def test_harmonic_limit(): n = Symbol('n') m = Symbol('m', positive=True) assert limit(harmonic(n, m + 1), n, oo) == zeta(m + 1) assert limit(harmonic(n, 2), n, oo) == pi**2 / 6 assert limit(harmonic(n, 3), n, oo) == -polygamma(2, 1) / 2
def test_harmonic_rewrite_sum_fail(): n = Symbol('n') m = Symbol('m') _k = Dummy('k') assert harmonic(n).rewrite(Sum) == Sum(1 / _k, (_k, 1, n)) assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))
def test_harmonic_rewrite_sum(): n = Symbol('n') m = Symbol('m') _k = Dummy('k') assert replace_dummy(harmonic(n).rewrite(Sum), _k) == Sum(1 / _k, (_k, 1, n)) assert replace_dummy(harmonic(n, m).rewrite(Sum), _k) == Sum(_k**(-m), (_k, 1, n))
def test_harmonic_evalf(): assert str(harmonic(1.5).evalf(10)) == '1.280372306' assert str(harmonic( 1.5, 2).evalf(10)) == '1.154576311' # issue sympy/sympy#7443
def test_harmonic_rewrite_polygamma(): n = Symbol('n') m = Symbol('m') assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma assert harmonic( n, 3).rewrite(polygamma) == polygamma(2, n + 1) / 2 - polygamma(2, 1) / 2 assert harmonic(n, m).rewrite(polygamma) == (-1)**m * ( polygamma(m - 1, 1) - polygamma(m - 1, n + 1)) / factorial(m - 1) assert expand_func( harmonic(n + 4) ) == harmonic(n) + 1 / (n + 4) + 1 / (n + 3) + 1 / (n + 2) + 1 / (n + 1) assert expand_func(harmonic( n - 4)) == harmonic(n) - 1 / (n - 1) - 1 / (n - 2) - 1 / (n - 3) - 1 / n assert harmonic(n, m).rewrite('tractable') == harmonic( n, m).rewrite(polygamma).rewrite(gamma).rewrite('tractable') assert isinstance(expand_func(harmonic(n, 2)), harmonic) assert expand_func(harmonic(n + Rational(1, 2))) == expand_func( harmonic(n + Rational(1, 2))) assert expand_func(harmonic(Rational(-1, 2))) == harmonic(Rational(-1, 2)) assert expand_func(harmonic(x)) == harmonic(x)
def test_harmonic(): n = Symbol('n') assert harmonic(n, 0) == n assert harmonic(n).evalf() == harmonic(n) assert harmonic(n, 1) == harmonic(n) assert harmonic(1, n).evalf() == harmonic(1, n) assert harmonic(0, 1) == 0 assert harmonic(1, 1) == 1 assert harmonic(2, 1) == Rational(3, 2) assert harmonic(3, 1) == Rational(11, 6) assert harmonic(4, 1) == Rational(25, 12) assert harmonic(0, 2) == 0 assert harmonic(1, 2) == 1 assert harmonic(2, 2) == Rational(5, 4) assert harmonic(3, 2) == Rational(49, 36) assert harmonic(4, 2) == Rational(205, 144) assert harmonic(0, 3) == 0 assert harmonic(1, 3) == 1 assert harmonic(2, 3) == Rational(9, 8) assert harmonic(3, 3) == Rational(251, 216) assert harmonic(4, 3) == Rational(2035, 1728) assert harmonic(oo, -1) == nan assert harmonic(oo, 0) == oo assert harmonic(oo, Rational(1, 2)) == oo assert harmonic(oo, 1) == oo assert harmonic(oo, 2) == (pi**2) / 6 assert harmonic(oo, 3) == zeta(3) assert harmonic(oo, x) == harmonic(oo, x, evaluate=False)
def test_harmonic_rational(): ne = Integer(6) no = Integer(5) pe = Integer(8) po = Integer(9) qe = Integer(10) qo = Integer(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * (Rational(1, 4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + Rational(5, 8))) + Rational(13944145, 4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + Rational(2422020029, 702257080)) Heoe = harmonic(ne + po / qe) Aeoe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(11818877030, 4286604231) + pi * (sqrt(5) / 8 + Rational(5, 8)) / sqrt(-sqrt(5) / 8 + Rational(5, 8))) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + Rational(11669332571, 3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * (Rational(1, 4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + Rational(5, 8))) + Rational(779405, 277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + Rational(53857323, 16331560)) Hooe = harmonic(no + po / qe) Aooe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(486853480, 186374097) + pi * (sqrt(5) / 8 + Rational(5, 8)) / sqrt(-sqrt(5) / 8 + Rational(5, 8))) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + Rational(383693479, 125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert h.evalf() == a.evalf()
