from sympy.core.function import expand_func, Subs from sympy.core import EulerGamma from sympy.core.numbers import (I, Rational, nan, oo, pi, zoo) from sympy.core.singleton import S from sympy.core.symbol import (Dummy, Symbol) from sympy.functions.combinatorial.factorials import factorial from sympy.functions.combinatorial.numbers import harmonic from sympy.functions.elementary.complexes import (Abs, conjugate, im, re) from sympy.functions.elementary.exponential import (exp, exp_polar, log) from sympy.functions.elementary.hyperbolic import tanh from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin, atan) from sympy.functions.special.error_functions import (Ei, erf, erfc) from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma, lowergamma, multigamma, polygamma, trigamma, uppergamma) from sympy.functions.special.zeta_functions import zeta from sympy.series.order import O from sympy.core.expr import unchanged from sympy.core.function import ArgumentIndexError from sympy.testing.pytest import raises from sympy.core.random import (test_derivative_numerically as td, random_complex_number as randcplx, verify_numerically as tn) x = Symbol('x') y = Symbol('y') n = Symbol('n', integer=True) w = Symbol('w', real=True) def test_gamma(): assert gamma(nan) is nan assert gamma(oo) is oo assert gamma(-100) is zoo assert gamma(0) is zoo assert gamma(-100.0) is zoo assert gamma(1) == 1 assert gamma(2) == 1 assert gamma(3) == 2 assert gamma(102) == factorial(101) assert gamma(S.Half) == sqrt(pi) assert gamma(Rational(3, 2)) == sqrt(pi)*S.Half assert gamma(Rational(5, 2)) == sqrt(pi)*Rational(3, 4) assert gamma(Rational(7, 2)) == sqrt(pi)*Rational(15, 8) assert gamma(Rational(-1, 2)) == -2*sqrt(pi) assert gamma(Rational(-3, 2)) == sqrt(pi)*Rational(4, 3) assert gamma(Rational(-5, 2)) == sqrt(pi)*Rational(-8, 15) assert gamma(Rational(-15, 2)) == sqrt(pi)*Rational(256, 2027025) assert gamma(Rational( -11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8)) assert gamma(Rational( -10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3)) assert gamma(Rational( 14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3)) assert gamma(Rational( 17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7)) assert gamma(Rational( 19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8)) assert gamma(x).diff(x) == gamma(x)*polygamma(0, x) assert gamma(x - 1).expand(func=True) == gamma(x)/(x - 1) assert gamma(x + 2).expand(func=True, mul=False) == x*(x + 1)*gamma(x) assert conjugate(gamma(x)) == gamma(conjugate(x)) assert expand_func(gamma(x + Rational(3, 2))) == \ (x + S.Half)*gamma(x + S.Half) assert expand_func(gamma(x - S.Half)) == \ gamma(S.Half + x)/(x - S.Half) # Test a bug: assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4)) # XXX: Not sure about these tests. I can fix them by defining e.g. # exp_polar.is_integer but I'm not sure if that makes sense. assert gamma(3*exp_polar(I*pi)/4).is_nonnegative is False assert gamma(3*exp_polar(I*pi)/4).is_extended_nonpositive is True y = Symbol('y', nonpositive=True, integer=True) assert gamma(y).is_real == False y = Symbol('y', positive=True, noninteger=True) assert gamma(y).is_real == True assert gamma(-1.0, evaluate=False).is_real == False assert gamma(0, evaluate=False).is_real == False assert gamma(-2, evaluate=False).is_real == False def test_gamma_rewrite(): assert gamma(n).rewrite(factorial) == factorial(n - 1) def test_gamma_series(): assert gamma(x + 1).series(x, 0, 3) == \ 1 - EulerGamma*x + x**2*(EulerGamma**2/2 + pi**2/12) + O(x**3) assert gamma(x).series(x, -1, 3) == \ -1/(x + 1) + EulerGamma - 1 + (x + 1)*(-1 - pi**2/12 - EulerGamma**2/2 + \ EulerGamma) + (x + 1)**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 - \ polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O((x + 1)**3, (x, -1)) def tn_branch(s, func): from sympy.core.random import uniform c = uniform(1, 5) expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi)) eps = 1e-15 expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I) return abs(expr.n() - expr2.n()).n() < 1e-10 def test_lowergamma(): from sympy.functions.special.error_functions import expint from sympy.functions.special.hyper import meijerg 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_uppergamma(): from sympy.functions.special.error_functions import expint from sympy.functions.special.hyper import meijerg 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) p = Symbol('p', positive=True) assert uppergamma(0, p) == -Ei(-p) assert uppergamma(p, 0) == gamma(p) 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 unchanged(uppergamma, x, -oo) assert unchanged(uppergamma, x, 0) assert tn_branch(-3, uppergamma) assert tn_branch(-4, uppergamma) assert tn_branch(Rational(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 unchanged(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_polygamma(): assert polygamma(n, nan) is nan assert polygamma(0, oo) is oo assert polygamma(0, -oo) is oo assert polygamma(0, I*oo) is oo assert polygamma(0, -I*oo) is oo assert polygamma(1, oo) == 0 assert polygamma(5, oo) == 0 assert polygamma(0, -9) is zoo assert polygamma(0, -9) is zoo assert polygamma(0, -1) is zoo assert polygamma(Rational(3, 2), -1) is zoo assert polygamma(0, 0) is zoo assert polygamma(0, 1) == -EulerGamma assert polygamma(0, 7) == Rational(49, 20) - EulerGamma assert polygamma(1, 1) == pi**2/6 assert polygamma(1, 2) == pi**2/6 - 1 assert polygamma(1, 3) == pi**2/6 - Rational(5, 4) assert polygamma(3, 1) == pi**4 / 15 assert polygamma(3, 5) == 6*(Rational(-22369, 20736) + pi**4/90) assert polygamma(5, 1) == 8 * pi**6 / 63 assert polygamma(1, S.Half) == pi**2 / 2 assert polygamma(2, S.Half) == -14*zeta(3) assert polygamma(11, S.Half) == 176896*pi**12 def t(m, n): x = S(m)/n r = polygamma(0, x) if r.has(polygamma): return False return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10 assert t(1, 2) assert t(3, 2) assert t(-1, 2) assert t(1, 4) assert t(-3, 4) assert t(1, 3) assert t(4, 3) assert t(3, 4) assert t(2, 3) assert t(123, 5) assert polygamma(0, x).rewrite(zeta) == polygamma(0, x) assert polygamma(1, x).rewrite(zeta) == zeta(2, x) assert polygamma(2, x).rewrite(zeta) == -2*zeta(3, x) assert polygamma(I, 2).rewrite(zeta) == polygamma(I, 2) n1 = Symbol('n1') n2 = Symbol('n2', real=True) n3 = Symbol('n3', integer=True) n4 = Symbol('n4', positive=True) n5 = Symbol('n5', positive=True, integer=True) assert polygamma(n1, x).rewrite(zeta) == polygamma(n1, x) assert polygamma(n2, x).rewrite(zeta) == polygamma(n2, x) assert polygamma(n3, x).rewrite(zeta) == polygamma(n3, x) assert polygamma(n4, x).rewrite(zeta) == polygamma(n4, x) assert polygamma(n5, x).rewrite(zeta) == (-1)**(n5 + 1) * factorial(n5) * zeta(n5 + 1, x) assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x) assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma assert polygamma(2, x).rewrite(harmonic) == 2*harmonic(x - 1, 3) - 2*zeta(3) ni = Symbol("n", integer=True) assert polygamma(ni, x).rewrite(harmonic) == (-1)**(ni + 1)*(-harmonic(x - 1, ni + 1) + zeta(ni + 1))*factorial(ni) # Polygamma of non-negative integer order is unbranched: k = Symbol('n', integer=True, nonnegative=True) assert polygamma(k, exp_polar(2*I*pi)*x) == polygamma(k, x) # but negative integers are branched! k = Symbol('n', integer=True) assert polygamma(k, exp_polar(2*I*pi)*x).args == (k, exp_polar(2*I*pi)*x) # Polygamma of order -1 is loggamma: assert polygamma(-1, x) == loggamma(x) - log(2*pi) / 2 # But smaller orders are iterated integrals and don't have a special name assert polygamma(-2, x).func is polygamma # Test a bug assert polygamma(0, -x).expand(func=True) == polygamma(0, -x) assert polygamma(2, 2.5).is_positive == False assert polygamma(2, -2.5).is_positive == False assert polygamma(3, 2.5).is_positive == True assert polygamma(3, -2.5).is_positive is True assert polygamma(-2, -2.5).is_positive is None assert polygamma(-3, -2.5).is_positive is None assert polygamma(2, 2.5).is_negative == True assert polygamma(3, 2.5).is_negative == False assert polygamma(3, -2.5).is_negative == False assert polygamma(2, -2.5).is_negative is True assert polygamma(-2, -2.5).is_negative is None assert polygamma(-3, -2.5).is_negative is None assert polygamma(I, 2).is_positive is None assert polygamma(I, 3).is_negative is None # issue 17350 assert (I*polygamma(I, pi)).as_real_imag() == \ (-im(polygamma(I, pi)), re(polygamma(I, pi))) assert (tanh(polygamma(I, 1))).rewrite(exp) == \ (exp(polygamma(I, 1)) - exp(-polygamma(I, 1)))/(exp(polygamma(I, 1)) + exp(-polygamma(I, 1))) assert (I / polygamma(I, 4)).rewrite(exp) == \ I*exp(-I*atan(im(polygamma(I, 4))/re(polygamma(I, 4))))/Abs(polygamma(I, 4)) # issue 12569 assert unchanged(im, polygamma(0, I)) assert polygamma(Symbol('a', positive=True), Symbol('b', positive=True)).is_real is True assert polygamma(0, I).is_real is None assert str(polygamma(pi, 3).evalf(n=10)) == "0.1169314564" assert str(polygamma(2.3, 1.0).evalf(n=10)) == "-3.003302909" assert str(polygamma(-1, 1).evalf(n=10)) == "-0.9189385332" # not zero assert str(polygamma(I, 1).evalf(n=10)) == "-3.109856569 + 1.89089016*I" assert str(polygamma(1, I).evalf(n=10)) == "-0.5369999034 - 0.7942335428*I" assert str(polygamma(I, I).evalf(n=10)) == "6.332362889 + 45.92828268*I" def test_polygamma_expand_func(): assert polygamma(0, x).expand(func=True) == polygamma(0, x) assert polygamma(0, 2*x).expand(func=True) == \ polygamma(0, x)/2 + polygamma(0, S.Half + x)/2 + log(2) assert polygamma(1, 2*x).expand(func=True) == \ polygamma(1, x)/4 + polygamma(1, S.Half + x)/4 assert polygamma(2, x).expand(func=True) == \ polygamma(2, x) assert polygamma(0, -1 + x).expand(func=True) == \ polygamma(0, x) - 1/(x - 1) assert polygamma(0, 1 + x).expand(func=True) == \ 1/x + polygamma(0, x ) assert polygamma(0, 2 + x).expand(func=True) == \ 1/x + 1/(1 + x) + polygamma(0, x) assert polygamma(0, 3 + x).expand(func=True) == \ polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) assert polygamma(0, 4 + x).expand(func=True) == \ polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) + 1/(3 + x) assert polygamma(1, 1 + x).expand(func=True) == \ polygamma(1, x) - 1/x**2 assert polygamma(1, 2 + x).expand(func=True, multinomial=False) == \ polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 assert polygamma(1, 3 + x).expand(func=True, multinomial=False) == \ polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - 1/(2 + x)**2 assert polygamma(1, 4 + x).expand(func=True, multinomial=False) == \ polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - \ 1/(2 + x)**2 - 1/(3 + x)**2 assert polygamma(0, x + y).expand(func=True) == \ polygamma(0, x + y) assert polygamma(1, x + y).expand(func=True) == \ polygamma(1, x + y) assert polygamma(1, 3 + 4*x + y).expand(func=True, multinomial=False) == \ polygamma(1, y + 4*x) - 1/(y + 4*x)**2 - \ 1/(1 + y + 4*x)**2 - 1/(2 + y + 4*x)**2 assert polygamma(3, 3 + 4*x + y).expand(func=True, multinomial=False) == \ polygamma(3, y + 4*x) - 6/(y + 4*x)**4 - \ 6/(1 + y + 4*x)**4 - 6/(2 + y + 4*x)**4 assert polygamma(3, 4*x + y + 1).expand(func=True, multinomial=False) == \ polygamma(3, y + 4*x) - 6/(y + 4*x)**4 e = polygamma(3, 4*x + y + Rational(3, 2)) assert e.expand(func=True) == e e = polygamma(3, x + y + Rational(3, 4)) assert e.expand(func=True, basic=False) == e assert polygamma(-1, x, evaluate=False).expand(func=True) == \ loggamma(x) - log(pi)/2 - log(2)/2 p2 = polygamma(-2, x).expand(func=True) + x**2/2 - x/2 + S(1)/12 assert isinstance(p2, Subs) assert p2.point == (-1,) def test_digamma(): assert digamma(nan) == nan assert digamma(oo) == oo assert digamma(-oo) == oo assert digamma(I*oo) == oo assert digamma(-I*oo) == oo assert digamma(-9) == zoo assert digamma(-9) == zoo assert digamma(-1) == zoo assert digamma(0) == zoo assert digamma(1) == -EulerGamma assert digamma(7) == Rational(49, 20) - EulerGamma def t(m, n): x = S(m)/n r = digamma(x) if r.has(digamma): return False return abs(digamma(x.n()).n() - r.n()).n() < 1e-10 assert t(1, 2) assert t(3, 2) assert t(-1, 2) assert t(1, 4) assert t(-3, 4) assert t(1, 3) assert t(4, 3) assert t(3, 4) assert t(2, 3) assert t(123, 5) assert digamma(x).rewrite(zeta) == polygamma(0, x) assert digamma(x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma assert digamma(I).is_real is None assert digamma(x,evaluate=False).fdiff() == polygamma(1, x) assert digamma(x,evaluate=False).is_real is None assert digamma(x,evaluate=False).is_positive is None assert digamma(x,evaluate=False).is_negative is None assert digamma(x,evaluate=False).rewrite(polygamma) == polygamma(0, x) def test_digamma_expand_func(): assert digamma(x).expand(func=True) == polygamma(0, x) assert digamma(2*x).expand(func=True) == \ polygamma(0, x)/2 + polygamma(0, Rational(1, 2) + x)/2 + log(2) assert digamma(-1 + x).expand(func=True) == \ polygamma(0, x) - 1/(x - 1) assert digamma(1 + x).expand(func=True) == \ 1/x + polygamma(0, x ) assert digamma(2 + x).expand(func=True) == \ 1/x + 1/(1 + x) + polygamma(0, x) assert digamma(3 + x).expand(func=True) == \ polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) assert digamma(4 + x).expand(func=True) == \ polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) + 1/(3 + x) assert digamma(x + y).expand(func=True) == \ polygamma(0, x + y) def test_trigamma(): assert trigamma(nan) == nan assert trigamma(oo) == 0 assert trigamma(1) == pi**2/6 assert trigamma(2) == pi**2/6 - 1 assert trigamma(3) == pi**2/6 - Rational(5, 4) assert trigamma(x, evaluate=False).rewrite(zeta) == zeta(2, x) assert trigamma(x, evaluate=False).rewrite(harmonic) == \ trigamma(x).rewrite(polygamma).rewrite(harmonic) assert trigamma(x,evaluate=False).fdiff() == polygamma(2, x) assert trigamma(x,evaluate=False).is_real is None assert trigamma(x,evaluate=False).is_positive is None assert trigamma(x,evaluate=False).is_negative is None assert trigamma(x,evaluate=False).rewrite(polygamma) == polygamma(1, x) def test_trigamma_expand_func(): assert trigamma(2*x).expand(func=True) == \ polygamma(1, x)/4 + polygamma(1, Rational(1, 2) + x)/4 assert trigamma(1 + x).expand(func=True) == \ polygamma(1, x) - 1/x**2 assert trigamma(2 + x).expand(func=True, multinomial=False) == \ polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 assert trigamma(3 + x).expand(func=True, multinomial=False) == \ polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - 1/(2 + x)**2 assert trigamma(4 + x).expand(func=True, multinomial=False) == \ polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - \ 1/(2 + x)**2 - 1/(3 + x)**2 assert trigamma(x + y).expand(func=True) == \ polygamma(1, x + y) assert trigamma(3 + 4*x + y).expand(func=True, multinomial=False) == \ polygamma(1, y + 4*x) - 1/(y + 4*x)**2 - \ 1/(1 + y + 4*x)**2 - 1/(2 + y + 4*x)**2 def test_loggamma(): raises(TypeError, lambda: loggamma(2, 