def rotation_list(tilt, turn):
from sage.functions.all import sin, cos
return [
sin(tilt * pi / 180.0),
sin(turn * pi / 180.0),
cos(tilt * pi / 180.0),
cos(turn * pi / 180.0)
]
def _derivative_(self, n, z, m, diff_param):
"""
EXAMPLES::
sage: n,z,m = var('n,z,m')
sage: elliptic_pi(n,z,m).diff(n)
1/4*(sqrt(-m*sin(z)^2 + 1)*n*sin(2*z)/(n*sin(z)^2 - 1)
+ 2*(m - n)*elliptic_f(z, m)/n
+ 2*(n^2 - m)*elliptic_pi(n, z, m)/n
+ 2*elliptic_e(z, m))/((m - n)*(n - 1))
sage: elliptic_pi(n,z,m).diff(z)
-1/(sqrt(-m*sin(z)^2 + 1)*(n*sin(z)^2 - 1))
sage: elliptic_pi(n,z,m).diff(m)
1/4*(m*sin(2*z)/(sqrt(-m*sin(z)^2 + 1)*(m - 1))
- 2*elliptic_e(z, m)/(m - 1)
- 2*elliptic_pi(n, z, m))/(m - n)
"""
if diff_param == 0:
return ((Integer(1) / (Integer(2) * (m - n) * (n - Integer(1)))) *
(elliptic_e(z, m) + ((m - n) / n) * elliptic_f(z, m) +
((n**Integer(2) - m) / n) * elliptic_pi(n, z, m) -
(n * sqrt(Integer(1) - m * sin(z)**Integer(2)) *
sin(Integer(2) * z)) /
(Integer(2) * (Integer(1) - n * sin(z)**Integer(2)))))
elif diff_param == 1:
return (Integer(1) /
(sqrt(Integer(1) - m * sin(z)**Integer(Integer(2))) *
(Integer(1) - n * sin(z)**Integer(2))))
elif diff_param == 2:
return ((Integer(1) / (Integer(2) * (n - m))) * (
elliptic_e(z, m) / (m - Integer(1)) + elliptic_pi(n, z, m) -
(m * sin(Integer(2) * z)) /
(Integer(2) *
(m - Integer(1)) * sqrt(Integer(1) - m * sin(z)**Integer(2))))
)
def _derivative_(self, n, z, m, diff_param):
"""
EXAMPLES::
sage: n,z,m = var('n,z,m')
sage: elliptic_pi(n,z,m).diff(n)
1/4*(sqrt(-m*sin(z)^2 + 1)*n*sin(2*z)/(n*sin(z)^2 - 1)
+ 2*(m - n)*elliptic_f(z, m)/n
+ 2*(n^2 - m)*elliptic_pi(n, z, m)/n
+ 2*elliptic_e(z, m))/((m - n)*(n - 1))
sage: elliptic_pi(n,z,m).diff(z)
-1/(sqrt(-m*sin(z)^2 + 1)*(n*sin(z)^2 - 1))
sage: elliptic_pi(n,z,m).diff(m)
1/4*(m*sin(2*z)/(sqrt(-m*sin(z)^2 + 1)*(m - 1))
- 2*elliptic_e(z, m)/(m - 1)
- 2*elliptic_pi(n, z, m))/(m - n)
"""
if diff_param == 0:
return ((Integer(1) / (Integer(2) * (m - n) * (n - Integer(1)))) *
(elliptic_e(z, m) + ((m - n) / n) * elliptic_f(z, m) +
((n ** Integer(2) - m) / n) * elliptic_pi(n, z, m) -
(n * sqrt(Integer(1) - m * sin(z) ** Integer(2)) *
sin(Integer(2) * z)) /
(Integer(2) * (Integer(1) - n * sin(z) ** Integer(2)))))
elif diff_param == 1:
return (Integer(1) /
(sqrt(Integer(1) - m * sin(z) ** Integer(Integer(2))) *
(Integer(1) - n * sin(z) ** Integer(2))))
elif diff_param == 2:
return ((Integer(1) / (Integer(2) * (n - m))) *
(elliptic_e(z, m) / (m - Integer(1)) +
elliptic_pi(n, z, m) - (m * sin(Integer(2) * z)) /
(Integer(2) * (m - Integer(1)) *
sqrt(Integer(1) - m * sin(z) ** Integer(2)))))
def _derivative_(self, z, m, diff_param):
"""
EXAMPLES::
sage: x,m = var('x,m')
sage: elliptic_f(x,m).diff(x)
1/sqrt(-m*sin(x)^2 + 1)
sage: elliptic_f(x,m).diff(m)
-1/2*elliptic_f(x, m)/m
+ 1/4*sin(2*x)/(sqrt(-m*sin(x)^2 + 1)*(m - 1))
- 1/2*elliptic_e(x, m)/((m - 1)*m)
"""
if diff_param == 0:
return Integer(1) / sqrt(Integer(1) - m * sin(z)**Integer(2))