def test_harmonic_sums(): assert summation(1/k, (k, 0, n)) == Sum(1/k, (k, 0, n)) assert summation(1/k, (k, 1, n)) == harmonic(n) assert summation(n/k, (k, 1, n)) == n*harmonic(n) assert summation(1/k, (k, 5, n)) == harmonic(n) - harmonic(4)
def test_harmonic_sums(): assert summation(1 / k, (k, 0, n)) == Sum(1 / k, (k, 0, n)) assert summation(1 / k, (k, 1, n)) == harmonic(n) assert summation(n / k, (k, 1, n)) == n * harmonic(n) assert summation(1 / k, (k, 5, n)) == harmonic(n) - harmonic(4)
def test_polygamma(): assert polygamma(n, nan) == nan assert polygamma(0, oo) == oo assert polygamma(0, -oo) == oo assert polygamma(0, I*oo) == oo assert polygamma(0, -I*oo) == oo assert polygamma(1, oo) == 0 assert polygamma(5, oo) == 0 assert polygamma(n, oo) == polygamma(n, oo, evaluate=False) assert polygamma(0, -9) == zoo assert polygamma(0, -9) == zoo assert polygamma(0, -1) == zoo assert polygamma(0, 0) == zoo assert polygamma(0, 1) == -EulerGamma assert polygamma(0, 7) == Rational(49, 20) - EulerGamma assert polygamma(0, -Rational(3, 11)) == polygamma(0, -Rational(3, 11), evaluate=False) 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 trigamma(x) == polygamma(1, x) def t(m, n): x = Integer(m)/n r = polygamma(0, x) if r.has(polygamma): return False return abs(polygamma(0, x.evalf()).evalf(strict=False) - r.evalf()).evalf(strict=False) < 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 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(3, 7*x).diff(x) == 7*polygamma(4, 7*x) pytest.raises(ArgumentIndexError, lambda: polygamma(3, 7*x).fdiff(3)) 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) assert polygamma(x, y).rewrite(harmonic) == polygamma(x, y) # 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) # But smaller orders are iterated integrals and don't have a special name assert isinstance(polygamma(-2, x), polygamma) # Test a bug assert polygamma(0, -x).expand(func=True) == polygamma(0, -x) assert polygamma(1, x).as_leading_term(x) == polygamma(1, x)
def test_polygamma(): from diofant import I assert polygamma(n, nan) == nan assert polygamma(0, oo) == oo assert polygamma(0, -oo) == oo assert polygamma(0, I * oo) == oo assert polygamma(0, -I * oo) == oo assert polygamma(1, oo) == 0 assert polygamma(5, oo) == 0 assert polygamma(0, -9) == zoo assert polygamma(0, -9) == zoo assert polygamma(0, -1) == zoo assert polygamma(0, 0) == 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 trigamma(x) == polygamma(1, x) def t(m, n): x = Integer(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 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(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: from diofant import exp_polar 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) # 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(1, x).as_leading_term(x) == polygamma(1, x)
def test_harmonic_rewrite_sum(): assert harmonic(n).rewrite(Sum) == Sum(1 / k, (k, 1, n)) assert harmonic(n, m).rewrite(Sum) == Sum(k**(-m), (k, 1, n))