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_addition_theorems(x, y, z, a, p):
xa, ya, za, pa = map(partial(add, a), (x, y, z, p))
# https://dlmf.nist.gov/19.26.E12
b = (xa**0.5 * y + x**0.5 * ya)**2 / a**2 - x
# https://dlmf.nist.gov/19.26.E11
assert nice_and_close(elliprc(x, y),
elliprc(xa, ya) + elliprc(x + b, y + b))
# https://dlmf.nist.gov/19.26.E5
b = ((x * y * z)**0.5 +
(xa * ya * za)**0.5)**2 / a**2 - (x + y + z + a)
xb, yb, zb, pb = map(partial(add, b), (x, y, z, p))
# https://dlmf.nist.gov/19.26.E10
c = p * pa * pb
d = (p - x) * (p - y) * (p - z)
# # https://dlmf.nist.gov/19.26.E1
assert nice_and_close(elliprf(x, y, z),
elliprf(xa, ya, za) + elliprf(xb, yb, zb))
# https://dlmf.nist.gov/19.26.E7
assert nice_and_close(
elliprd(x, y, z) - 3 / (z * za * zb)**0.5,
elliprd(xa, ya, za) + elliprd(xb, yb, zb))
# https://dlmf.nist.gov/19.26.E8
assert nice_and_close(
2 * elliprg(x, y, z) + a * elliprf(xa, ya, za) +
b * elliprf(xb, yb, zb) + (x + y + z + a + b)**0.5,
2 * (elliprg(xa, ya, za) + elliprg(xb, yb, zb)))
# https://dlmf.nist.gov/19.26.E9
assert nice_and_close(
elliprj(x, y, z, p) - 3 * elliprc(c - d, c),
elliprj(xa, ya, za, pa) + elliprj(xb, yb, zb, pb))
def test_ellippi_m_equals_n(m):
# https://dlmf.nist.gov/19.6.E1, line 4
assert nice_and_close(ellippi(m, m), ellipe(m) / (1 - m))
# https://dlmf.nist.gov/19.6.E2
assert nice_and_close(ellippi(-m**0.5, m),
pi / 4 / (1 + m**0.5) + ellipk(m) / 2)
def test_elliprg(x, y, z, l):
assert nice_and_close(elliprg(x, x, x), x**0.5)
assert nice_and_close(elliprg(l * x, l * y, l * z),
elliprg(x, y, z) *
l**0.5) # TODO: elliprc sign issue
assert nice_and_close(elliprg(0, y, y), pi / 4 * y**0.5)
assert nice_and_close(elliprg(0, 0, z), z**0.5 / 2)
assert nice_and_close(2 * elliprg(x, y, y), y * elliprc(x, y) + x**0.5)
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_complete_1(y, z):
# https://dlmf.nist.gov/19.21.E2
assert nice_and_close(3 * elliprf(0, y, z),
z * elliprd(0, y, z) + y * elliprd(0, z, y))
# https://dlmf.nist.gov/19.21.E3
assert nice_and_close(6 * elliprg(0, y, z),
y * z * (elliprd(0, y, z) + elliprd(0, z, y)))
assert nice_and_close(
6 * elliprg(0, y, z),
3 * z * elliprf(0, y, z) + z * (y - z) * elliprd(0, y, z))
def test_complete_2(z: complex):
assume(not z.imag == 0)
sign = (1 if phase(z) > 0 else -1) * 1j
# https://dlmf.nist.gov/19.21.E4
assert nice_and_close(elliprf(0, z - 1, z),
elliprf(0, 1 - z, 1) - sign * elliprf(0, z, 1))
# https://dlmf.nist.gov/19.21.E5
assert nice_and_close(
2 * elliprg(0, z - 1, z),
2 * (elliprg(0, 1 - z, 1) + sign * elliprg(0, z, 1)) +
(z - 1) * elliprf(0, 1 - z, 1) - sign * z * elliprf(0, z, 1))
def test_legendre_relation(m: complex):
k = m**0.5
mm = 1 - m
kp = mm**0.5
# https://dlmf.nist.gov/19.7.E2
assert nice_and_close(ellipk((+1j * k / kp)**2), kp * ellipk(m))
assert nice_and_close(ellipk((-1j * kp / k)**2), k * ellipk(mm))
assert nice_and_close(ellipe((+1j * k / kp)**2), ellipe(m) / kp)
assert nice_and_close(ellipe((-1j * kp / k)**2), ellipe(mm) / k)
# https://dlmf.nist.gov/19.7.E3
assert nice_and_close(
ellipk(1 / m),
k * (ellipk(m) - (1 if m.imag > 0 else -1) * 1j * ellipk(mm)))