elif diff_param == 1:
return (elliptic_e(z, m) / (Integer(2) * (Integer(1) - m) * m) -
elliptic_f(z, m) / (Integer(2) * m) -
(sin(Integer(2) * z) /
(Integer(4) * (Integer(1) - m) *
sqrt(Integer(1) - m * sin(z)**Integer(2)))))
def _derivative_(self, z, m, diff_param):
"""
EXAMPLES::
sage: x,m = var('x,m')
sage: elliptic_f(x,m).diff(x)
1/sqrt(-m*sin(x)^2 + 1)
sage: elliptic_f(x,m).diff(m)
-1/2*elliptic_f(x, m)/m
+ 1/4*sin(2*x)/(sqrt(-m*sin(x)^2 + 1)*(m - 1))
- 1/2*elliptic_e(x, m)/((m - 1)*m)
"""
if diff_param == 0:
return Integer(1) / sqrt(Integer(1) - m * sin(z) ** Integer(2))
elif diff_param == 1:
return (elliptic_e(z, m) / (Integer(2) * (Integer(1) - m) * m) -
elliptic_f(z, m) / (Integer(2) * m) -
(sin(Integer(2) * z) /
(Integer(4) * (Integer(1) - m) *
sqrt(Integer(1) - m * sin(z) ** Integer(2)))))
def _derivative_(self, z, m, diff_param):
"""
EXAMPLES::
sage: x,z = var('x,z')
sage: elliptic_e(z, x).diff(z, 1)
sqrt(-x*sin(z)^2 + 1)
sage: elliptic_e(z, x).diff(x, 1)
1/2*(elliptic_e(z, x) - elliptic_f(z, x))/x
"""
if diff_param == 0:
return sqrt(Integer(1) - m * sin(z)**Integer(2))
elif diff_param == 1:
return (elliptic_e(z, m) - elliptic_f(z, m)) / (Integer(2) * m)
def _derivative_(self, z, m, diff_param):
"""
EXAMPLES::
sage: x,z = var('x,z')
sage: elliptic_e(z, x).diff(z, 1)
sqrt(-x*sin(z)^2 + 1)
sage: elliptic_e(z, x).diff(x, 1)
1/2*(elliptic_e(z, x) - elliptic_f(z, x))/x
"""
if diff_param == 0:
return sqrt(Integer(1) - m * sin(z) ** Integer(2))
elif diff_param == 1:
return (elliptic_e(z, m) - elliptic_f(z, m)) / (Integer(2) * m)
def dst(self):
"""
Implements a discrete Sine transform
EXAMPLES:
sage: J = range(5)
sage: I = CC.0; pi = CC(pi)
sage: A = [exp(-2*pi*i*I/5) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dst() # discrete sine
Indexed sequence: [1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I]
indexed by [0, 1, 2, 3, 4]
"""
from sage.symbolic.constants import pi
F = self.base_ring() ## elements must be coercible into RR
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
PI = F(pi)
FT = [sum([S[i]*sin(2*PI*i/N) for i in J]) for j in J]
return IndexedSequence(FT,J)
def dst(self):
"""
Implements a discrete Sine transform
EXAMPLES:
sage: J = range(5)
sage: I = CC.0; pi = CC(pi)
sage: A = [exp(-2*pi*i*I/5) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dst() # discrete sine
Indexed sequence: [1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I, 1.11022302462516e-16 - 2.50000000000000*I]
indexed by [0, 1, 2, 3, 4]
"""
from sage.symbolic.constants import pi
F = self.base_ring() ## elements must be coercible into RR
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
PI = F(pi)
FT = [sum([S[i]*sin(2*PI*i/N) for i in J]) for j in J]
return IndexedSequence(FT,J)
def rotation_list(tilt,turn):
from sage.functions.all import sin, cos
return [ sin(tilt*pi/180.0), sin(turn*pi/180.0), cos(tilt*pi/180.0), cos(turn*pi/180.0) ]