3)) raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2)) assert loggamma(-1) is oo assert loggamma(-2) is oo assert loggamma(0) is oo assert loggamma(1) == 0 assert loggamma(2) == 0 assert loggamma(3) == log(2) assert loggamma(4) == log(6) n = Symbol("n", integer=True, positive=True) assert loggamma(n) == log(gamma(n)) assert loggamma(-n) is oo assert loggamma(n/2) == log(2**(-n + 1)*sqrt(pi)*gamma(n)/gamma(n/2 + S.Half)) assert loggamma(oo) is oo assert loggamma(-oo) is zoo assert loggamma(I*oo) is zoo assert loggamma(-I*oo) is zoo assert loggamma(zoo) is zoo assert loggamma(nan) is nan L = loggamma(Rational(16, 3)) E = -5*log(3) + loggamma(Rational(1, 3)) + log(4) + log(7) + log(10) + log(13) assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(Rational(19, 4)) E = -4*log(4) + loggamma(Rational(3, 4)) + log(3) + log(7) + log(11) + log(15) assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(Rational(23, 7)) E = -3*log(7) + log(2) + loggamma(Rational(2, 7)) + log(9) + log(16) assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(Rational(19, 4) - 7) E = -log(9) - log(5) + loggamma(Rational(3, 4)) + 3*log(4) - 3*I*pi assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(Rational(23, 7) - 6) E = -log(19) - log(12) - log(5) + loggamma(Rational(2, 7)) + 3*log(7) - 3*I*pi assert expand_func(L).doit() == E assert L.n() == E.n() assert loggamma(x).diff(x) == polygamma(0, x) s1 = loggamma(1/(x + sin(x)) + cos(x)).nseries(x, n=4) s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \ log(x)*x**2/2 assert (s1 - s2).expand(force=True).removeO() == 0 s1 = loggamma(1/x).series(x) s2 = (1/x - S.Half)*log(1/x) - 1/x + log(2*pi)/2 + \ x/12 - x**3/360 + x**5/1260 + O(x**7) assert ((s1 - s2).expand(force=True)).removeO() == 0 assert loggamma(x).rewrite('intractable') == log(gamma(x)) s1 = loggamma(x).series(x).cancel() assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \ pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6) assert s1 == loggamma(x).rewrite('intractable').series(x).cancel() assert conjugate(loggamma(x)) == loggamma(conjugate(x)) assert conjugate(loggamma(0)) is oo assert conjugate(loggamma(1)) == loggamma(conjugate(1)) assert conjugate(loggamma(-oo)) == conjugate(zoo) assert loggamma(Symbol('v', positive=True)).is_real is True assert loggamma(Symbol('v', zero=True)).is_real is False assert loggamma(Symbol('v', negative=True)).is_real is False assert loggamma(Symbol('v', nonpositive=True)).is_real is False assert loggamma(Symbol('v', nonnegative=True)).is_real is None assert loggamma(Symbol('v', imaginary=True)).is_real is None assert loggamma(Symbol('v', real=True)).is_real is None assert loggamma(Symbol('v')).is_real is None assert loggamma(S.Half).is_real is True assert loggamma(0).is_real is False assert loggamma(Rational(-1, 2)).is_real is False assert loggamma(I).is_real is None assert loggamma(2 + 3*I).is_real is None def tN(N, M): assert loggamma(1/x)._eval_nseries(x, n=N).getn() == M tN(0, 0) tN(1, 1) tN(2, 2) tN(3, 3) tN(4, 4) tN(5, 5) def test_polygamma_expansion(): # A. & S., pa. 259 and 260 assert polygamma(0, 1/x).nseries(x, n=3) == \ -log(x) - x/2 - x**2/12 + O(x**3) assert polygamma(1, 1/x).series(x, n=5) == \ x + x**2/2 + x**3/6 + O(x**5) assert polygamma(3, 1/x).nseries(x, n=11) == \ 2*x**3 + 3*x**4 + 2*x**5 - x**7 + 4*x**9/3 + O(x**11) def test_polygamma_leading_term(): expr = -log(1/x) + polygamma(0, 1 + 1/x) + S.EulerGamma assert expr.as_leading_term(x, logx=-y) == S.EulerGamma def test_issue_8657(): n = Symbol('n', negative=True, integer=True) m = Symbol('m', integer=True) o = Symbol('o', positive=True) p = Symbol('p', negative=True, integer=False) assert gamma(n).is_real is False assert gamma(m).is_real is None