def test_modulus_transformations(phi, n, m):
assume(m != 0 and sin(phi)**2 != 0)
# https://dlmf.nist.gov/19.7.E6
kappap = 1 / (1 + m)**0.5
mu = m * kappap**2
ctheta = (1 + m * sin(phi)**2) * kappap**2 / sin(phi)**2
n1 = (n + m) * kappap**2
# https://dlmf.nist.gov/19.7.E4
cbeta = m / sin(phi + 0j)**2
for left, right in ((ellipkinc(phi, 1 / m),
(m**0.5 * ellipkinc(None, m, cbeta))),
(ellipeinc(phi, 1 / m),
(ellipeinc(None, m, cbeta) -
(1 - m) * ellipkinc(None, m, cbeta)) / m**0.5),
(ellippiinc(n, phi, 1 / m),
m**0.5 * ellippiinc(m * n, None, m, cbeta))):
# TODO: properly analytically continue
assert nice_and_close(left.real, right.real)
# https://dlmf.nist.gov/19.7.E5
assert nice_and_close(ellipkinc(phi, -m),
kappap * ellipkinc(None, mu, ctheta))
assert nice_and_close(ellipeinc(
phi, -m), (ellipeinc(None, mu, ctheta) - mu *
((ctheta - 1) / (ctheta - mu) / ctheta)**0.5) / kappap)
assert nice_and_close(
ellippiinc(n, phi, -m),
(kappap / n1) * (mu * ellipkinc(None, mu, ctheta) +
kappap**2 * n * ellippiinc(n1, None, mu, ctheta)))
def test_every_phi_but_zero(phi):
assume(sin(phi) != 0 and not isnan(ellippiinc(1, phi, 1)))
c = 1 / sin(phi)**2
# https://dlmf.nist.gov/19.6.E12
assert nice_and_close(ellippiinc(1, phi, 1),
(elliprc(c, c - 1) + c**0.5 / (c - 1)) / 2)
def test_argument_transformations(phi, n, m):
# https://dlmf.nist.gov/19.7.E7
assume(sinh(phi) != 0)
cpsi = 1 + sinh(phi)**(-2)
for left, right in ((ellipkinc(1j * phi,
m), ellipkinc(None, 1 - m, cpsi)),
(ellipeinc(1j * phi,
m), ellipkinc(None, 1 - m, cpsi) -
ellipeinc(None, 1 - m, cpsi) +
(1 if phi >= 0 else -1) * sinh(phi) *
(1 - (1 - m) / cpsi)**0.5),
(ellippiinc(n, 1j * phi, m),
(ellipkinc(None, 1 - m, cpsi) -
n * ellippiinc(1 - n, None, 1 - m, cpsi)) /
(1 - n))):
# TODO: properly analytically continue
assert nice_and_close(abs(left.imag), abs(right.real))
def test_ellippiinc_every_phi_m(phi, m):
assume(sin(phi) != 0)
c = 1 / sin(phi)**2
D = (1 - m / c)**0.5
# https://dlmf.nist.gov/19.6.E12
assert nice_and_close(ellippiinc(m, phi, 0), elliprc(c - 1, c - m))
assert nice_and_close(ellippiinc(
m, phi, 1), (elliprc(c, c - 1) - m * elliprc(c, c - m)) / (1 - m))
assert nice_and_close(ellippiinc(0, phi, m), ellipkinc(phi, m))
# https://dlmf.nist.gov/19.6.E13
assert nice_and_close(
ellippiinc(m, phi, m),
(ellipeinc(phi, m) - m / D * sin(phi) * cos(phi)) / (1 - m))
assert nice_and_close(
ellippiinc(1, phi, m),
ellipkinc(phi, m) - (ellipeinc(phi, m) - D * tan(phi)) / (1 - m))
def test_every_phi(phi):
# https://dlmf.nist.gov/19.6.E8
assert nice_and_close(ellipkinc(phi, 1), asinh(tan(phi)))
# https://dlmf.nist.gov/19.6.E9, line 4
assert nice_and_close(ellipeinc(phi, 1), sin(phi))
#
# https://dlmf.nist.gov/19.6.E11, line 3
assert nice_and_close(ellippiinc(1, phi, 0), tan(phi))
def test_ellippi(n, m):
assert isclose(ellippi(n, m), ellippiinc(n, pi / 2, m))