def z_am(u, m):
"""Jacobi amplitude function (eq 16.1.5, [Abramowitz]_)."""
snM = ellipfun('sn', u=u, m=m)
cnM = ellipfun('cn', u=u, m=m)
if (0 <= cnM <= 1):
phi = mp.asin(snM)
elif (-1 <= cnM < 0):
if (snM >= 0):
phi = mp.pi - mp.asin(snM)
else:
phi = mp.asin(snM) - mp.pi
else:
print("This function only handles real 'phi' values.")
return phi
def z_am(u, m):
"""Jacobi amplitude function (eq 16.1.5, [Abramowitz]_)."""
snM = ellipfun('sn', u=u, m=m)
cnM = ellipfun('cn', u=u, m=m)
if (0 <= cnM <= 1):
phi = mp.asin(snM)
elif (-1 <= cnM < 0):
if (snM >= 0):
phi = mp.pi - mp.asin(snM)
else:
phi = mp.asin(snM) - mp.pi
else:
print "This function only handles real 'phi' values."
return phi
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u=-M, m=m)
xbar = x3 * mp.sqrt(
(x**2 - 1) / (x**2 - x3**2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar),
m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if f.imag / f.real > 1e-10:
print("imaginary part of the Zolotarev function is not negligible!")
print("f_imaginary = ", f.imag)
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def z_am(u, m):
r"""
A function evaluate Jacobi amplitude function (eq 16.1.5, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi amplitude function evaluated for the argument `u` and parameter `m`
"""
snM = ellipfun('sn', u=u, m=m)
cnM = ellipfun('cn', u=u, m=m)
if (0 <= cnM <= 1):
phi = mp.asin(snM)
elif (-1 <= cnM < 0):
if (snM >= 0):
phi = mp.pi - mp.asin(snM)
else:
phi = mp.asin(snM) - mp.pi
else:
print("This function only handles real 'phi' values.")
return phi
def z_am(u, m):
r"""
A function evaluate Jacobi amplitude function (eq 16.1.5, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi amplitude function evaluated for the argument `u` and parameter `m`
"""
snM = ellipfun('sn', u=u, m=m)
cnM = ellipfun('cn', u=u, m=m)
if (0 <= cnM <= 1):
phi = mp.asin(snM)
elif (-1 <= cnM < 0):
if (snM >= 0):
phi = mp.pi - mp.asin(snM)
else:
phi = mp.asin(snM) - mp.pi
else:
print "This function only handles real 'phi' values."
return phi
def z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u=-M, m=m)
xbar = x3 * mp.sqrt(
(x**2 - 1) / (x**2 - x3**2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar),
m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print("imaginary part of the Zolotarev function is not negligible!")
print("f_imaginary = ", f.imag)
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f