assert nice_and_close(
ellipk(1 / mm),
kp * (ellipk(mm) + (1 if m.imag >= 0 else -1) * 1j * ellipk(m)))
assert nice_and_close(ellipe(
1 / m), (ellipe(m) + (1 if m.imag > 0 else -1) * 1j * ellipe(mm) -
(mm * ellipk(m) +
(1 if m.imag > 0 else -1) * 1j * m * ellipk(mm))) / k)
assert nice_and_close(ellipe(
1 / mm), (ellipe(mm) - (1 if m.imag > 0 else -1) * 1j * ellipe(m) -
(m * ellipk(mm) -
(1 if m.imag > 0 else -1) * 1j * mm * ellipk(m))) / kp)
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_trivial_zeros(n):
try:
# these are hard-coded, but we want it to stay like that,
# if we un-hard-code them
assert nice_and_close(abs(zeta(-2 * n)), 0)
except RuntimeError: # overflow
assume(False)
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_elliprj(x, y, z, p):
# TODO: properly analytically continue
assume(distinct(x, y, z, p))
# https://dlmf.nist.gov/19.21.E13
q = (y - x) * (z - x) / (p - x) + x
# # https://dlmf.nist.gov/19.21.E12
# TODO: elliprc sign issue
assert nice_and_close(
((p - x) * elliprj(x, y, z, p) +
(q - x) * elliprj(x, y, z, q)).real,
(3 * (elliprf(x, y, z) - elliprc(y * z / x, p * q / x))).real)
# https://dlmf.nist.gov/19.21.E15
# special case of above with x=0
q = y * z / p
assert nice_and_close(
(p * elliprj(0, y, z, p) + q * elliprj(0, y, z, q)).real,
3 * elliprf(0, y, z).real)
def test_incomplete(x, y, z):
# TODO: sign of sqrt(x, y, z)
assume(distinct(x, y, z))
# https://dlmf.nist.gov/19.21.E7
assert nice_and_close(
abs(3 * elliprf(x, y, z) - ((x - y) * elliprd(y, z, x) +
(z - y) * elliprd(x, y, z))),
abs(3 * (y / x / z)**0.5))
# https://dlmf.nist.gov/19.21.E8
assert nice_and_close(
abs(elliprd(y, z, x) + elliprd(z, x, y) + elliprd(x, y, z)),
abs(3 * (x * y * z)**(-0.5)))
# https://dlmf.nist.gov/19.21.E9
assert nice_and_close(
x * elliprd(y, z, x) + y * elliprd(z, x, y) + z * elliprd(x, y, z),
3 * elliprf(x, y, z))
# https://dlmf.nist.gov/19.21.E10
assert nice_and_close(
abs(2 * elliprg(x, y, z) - (z * elliprf(x, y, z) - (x - z) *
(y - z) * elliprd(x, y, z) / 3)),
abs((x * y / z)**0.5))
# https://dlmf.nist.gov/19.21.E11
assert nice_and_close(
6 * elliprg(x, y, z), 3 * (x + y + z) * elliprf(x, y, z) -
sum(_x**2 * elliprd(_y, _z, _x)
for _x, _y, _z in circular_shifts((x, y, z))))
assert nice_and_close(
6 * elliprg(x, y, z),
sum(_x * (_y + _z) * elliprd(_y, _z, _x)
for _x, _y, _z in circular_shifts((x, y, z))))
def test_duplication_formulas(x, y, z, p):
xs, ys, zs = (_**0.5 for _ in (x, y, z))
ss = xs + ys + zs
# https://dlmf.nist.gov/19.26.E25
a = y + 2 * xs * ys
assert nice_and_close(elliprc(x, y), 2 * elliprc(x + a, y + a))
# https://dlmf.nist.gov/19.26.E19
a = sum(map(product, combinations((xs, ys, zs), 2)))
xa, ya, za, pa = map(partial(add, a), (x, y, z, p))
# https://dlmf.nist.gov/19.26.E18
assert nice_and_close(elliprf(x, y, z), 2 * elliprf(xa, ya, za))
assert nice_and_close(elliprf(x, y, z),
elliprf(xa / 4, ya / 4, za / 4))
# https://dlmf.nist.gov/19.26.E20
assert nice_and_close(elliprd(x, y, z),
2 * elliprd(xa, ya, za) + 3 / z**0.5 / za)
# https://dlmf.nist.gov/19.26.E21
assert nice_and_close(
2 * elliprg(x, y, z),
4 * elliprg(xa, ya, za) - a * elliprf(x, y, z) - ss)
