def test_P():
assert P(0, z, m) == F(z, m)
assert P(1, z, m) == F(z, m) + \
(sqrt(1 - m*sin(z)**2)*tan(z) - E(z, m))/(1 - m)
assert P(n, i * pi / 2, m) == i * P(n, m)
assert P(n, z, 0) == atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1)
assert P(n, z,
n) == F(z, n) - P(1, z, n) + tan(z) / sqrt(1 - n * sin(z)**2)
assert P(oo, z, m) == 0
assert P(-oo, z, m) == 0
assert P(n, z, oo) == 0
assert P(n, z, -oo) == 0
assert P(0, m) == K(m)
assert P(1, m) is zoo
assert P(n, 0) == pi / (2 * sqrt(1 - n))
assert P(2, 1) is -oo
assert P(-1, 1) is oo
assert P(n, n) == E(n) / (1 - n)
assert P(n, -z, m) == -P(n, z, m)
ni, mi = Symbol('n', real=False), Symbol('m', real=False)
assert P(ni, z, mi).conjugate() == \
P(ni.conjugate(), z.conjugate(), mi.conjugate())
nr, mr = Symbol('n', real=True, negative=True), \
Symbol('m', real=True, negative=True)
assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())
assert P(n, z, m).diff(n) == (E(z, m) + (m - n) * F(z, m) / n +
(n**2 - m) * P(n, z, m) / n -
n * sqrt(1 - m * sin(z)**2) * sin(2 * z) /
(2 * (1 - n * sin(z)**2))) / (2 * (m - n) *
(n - 1))
assert P(n, z,
m).diff(z) == 1 / (sqrt(1 - m * sin(z)**2) * (1 - n * sin(z)**2))
assert P(
n, z,
m).diff(m) == (E(z, m) / (m - 1) + P(n, z, m) - m * sin(2 * z) /
(2 * (m - 1) * sqrt(1 - m * sin(z)**2))) / (2 * (n - m))
assert P(n,
m).diff(n) == (E(m) + (m - n) * K(m) / n +
(n**2 - m) * P(n, m) / n) / (2 * (m - n) * (n - 1))
assert P(n, m).diff(m) == (E(m) / (m - 1) + P(n, m)) / (2 * (n - m))
rx, ry = randcplx(), randcplx()
assert td(P(n, rx, ry), n)
assert td(P(rx, z, ry), z)
assert td(P(rx, ry, m), m)
assert P(n, z, m).series(z) == z + z**3*(m/6 + n/3) + \
z**5*(3*m**2/40 + m*n/10 - m/30 + n**2/5 - n/15) + O(z**6)
assert P(n, z, m).rewrite(Integral).dummy_eq(
Integral(1 / ((1 - n * sin(t)**2) * sqrt(1 - m * sin(t)**2)),
(t, 0, z)))
assert P(n, m).rewrite(Integral).dummy_eq(
Integral(1 / ((1 - n * sin(t)**2) * sqrt(1 - m * sin(t)**2)),
(t, 0, pi / 2)))
def test_P():
assert P(0, z, m) == F(z, m)
assert P(1, z, m) == F(z, m) + \
(sqrt(1 - m*sin(z)**2)*tan(z) - E(z, m))/(1 - m)
assert P(n, i*pi/2, m) == i*P(n, m)
assert P(n, z, 0) == atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1)
assert P(n, z, n) == F(z, n) - P(1, z, n) + tan(z)/sqrt(1 - n*sin(z)**2)
assert P(oo, z, m) == 0
assert P(-oo, z, m) == 0
assert P(n, z, oo) == 0
assert P(n, z, -oo) == 0
assert P(0, m) == K(m)
assert P(1, m) is zoo
assert P(n, 0) == pi/(2*sqrt(1 - n))
assert P(2, 1) is -oo
assert P(-1, 1) is oo
assert P(n, n) == E(n)/(1 - n)
assert P(n, -z, m) == -P(n, z, m)
ni, mi = Symbol('n', real=False), Symbol('m', real=False)
assert P(ni, z, mi).conjugate() == \
P(ni.conjugate(), z.conjugate(), mi.conjugate())
nr, mr = Symbol('n', real=True, negative=True), \
Symbol('m', real=True, negative=True)
assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())
assert P(n, z, m).diff(n) == (E(z, m) + (m - n)*F(z, m)/n +
(n**2 - m)*P(n, z, m)/n - n*sqrt(1 -
m*sin(z)**2)*sin(2*z)/(2*(1 - n*sin(z)**2)))/(2*(m - n)*(n - 1))
assert P(n, z, m).diff(z) == 1/(sqrt(1 - m*sin(z)**2)*(1 - n*sin(z)**2))
assert P(n, z, m).diff(m) == (E(z, m)/(m - 1) + P(n, z, m) -
m*sin(2*z)/(2*(m - 1)*sqrt(1 - m*sin(z)**2)))/(2*(n - m))
assert P(n, m).diff(n) == (E(m) + (m - n)*K(m)/n +
(n**2 - m)*P(n, m)/n)/(2*(m - n)*(n - 1))
assert P(n, m).diff(m) == (E(m)/(m - 1) + P(n, m))/(2*(n - m))
# These tests fail due to
# https://github.com/fredrik-johansson/mpmath/issues/571#issuecomment-777201962
# https://github.com/sympy/sympy/issues/20933#issuecomment-777080385
#
# rx, ry = randcplx(), randcplx()
# assert td(P(n, rx, ry), n)
# assert td(P(rx, z, ry), z)
# assert td(P(rx, ry, m), m)
assert P(n, z, m).series(z) == z + z**3*(m/6 + n/3) + \
