from sympy.core.numbers import (I, Rational, oo, pi, zoo) from sympy.core.singleton import S from sympy.core.symbol import (Dummy, Symbol) from sympy.functions.elementary.hyperbolic import atanh from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (sin, tan) from sympy.functions.special.gamma_functions import gamma from sympy.functions.special.hyper import (hyper, meijerg) from sympy.integrals.integrals import Integral from sympy.series.order import O from sympy.functions.special.elliptic_integrals import (elliptic_k as K, elliptic_f as F, elliptic_e as E, elliptic_pi as P) from sympy.core.random import (test_derivative_numerically as td, random_complex_number as randcplx, verify_numerically as tn) from sympy.abc import z, m, n i = Symbol('i', integer=True) j = Symbol('k', integer=True, positive=True) t = Dummy('t') def test_K(): assert K(0) == pi/2 assert K(S.Half) == 8*pi**Rational(3, 2)/gamma(Rational(-1, 4))**2 assert K(1) is zoo assert K(-1) == gamma(Rational(1, 4))**2/(4*sqrt(2*pi)) assert K(oo) == 0 assert K(-oo) == 0 assert K(I*oo) == 0 assert K(-I*oo) == 0 assert K(zoo) == 0 assert K(z).diff(z) == (E(z) - (1 - z)*K(z))/(2*z*(1 - z)) assert td(K(z), z) zi = Symbol('z', real=False) assert K(zi).conjugate() == K(zi.conjugate()) zr = Symbol('z', negative=True) assert K(zr).conjugate() == K(zr) assert K(z).rewrite(hyper) == \ (pi/2)*hyper((S.Half, S.Half), (S.One,), z) assert tn(K(z), (pi/2)*hyper((S.Half, S.Half), (S.One,), z)) assert K(z).rewrite(meijerg) == \ meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2 assert tn(K(z), meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2) assert K(z).series(z) == pi/2 + pi*z/8 + 9*pi*z**2/128 + \ 25*pi*z**3/512 + 1225*pi*z**4/32768 + 3969*pi*z**5/131072 + O(z**6) assert K(m).rewrite(Integral).dummy_eq( Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, pi/2))) def test_F(): assert F(z, 0) == z assert F(0, m) == 0 assert F(pi*i/2, m) == i*K(m) assert F(z, oo) == 0 assert F(z, -oo) == 0 assert F(-z, m) == -F(z, m) assert F(z, m).diff(z) == 1/sqrt(1 - m*sin(z)**2) assert F(z, m).diff(m) == E(z, m)/(2*m*(1 - m)) - F(z, m)/(2*m) - \ sin(2*z)/(4*(1 - m)*sqrt(1 - m*sin(z)**2)) r = randcplx() assert td(F(z, r), z) assert td(F(r, m), m) mi = Symbol('m', real=False) assert F(z, mi).conjugate() == F(z.conjugate(), mi.conjugate()) mr = Symbol('m', negative=True) assert F(z, mr).conjugate() == F(z.conjugate(), mr) assert F(z, m).series(z) == \ z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6) assert F(z, m).rewrite(Integral).dummy_eq( Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, z))) def test_E(): assert E(z, 0) == z assert E(0, m) == 0 assert E(i*pi/2, m) == i*E(m) assert E(z, oo) is zoo assert E(z, -oo) is zoo assert E(0) == pi/2 assert E(1) == 1 assert E(oo) == I*oo assert E(-oo) is oo assert E(zoo) is zoo assert E(-z, m) == -E(z, m) assert E(z, m).diff(z) == sqrt(1 - m*sin(z)**2) assert E(z, m).diff(m) == (E(z, m) - F(z, m))/(2*m) assert E(z).diff(z) == (E(z) - K(z))/(2*z) r = randcplx() assert td(E(r, m), m) assert td(E(z, r), z) assert td(E(z), z) mi = Symbol('m', real=False) assert E(z, mi).conjugate() == E(z.conjugate(), mi.conjugate()) assert E(mi).conjugate() == E(mi.conjugate()) mr = Symbol('m', negative=True) assert E(z, mr).conjugate() == E(z.conjugate(), mr) assert E(mr).conjugate() == E(mr) assert E(z).rewrite(hyper) == (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), z) assert tn(E(z), (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), z)) assert E(z).rewrite(meijerg) == \ -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4 assert tn(E(z), -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4) assert E(z, m).series(z) == \ z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6) assert E(z).series(z) == pi/2 - pi*z/8 - 3*pi*z**2/128 - \ 5*pi*z**3/512 - 175*pi*z**4/32768 - 441*pi*z**5/131072 + O(z**6) assert E(z, m).rewrite(Integral).dummy_eq( Integral(sqrt(1 - m*sin(t)**2), (t, 0, z))) assert E(m).rewrite(Integral).dummy_eq( Integral(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', negative=True), \ Symbol('m', 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)))