def _sympysage_polygamma(self):
"""
EXAMPLES::
sage: from sympy import Symbol, polygamma as pg
sage: _ = var('x, y')
sage: pgxy = pg(Symbol('x'), Symbol('y'))
sage: assert psi(x)._sympy_() == pg(0, Symbol('x'))
sage: assert psi(x) == pg(0, Symbol('x'))._sage_()
sage: assert psi(x,y)._sympy_() == pgxy
sage: assert psi(x,y) == pgxy._sage_()
sage: integrate(psi(x), x, algorithm='sympy')
integrate(psi(x), x)
"""
from sage.functions.other import psi
return psi(self.args[0]._sage_(), self.args[1]._sage_())
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1)**s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1) ** s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
co = get_coercion_model().canonical_coercion(s, x)[0]
if is_inexact(co) and not isinstance(co, Expression):
return self._evalf_(s, x, parent=parent(co))
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1)**s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
co = get_coercion_model().canonical_coercion(s, x)[0]
if is_inexact(co) and not isinstance(co, Expression):
return self._evalf_(s, x, parent=parent(co))
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1) ** s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return