def test_rewrite(): assert besselj(n, z).rewrite(jn) == sqrt(2 * z / pi) * jn(n - S.Half, z) assert bessely(n, z).rewrite(yn) == sqrt(2 * z / pi) * yn(n - S.Half, z) assert besseli(n, z).rewrite(besselj) == \ exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z) assert besselj(n, z).rewrite(besseli) == \ exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z) nu = randcplx() assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z) assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z) assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z) assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z) assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z) assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z) # check that a rewrite was triggered, when the order is set to a generic # symbol 'nu' assert yn(nu, z) != yn(nu, z).rewrite(jn) assert hn1(nu, z) != hn1(nu, z).rewrite(jn) assert hn2(nu, z) != hn2(nu, z).rewrite(jn) assert jn(nu, z) != jn(nu, z).rewrite(yn) assert hn1(nu, z) != hn1(nu, z).rewrite(yn) assert hn2(nu, z) != hn2(nu, z).rewrite(yn) # rewriting spherical bessel functions (SBFs) w.r.t. besselj, bessely is # not allowed if a generic symbol 'nu' is used as the order of the SBFs # to avoid inconsistencies (the order of bessel[jy] is allowed to be # complex-valued, whereas SBFs are defined only for integer orders) order = nu for f in (besselj, bessely): assert hn1(order, z) == hn1(order, z).rewrite(f) assert hn2(order, z) == hn2(order, z).rewrite(f) assert jn(order, z).rewrite(besselj) == sqrt(2) * sqrt(pi) * sqrt( 1 / z) * besselj(order + S.Half, z) / 2 assert jn(order, z).rewrite(bessely) == (-1)**nu * sqrt(2) * sqrt(pi) * sqrt( 1 / z) * bessely(-order - S.Half, z) / 2 # for integral orders rewriting SBFs w.r.t bessel[jy] is allowed N = Symbol('n', integer=True) ri = randint(-11, 10) for order in (ri, N): for f in (besselj, bessely): assert yn(order, z) != yn(order, z).rewrite(f) assert jn(order, z) != jn(order, z).rewrite(f) assert hn1(order, z) != hn1(order, z).rewrite(f) assert hn2(order, z) != hn2(order, z).rewrite(f) for func, refunc in product((yn, jn, hn1, hn2), (jn, yn, besselj, bessely)): assert tn(func(ri, z), func(ri, z).rewrite(refunc), z)
def test_rewrite(): from sympy import polar_lift, exp, I assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S(1)/2, z) assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S(1)/2, z) assert besseli(n, z).rewrite(besselj) == \ exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z) assert besselj(n, z).rewrite(besseli) == \ exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z) nu = randcplx() assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z) assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z) assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z) assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z) assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z) assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z) assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z) # check that a rewrite was triggered, when the order is set to a generic # symbol 'nu' assert yn(nu, z) != yn(nu, z).rewrite(jn) assert hn1(nu, z) != hn1(nu, z).rewrite(jn) assert hn2(nu, z) != hn2(nu, z).rewrite(jn) assert jn(nu, z) != jn(nu, z).rewrite(yn) assert hn1(nu, z) != hn1(nu, z).rewrite(yn) assert hn2(nu, z) != hn2(nu, z).rewrite(yn) # rewriting spherical bessel functions (SBFs) w.r.t. besselj, bessely is # not allowed if a generic symbol 'nu' is used as the order of the SBFs # to avoid inconsistencies (the order of bessel[jy] is allowed to be # complex-valued, whereas SBFs are defined only for integer orders) order = nu for f in (besselj, bessely): assert hn1(order, z) == hn1(order, z).rewrite(f) assert hn2(order, z) == hn2(order, z).rewrite(f) assert jn(order, z).rewrite(besselj) == sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(order + S(1)/2, z)/2 assert jn(order, z).rewrite(bessely) == (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-order - S(1)/2, z)/2 # for integral orders rewriting SBFs w.r.t bessel[jy] is allowed N = Symbol('n', integer=True) ri = randint(-11, 10) for order in (ri, N): for f in (besselj, bessely): assert yn(order, z) != yn(order, z).rewrite(f) assert jn(order, z) != jn(order, z).rewrite(f) assert hn1(order, z) != hn1(order, z).rewrite(f) assert hn2(order, z) != hn2(order, z).rewrite(f) for func, refunc in product((yn, jn, hn1, hn2), (jn, yn, besselj, bessely)): assert tn(func(ri, z), func(ri, z).rewrite(refunc), z)
def test_meromorphic(): assert besselj(2, x).is_meromorphic(x, 1) == True assert besselj(2, x).is_meromorphic(x, 0) == True assert besselj(2, x).is_meromorphic(x, oo) == False assert besselj(S(2) / 3, x).is_meromorphic(x, 1) == True assert besselj(S(2) / 3, x).is_meromorphic(x, 0) == False assert besselj(S(2) / 3, x).is_meromorphic(x, oo) == False assert besselj(x, 2 * x).is_meromorphic(x, 2) == False assert besselk(0, x).is_meromorphic(x, 1) == True assert besselk(2, x).is_meromorphic(x, 0) == True assert besseli(0, x).is_meromorphic(x, 1) == True assert besseli(2, x).is_meromorphic(x, 0) == True assert bessely(0, x).is_meromorphic(x, 1) == True assert bessely(0, x).is_meromorphic(x, 0) == False assert bessely(2, x).is_meromorphic(x, 0) == True assert hankel1(3, x**2 + 2 * x).is_meromorphic(x, 1) == True assert hankel1(0, x).is_meromorphic(x, 0) == False assert hankel2(11, 4).is_meromorphic(x, 5) == True assert hn1(6, 7 * x**3 + 4).is_meromorphic(x, 7) == True assert hn2(3, 2 * x).is_meromorphic(x, 9) == True assert jn(5, 2 * x + 7).is_meromorphic(x, 4) == True assert yn(8, x**2 + 11).is_meromorphic(x, 6) == True