2*pi assert periodicity(tan((3*x-2)%4), x) == Rational(4, 3) assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1) assert periodicity((x**2+1) % x, x) is None assert periodicity(sin(re(x)), x) == 2*pi assert periodicity(sin(x)**2 + cos(x)**2, x) is S.Zero assert periodicity(tan(x), y) is S.Zero assert periodicity(sin(x) + I*cos(x), x) == 2*pi assert periodicity(x - sin(2*y), y) == pi assert periodicity(exp(x), x) is None assert periodicity(exp(I*x), x) == 2*pi assert periodicity(exp(I*z), z) == 2*pi assert periodicity(exp(z), z) is None assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi assert periodicity(exp(sin(z)), z) == 2*pi assert periodicity(exp(2*I*z), z) == pi assert periodicity(exp(z + I*sin(z)), z) is None assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi assert periodicity(log(x), x) is None assert periodicity(exp(x)**sin(x), x) is None assert periodicity(sin(x)**y, y) is None assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi assert all(periodicity(Abs(f(x)), x) == pi for f in ( cos, sin, sec, csc, tan, cot)) assert periodicity(Abs(sin(tan(x))), x) == pi assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi assert periodicity(sin(x) > S.Half, x) == 2*pi assert periodicity(x > 2, x) is None assert periodicity(x**3 - x**2 + 1, x) is None assert periodicity(Abs(x), x) is None assert periodicity(Abs(x**2 - 1), x) is None assert periodicity((x**2 + 4)%2, x) is None assert periodicity((E**x)%3, x) is None assert periodicity(sin(expint(1, x))/expint(1, x), x) is None
def test_expint(): assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma), y**(x-1)*uppergamma(1 - x, y), x) assert mytd(expint(x, y), -y**(x - 1)*meijerg([], [1, 1], [0, 0, 1-x], [], y), x) assert mytd(expint(x, y), -expint(x - 1, y), y) assert mytn(expint(1, x), expint(1, x).rewrite(Ei), -Ei(x*polar_lift(-1)) + I*pi, x) assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \ + 24*exp(-x)/x**4 + 24*exp(-x)/x**5 assert expint(-S(3)/2, x) == \ exp(-x)/x + 3*exp(-x)/(2*x**2) - 3*sqrt(pi)*erf(sqrt(x))/(4*x**S('5/2')) \ + 3*sqrt(pi)/(4*x**S('5/2')) assert tn_branch(expint, 1) assert tn_branch(expint, 2) assert tn_branch(expint, 3) assert tn_branch(expint, 1.7) assert tn_branch(expint, pi) assert expint(y, x*exp_polar(2*I*pi)) \ == x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(y, x*exp_polar(-2*I*pi)) \ == x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x) assert expint(2, x*exp_polar(2*I*pi)) == 2*I*pi*x + expint(2, x) assert expint(2, x*exp_polar(-2*I*pi)) == -2*I*pi*x + expint(2, x) assert expint(1,x).rewrite(Ei).rewrite(expint) == expint(1, x) assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x) assert mytn(E1(polar_lift(I)*x), E1(polar_lift(I)*x).rewrite(Si), -Ci(x) + I*Si(x) - I*pi/2, x) assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint), -x*E1(x) + exp(-x), x) assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint), x**2*E1(x)/2 + (1-x)*exp(-x)/2, x)
def _eval_rewrite_as_expint(self, s, x, **kwargs): from sympy import expint return expint(1 - s, x) * x**s
def test_latex_functions(): assert latex(exp(x)) == "e^{x}" assert latex(exp(1)+exp(2)) == "e + e^{2}" f = Function('f') assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}' beta = Function('beta') assert latex(beta(x)) == r"\beta{\left (x \right )}" assert latex(sin(x)) == r"\sin{\left (x \right )}" assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" assert latex(sin(2*x**2), fold_func_brackets=True) == \ r"\sin {2 x^{2}}" assert latex(sin(x**2), fold_func_brackets=True) == \ r"\sin {x^{2}}" assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}" assert latex(asin(x)**2,inv_trig_style="full") == \ r"\arcsin^{2}{\left (x \right )}" assert latex(asin(x)**2,inv_trig_style="power") == \ r"\sin^{-1}{\left (x \right )}^{2}" assert latex(asin(x**2),inv_trig_style="power",fold_func_brackets=True) == \ r"\sin^{-1} {x^{2}}" assert latex(factorial(k)) == r"k!" assert latex(factorial(-k)) == r"\left(- k\right)!" assert latex(factorial2(k)) == r"k!!" assert latex(factorial2(-k)) == r"\left(- k\right)!!" assert latex(binomial(2,k)) == r"{\binom{2}{k}}" assert latex(FallingFactorial(3,k)) == r"{\left(3\right)}_{\left(k\right)}" assert latex(RisingFactorial(3,k)) == r"{\left(3\right)}^{\left(k\right)}" assert latex(floor(x)) == r"\lfloor{x}\rfloor" assert latex(ceiling(x)) == r"\lceil{x}\rceil" assert latex(Abs(x)) == r"\lvert{x}\rvert" assert latex(re(x)) == r"\Re{x}" assert latex(re(x+y)) == r"\Re {\left (x + y \right )}" assert latex(im(x)) == r"\Im{x}" assert latex(conjugate(x)) == r"\overline{x}" assert latex(gamma(x)) == r"\Gamma\left(x\right)" assert latex(Order(x)) == r"\mathcal{O}\left(x\right)" assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)' assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)' assert latex(cot(x)) == r'\cot{\left (x \right )}' assert latex(coth(x)) == r'\coth{\left (x \right )}' assert latex(re(x)) == r'\Re{x}' assert latex(im(x)) == r'\Im{x}' assert latex(root(x,y)) == r'x^{\frac{1}{y}}' assert latex(arg(x)) == r'\arg{\left (x \right )}' assert latex(zeta(x)) == r'\zeta\left(x\right)' assert latex(zeta(x)) == r"\zeta\left(x\right)" assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)" assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)" assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)" assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" assert latex(polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)" assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}' assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}' assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)' assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}' assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}' assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}' assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'