def _derivative_(self, u, m, diff_param):
"""
EXAMPLES::
sage: x,m = var('x,m')
sage: elliptic_eu(x,m).diff(x)
sqrt(-m*jacobi_sn(x, m)^2 + 1)*jacobi_dn(x, m)
sage: elliptic_eu(x,m).diff(m)
1/2*(elliptic_eu(x, m)
- elliptic_f(jacobi_am(x, m), m))/m
- 1/2*(m*jacobi_cn(x, m)*jacobi_sn(x, m)
- (m - 1)*x
- elliptic_eu(x, m)*jacobi_dn(x, m))*sqrt(-m*jacobi_sn(x, m)^2 + 1)/((m - 1)*m)
"""
from sage.functions.jacobi import jacobi, jacobi_am
if diff_param == 0:
return (sqrt(-m * jacobi('sn', u, m)**Integer(2) + Integer(1)) *
jacobi('dn', u, m))
elif diff_param == 1:
return (Integer(1) / Integer(2) *
(elliptic_eu(u, m) - elliptic_f(jacobi_am(u, m), m)) / m -
Integer(1) / Integer(2) *
sqrt(-m * jacobi('sn', u, m)**Integer(2) + Integer(1)) *
(m * jacobi('sn', u, m) * jacobi('cn', u, m) -
(m - Integer(1)) * u -
elliptic_eu(u, m) * jacobi('dn', u, m)) /
((m - Integer(1)) * m))
def _derivative_(self, u, m, diff_param):
"""
EXAMPLES::
sage: x,m = var('x,m')
sage: elliptic_eu(x,m).diff(x)
sqrt(-m*jacobi_sn(x, m)^2 + 1)*jacobi_dn(x, m)
sage: elliptic_eu(x,m).diff(m)
1/2*(elliptic_eu(x, m)
- elliptic_f(jacobi_am(x, m), m))/m
- 1/2*(m*jacobi_cn(x, m)*jacobi_sn(x, m)
- (m - 1)*x
- elliptic_eu(x, m)*jacobi_dn(x, m))*sqrt(-m*jacobi_sn(x, m)^2 + 1)/((m - 1)*m)
"""
from sage.functions.jacobi import jacobi, jacobi_am
if diff_param == 0:
return (sqrt(-m * jacobi('sn', u, m) ** Integer(2) +
Integer(1)) * jacobi('dn', u, m))
elif diff_param == 1:
return (Integer(1) / Integer(2) *
(elliptic_eu(u, m) - elliptic_f(jacobi_am(u, m), m)) / m -
Integer(1) / Integer(2) * sqrt(-m * jacobi('sn', u, m) **
Integer(2) + Integer(1)) * (m * jacobi('sn', u, m) *
jacobi('cn', u, m) - (m - Integer(1)) * u -
elliptic_eu(u, m) * jacobi('dn', u, m)) /
((m - Integer(1)) * m))