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_yn():
z = symbols("z")
assert myn(0, z) == -cos(z)/z
assert myn(1, z) == -cos(z)/z**2 - sin(z)/z
assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z))
assert expand_func(yn(n, z)) == yn(n, z)
# SBFs not defined for complex-valued orders
assert yn(2+3j, 5.2+0.3j).evalf() == yn(2+3j, 5.2+0.3j)
assert eq([yn(2, 5.2+0.3j).evalf(10)],
[0.185250342 + 0.01489557397*I])
def test_bessel_rand():
assert td(besselj(randcplx(), z), z)
assert td(bessely(randcplx(), z), z)
assert td(besseli(randcplx(), z), z)
assert td(besselk(randcplx(), z), z)
assert td(hankel1(randcplx(), z), z)
assert td(hankel2(randcplx(), z), z)
assert td(jn(randcplx(), z), z)
assert td(yn(randcplx(), z), 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(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)
def test_yn():
z = symbols("z")
assert myn(0, z) == -cos(z)/z
assert myn(1, z) == -cos(z)/z**2 - sin(z)/z
assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z))
assert expand_func(yn(n, z)) == yn(n, z)
def myn(n, z):
return expand_func(yn(n, z))
def myn(n, z): return expand_func(yn(n,z)) def test_jn():
def test_rewrite():
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)
def test_yn():
z = symbols("z")
assert yn(0, z) == -cos(z)/z
assert yn(1, z) == -cos(z)/z**2-sin(z)/z
assert yn(2, z) == -((3/z**3-1/z)*cos(z)+(3/z**2)*sin(z))
def test_sympify_yn():
assert S(15) in yn(3, pi).atoms()
assert yn(3, pi) == 15/pi**4 - 6/pi**2
def test_yn():
z = symbols("z")
assert myn(0, z) == -cos(z) / z
assert myn(1, z) == -cos(z) / z**2 - sin(z) / z
assert myn(2, z) == -((3 / z**3 - 1 / z) * cos(z) + (3 / z**2) * sin(z))
assert expand_func(yn(n, z)) == yn(n, z)
def myn(n, z):
return expand_func(yn(n, z))
def test_sympify_yn():
assert S(15) in yn(3, pi).atoms()
assert yn(3, pi) == 15 / pi**4 - 6 / pi**2
def test_yn():
z = symbols("z")
assert yn(0, z) == -cos(z) / z
assert yn(1, z) == -cos(z) / z**2 - sin(z) / z
assert yn(2, z) == -((3 / z**3 - 1 / z) * cos(z) + (3 / z**2) * sin(z))
def test_rewrite():
from sympy import polar_lift, exp, I
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)
