def fdiff(self, argindex):
x, y = self.args
if argindex == 1:
# Diff wrt x
return beta(x, y)*(digamma(x) - digamma(x + y))
elif argindex == 2:
# Diff wrt y
return beta(x, y)*(digamma(y) - digamma(x + y))
else:
raise ArgumentIndexError(self, argindex)
def fdiff(self, argindex):
x, y = self.args
if argindex == 1:
# Diff wrt x
return beta(x, y) * (digamma(x) - digamma(x + y))
elif argindex == 2:
# Diff wrt y
return beta(x, y) * (digamma(y) - digamma(x + y))
else:
raise ArgumentIndexError(self, argindex)
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_gruntz_eval_special():
# Gruntz, p. 126
assert gruntz(
exp(x) * (sin(1 / x + exp(-x)) - sin(1 / x + exp(-x**2))), x, oo) == 1
assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2),
x, oo) == -2 / sqrt(pi)
assert gruntz(
exp(exp(x)) * (exp(sin(1 / x + exp(-exp(x)))) - exp(sin(1 / x))), x,
oo) == 1
assert gruntz(exp(x) * (gamma(x + exp(-x)) - gamma(x)), x, oo) is oo
assert gruntz(exp(exp(digamma(digamma(x)))) / x, x,
oo) == exp(Rational(-1, 2))
assert gruntz(exp(exp(digamma(log(x)))) / x, x, oo) == exp(Rational(-1, 2))
assert gruntz(digamma(digamma(digamma(x))), x, oo) is oo
assert gruntz(loggamma(loggamma(x)), x, oo) is oo
assert gruntz(
((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x) - cos(1 / x)) * x *
log(x), x, oo) == Rational(-1, 2)
assert gruntz(x * (gamma(x - 1/gamma(x)) - gamma(x) + log(x)), x, oo) \
== S.Half
assert gruntz((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x), x, oo) == 1
def test_issue_15146():
e = (x/2) * (-2*x**3 - 2*(x**3 - 1) * x**2 * digamma(x**3 + 1) + \
2*(x**3 - 1) * x**2 * digamma(x**3 + x + 1) + x + 3)
assert limit(e, x, oo) == S(1)/3
def test_leading_term():
from sympy.functions.special.gamma_functions import digamma
assert O(1 / digamma(1 / x)) == O(1 / log(x))
def test_gruntz_eval_special_slow():
_sskip()
assert gruntz(
gamma(x + 1) / sqrt(2 * pi) - exp(-x) *
(x**(x + S.Half) + x**(x - S.Half) / 12), x, oo) is oo
assert gruntz(exp(exp(exp(digamma(digamma(digamma(x)))))) / x, x, oo) == 0
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
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_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