# https://dlmf.nist.gov/19.26.E22
assert nice_and_close(
elliprj(x, y, z, p), 2 * elliprj(xa, ya, za, pa) + 3 * elliprc(
(p * ss + xs * ys * zs)**2, p * pa**2))
def test_complete_3(y, z, p):
# https://dlmf.nist.gov/19.21.E6
assume(distinct(y, z, p))
if z < y < p or p < y < z:
y, z = z, y
r = (y - p) / (y - z)
assume(r > 0)
assert nice_and_close(
(r * p)**0.5 / z * elliprj(0, y, z, p),
(r - 1) * elliprf(0, y, z) * elliprd(p, r * z, z) +
elliprd(0, y, z) * elliprf(p, r * z, z))
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_elliprj(x, y, z, p, l):
assert nice_and_close(elliprj(x, x, x, x), x**(-1.5))
assert nice_and_close(elliprj(l * x, l * y, l * z, l * p),
elliprj(x, y, z, p) / l**1.5)
assert nice_and_close(elliprj(x, y, z, z), elliprd(x, y, z))
assert isinf(elliprj(0, 0, z, p))
assert nice_and_close(elliprj(x, x, x, p), elliprd(p, p, x))
assert nice_and_close(
elliprj(x, x, x, p).real,
3 * (elliprc(x, p) - x**(-0.5)) / (x - p))
assert nice_and_close(elliprj(x, y, y, y), elliprd(x, y, y))
# assert nice_and_close(elliprj(0, y, z, (y*z)**0.5), 1.5/(y*z)**0.5 * elliprf(0, y, z))
# assert nice_and_close(elliprj(0, y, z, -(y*z)**0.5), -1.5/(y*z)**0.5 * elliprf(0, y, z))
# TODO: elliprc sign issue
assume(distinct(x, y, z))
p = x + ((y - x) * (z - x))**0.5
assert nice_and_close(
(p - x) * elliprj(x, y, z, p),
1.5 * (elliprf(x, y, z) - x**0.5 * elliprc(y * z, p**2)))
def test_ellippi_every_n(n):
# https://dlmf.nist.gov/19.6.E3
assert nice_and_close(ellippi(n, 0), pi / 2 / (1 - n)**0.5)
def test_values(self, ours, theirs, nargs, x, y, z, p):
args = (x, y, z, p)[:nargs]
assert nice_and_close(ours(*args), theirs(*args))
def test_legendre_relation(z):
# https://dlmf.nist.gov/19.21.E1
assert nice_and_close(
elliprf(0, z + 1, z) * elliprd(0, z + 1, 1) +
elliprd(0, z + 1, z) * elliprf(0, z + 1, 1), 3 * pi / 2 / z)
def test_elliprj2(y, p):
assume(distinct(y, p) and y != 0)
assert nice_and_close(
elliprj(0, y, y, -p).real, -1.5 * pi / y**0.5 / (y + p))
def test_elliprj1(x, y, p):
assume(distinct(y, p))
assert nice_and_close(elliprj(0, y, y, p),
1.5 * pi / (y * p**0.5 + p * y**0.5))
assert nice_and_close(elliprj(x, y, y, p),
3 * (elliprc(x, y) - elliprc(x, p)) / (p - y))
def test_elliprc(x, y, l):
assert nice_and_close(elliprc(x, x), x**(-0.5))
assert nice_and_close(elliprc(0, x), pi / 2 / x**0.5)
assert nice_and_close(elliprc(l * x, l * y), elliprc(x, y) / l**0.5)
def test_elliprd1(x, y, z):
assume(x != y and y != 0 and x != z and x * z != 0)
assert nice_and_close(elliprd(x, y, y),
1.5 * (elliprc(x, y) - x**0.5 / y) / (y - x))
assert nice_and_close(elliprd(x, x, z),
3 * (elliprc(z, x) - z**(-0.5)) / (z - x))
def test_elliprd(x, y, z, l):
assert nice_and_close(elliprd(x, x, x), x**(-1.5))
assert nice_and_close(elliprd(l * x, l * y, l * z),
elliprd(x, y, z) / l**1.5)
assert nice_and_close(elliprd(0, y, y), 3 * pi / 4 / y**1.5)
assert isinf(elliprd(0, 0, z))
def test_elliprf(x, y, z, l):
assert nice_and_close(elliprf(x, x, x), x**(-0.5))
assert nice_and_close(elliprf(0, y, y), pi / 2 / y**0.5)
assert nice_and_close(elliprf(l * x, l * y, l * z),
elliprf(x, y, z) / l**0.5)
assert isinf(elliprf(0, 0, z))