z**5*(3*m**2/40 + m*n/10 - m/30 + n**2/5 - n/15) + O(z**6)
assert P(n, z, m).rewrite(Integral).dummy_eq(
Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z)))
assert P(n, m).rewrite(Integral).dummy_eq(
Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, pi/2)))
def test_P():
assert P(0, z, m) == F(z, m)
assert P(1, z, m) == F(z, m) + \
(sqrt(1 - m*sin(z)**2)*tan(z) - E(z, m))/(1 - m)
assert P(n, i * pi / 2, m) == i * P(n, m)
assert P(n, z, 0) == atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1)
assert P(n, z,
n) == F(z, n) - P(1, z, n) + tan(z) / sqrt(1 - n * sin(z)**2)
assert P(oo, z, m) == 0
assert P(-oo, z, m) == 0
assert P(n, z, oo) == 0
assert P(n, z, -oo) == 0
assert P(0, m) == K(m)
assert P(1, m) == zoo
assert P(n, 0) == pi / (2 * sqrt(1 - n))
assert P(2, 1) == -oo
assert P(-1, 1) == oo
assert P(n, n) == E(n) / (1 - n)
assert P(n, -z, m) == -P(n, z, m)
ni, mi = Symbol('n', real=False), Symbol('m', real=False)
assert P(ni, z, mi).conjugate() == \
P(ni.conjugate(), z.conjugate(), mi.conjugate())
nr, mr = Symbol('n', real=True, negative=True), \
Symbol('m', real=True, negative=True)
assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())
assert P(n, z, m).diff(n) == (E(z, m) + (m - n) * F(z, m) / n +
(n**2 - m) * P(n, z, m) / n -
n * sqrt(1 - m * sin(z)**2) * sin(2 * z) /
(2 * (1 - n * sin(z)**2))) / (2 * (m - n) *
(n - 1))
assert P(n, z,
m).diff(z) == 1 / (sqrt(1 - m * sin(z)**2) * (1 - n * sin(z)**2))
assert P(
n, z,
m).diff(m) == (E(z, m) / (m - 1) + P(n, z, m) - m * sin(2 * z) /
(2 * (m - 1) * sqrt(1 - m * sin(z)**2))) / (2 * (n - m))
assert P(n,
m).diff(n) == (E(m) + (m - n) * K(m) / n +
(n**2 - m) * P(n, m) / n) / (2 * (m - n) * (n - 1))
assert P(n, m).diff(m) == (E(m) / (m - 1) + P(n, m)) / (2 * (n - m))
rx, ry = randcplx(), randcplx()
assert td(P(n, rx, ry), n)
assert td(P(rx, z, ry), z)
assert td(P(rx, ry, m), m)
def test_P():
assert P(0, z, m) == F(z, m)
assert P(1, z, m) == F(z, m) + (sqrt(1 - m * sin(z) ** 2) * tan(z) - E(z, m)) / (1 - m)
assert P(n, i * pi / 2, m) == i * P(n, m)
assert P(n, z, 0) == atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1)
assert P(n, z, n) == F(z, n) - P(1, z, n) + tan(z) / sqrt(1 - n * sin(z) ** 2)
assert P(oo, z, m) == 0
assert P(-oo, z, m) == 0
assert P(n, z, oo) == 0
assert P(n, z, -oo) == 0
assert P(0, m) == K(m)
assert P(1, m) == zoo
assert P(n, 0) == pi / (2 * sqrt(1 - n))
assert P(2, 1) == -oo
assert P(-1, 1) == oo
assert P(n, n) == E(n) / (1 - n)
assert P(n, -z, m) == -P(n, z, m)
ni, mi = Symbol("n", real=False), Symbol("m", real=False)
assert P(ni, z, mi).conjugate() == P(ni.conjugate(), z.conjugate(), mi.conjugate())
nr, mr = Symbol("n", real=True, negative=True), Symbol("m", real=True, negative=True)
assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())
assert P(n, z, m).diff(n) == (
E(z, m)
+ (m - n) * F(z, m) / n
+ (n ** 2 - m) * P(n, z, m) / n
- n * sqrt(1 - m * sin(z) ** 2) * sin(2 * z) / (2 * (1 - n * sin(z) ** 2))
) / (2 * (m - n) * (n - 1))
assert P(n, z, m).diff(z) == 1 / (sqrt(1 - m * sin(z) ** 2) * (1 - n * sin(z) ** 2))
assert P(n, z, m).diff(m) == (
E(z, m) / (m - 1) + P(n, z, m) - m * sin(2 * z) / (2 * (m - 1) * sqrt(1 - m * sin(z) ** 2))
) / (2 * (n - m))
assert P(n, m).diff(n) == (E(m) + (m - n) * K(m) / n + (n ** 2 - m) * P(n, m) / n) / (2 * (m - n) * (n - 1))
assert P(n, m).diff(m) == (E(m) / (m - 1) + P(n, m)) / (2 * (n - m))
rx, ry = randcplx(), randcplx()
assert td(P(n, rx, ry), n)
assert td(P(rx, z, ry), z)
assert td(P(rx, ry, m), m)
assert P(n, z, m).series(z) == z + z ** 3 * (m / 6 + n / 3) + z ** 5 * (
3 * m ** 2 / 40 + m * n / 10 - m / 30 + n ** 2 / 5 - n / 15
) + O(z ** 6)