assert gamma(o).is_real is True assert gamma(p).is_real is True assert gamma(w).is_real is None def test_issue_8524(): x = Symbol('x', positive=True) y = Symbol('y', negative=True) z = Symbol('z', positive=False) p = Symbol('p', negative=False) q = Symbol('q', integer=True) r = Symbol('r', integer=False) e = Symbol('e', even=True, negative=True) assert gamma(x).is_positive is True assert gamma(y).is_positive is None assert gamma(z).is_positive is None assert gamma(p).is_positive is None assert gamma(q).is_positive is None assert gamma(r).is_positive is None assert gamma(e + S.Half).is_positive is True assert gamma(e - S.Half).is_positive is False def test_issue_14450(): assert uppergamma(Rational(3, 8), x).evalf() == uppergamma(Rational(3, 8), x) assert lowergamma(x, Rational(3, 8)).evalf() == lowergamma(x, Rational(3, 8)) # some values from Wolfram Alpha for comparison assert abs(uppergamma(Rational(3, 8), 2).evalf() - 0.07105675881) < 1e-9 assert abs(lowergamma(Rational(3, 8), 2).evalf() - 2.2993794256) < 1e-9 def test_issue_14528(): k = Symbol('k', integer=True, nonpositive=True) assert isinstance(gamma(k), gamma) def test_multigamma(): from sympy.concrete.products import Product p = Symbol('p') _k = Dummy('_k') assert multigamma(x, p).dummy_eq(pi**(p*(p - 1)/4)*\ Product(gamma(x + (1 - _k)/2), (_k, 1, p))) assert conjugate(multigamma(x, p)).dummy_eq(pi**((conjugate(p) - 1)*\ conjugate(p)/4)*Product(gamma(conjugate(x) + (1-conjugate(_k))/2), (_k, 1, p))) assert conjugate(multigamma(x, 1)) == gamma(conjugate(x)) p = Symbol('p', positive=True) assert conjugate(multigamma(x, p)).dummy_eq(pi**((p - 1)*p/4)*\ Product(gamma(conjugate(x) + (1-conjugate(_k))/2), (_k, 1, p))) assert multigamma(nan, 1) is nan assert multigamma(oo, 1).doit() is oo assert multigamma(1, 1) == 1 assert multigamma(2, 1) == 1 assert multigamma(3, 1) == 2 assert multigamma(102, 1) == factorial(101) assert multigamma(S.Half, 1) == sqrt(pi) assert multigamma(1, 2) == pi assert multigamma(2, 2) == pi/2 assert multigamma(1, 3) is zoo assert multigamma(2, 3) == pi**2/2 assert multigamma(3, 3) == 3*pi**2/2 assert multigamma(x, 1).diff(x) == gamma(x)*polygamma(0, x) assert multigamma(x, 2).diff(x) == sqrt(pi)*gamma(x)*gamma(x - S.Half)*\ polygamma(0, x) + sqrt(pi)*gamma(x)*gamma(x - S.Half)*polygamma(0, x - S.Half) assert multigamma(x - 1, 1).expand(func=True) == gamma(x)/(x - 1) assert multigamma(x + 2, 1).expand(func=True, mul=False) == x*(x + 1)*\ gamma(x) assert multigamma(x - 1, 2).expand(func=True) == sqrt(pi)*gamma(x)*\ gamma(x + S.Half)/(x**3 - 3*x**2 + x*Rational(11, 4) - Rational(3, 4)) assert multigamma(x - 1, 3).expand(func=True) == pi**Rational(3, 2)*gamma(x)**2*\ gamma(x + S.Half)/(x**5 - 6*x**4 + 55*x**3/4 - 15*x**2 + x*Rational(31, 4) - Rational(3, 2)) assert multigamma(n, 1).rewrite(factorial) == factorial(n - 1) assert multigamma(n, 2).rewrite(factorial) == sqrt(pi)*\ factorial(n - Rational(3, 2))*factorial(n - 1) assert multigamma(n, 3).rewrite(factorial) == pi**Rational(3, 2)*\ factorial(n - 2)*factorial(n - Rational(3, 2))*factorial(n - 1) assert multigamma(Rational(-1, 2), 3, evaluate=False).is_real == False assert multigamma(S.Half, 3, evaluate=False).is_real == False assert multigamma(0, 1, evaluate=False).is_real == False assert multigamma(1, 3, evaluate=False).is_real == False assert multigamma(-1.0, 3, evaluate=False).is_real == False assert multigamma(0.7, 3, evaluate=False).is_real == True assert multigamma(3, 3, evaluate=False).is_real == True def test_gamma_as_leading_term(): assert gamma(x).as_leading_term(x) == 1/x assert gamma(2 + x).as_leading_term(x) == S(1) assert gamma(cos(x)).as_leading_term(x) == S(1) assert gamma(sin(x)).as_leading_term(x) == 1/x