def test_besselsimp():
assert besselsimp(exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))) == \
besselj(y, z)
assert besselsimp(exp(-I*pi*a/2)*besseli(a, 2*sqrt(x)*exp_polar(I*pi/2))) == \
besselj(a, 2*sqrt(x))
assert besselsimp(sqrt(2)*sqrt(pi)*root(x, 4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-Rational(1, 2), sqrt(x)*exp_polar(I*pi/2)) *
besseli(a, sqrt(x)*exp_polar(I*pi/2))/2) == \
besselj(a, sqrt(x)) * cos(sqrt(x))
assert besselsimp(besseli(Rational(-1, 2), z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(a, z*exp_polar(-I*pi/2))) == \
exp(-I*pi*a/2)*besselj(a, z)
assert cosine_transform(1/t*sin(a/t), t, y) == \
sqrt(2)*sqrt(pi)*besselj(0, 2*sqrt(a)*sqrt(y))/2
def test_expand():
assert expand_func(besselj(Rational(1, 2), z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(Rational(1, 2), z).rewrite(yn)) == \
-sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert expand_func(besselj(I, z)) == besselj(I, z)
# Test simplify helper
assert simplify(besselj(Rational(1, 2),
z)) == sqrt(2) * sin(z) / (sqrt(pi) * sqrt(z))
# XXX: teach sin/cos to work around arguments like
# x*exp_polar(I*pi*n/2). Then change besselsimp -> expand_func
assert besselsimp(besselj(Rational(1, 2),
z)) == sqrt(2) * sin(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(besselj(Rational(-1, 2),
z)) == sqrt(2) * cos(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(besselj(Rational(5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselj(-Rational(5, 2), z)) == \
-sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(1, 2), z)) == \
-(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(-1, 2),
z)) == sqrt(2) * sin(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(bessely(Rational(5, 2), z)) == \
sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(-5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(1, 2),
z)) == sqrt(2) * sinh(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(besseli(Rational(-1, 2), z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(5, 2), z)) == \
sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(-5, 2), z)) == \
sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselk(Rational(1, 2), z)) == \
besselsimp(besselk(Rational(-1, 2), z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
assert besselsimp(besselk(Rational(5, 2), z)) == \
besselsimp(besselk(Rational(-5, 2), z)) == \
sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**Rational(5, 2))
def check(eq, ans):
return tn(eq, ans) and eq == ans
rn = randcplx(a=1, b=0, d=0, c=2)
for besselx in [besselj, bessely, besseli, besselk]:
ri = Rational(2 * randint(-11, 10) + 1,
2) # half integer in [-21/2, 21/2]
assert tn(besselsimp(besselx(ri, z)), besselx(ri, z))
assert check(expand_func(besseli(rn, x)),
besseli(rn - 2, x) - 2 * (rn - 1) * besseli(rn - 1, x) / x)
assert check(expand_func(besseli(-rn, x)),
besseli(-rn + 2, x) + 2 * (-rn + 1) * besseli(-rn + 1, x) / x)
assert check(expand_func(besselj(rn, x)),
-besselj(rn - 2, x) + 2 * (rn - 1) * besselj(rn - 1, x) / x)
assert check(
expand_func(besselj(-rn, x)),
-besselj(-rn + 2, x) + 2 * (-rn + 1) * besselj(-rn + 1, x) / x)
assert check(expand_func(besselk(rn, x)),
besselk(rn - 2, x) + 2 * (rn - 1) * besselk(rn - 1, x) / x)
assert check(expand_func(besselk(-rn, x)),
besselk(-rn + 2, x) - 2 * (-rn + 1) * besselk(-rn + 1, x) / x)
assert check(expand_func(bessely(rn, x)),
-bessely(rn - 2, x) + 2 * (rn - 1) * bessely(rn - 1, x) / x)
assert check(
expand_func(bessely(-rn, x)),
-bessely(-rn + 2, x) + 2 * (-rn + 1) * bessely(-rn + 1, x) / x)
n = Symbol('n', integer=True, positive=True)
assert expand_func(besseli(n + 2, z)) == \
besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
assert expand_func(besselj(n + 2, z)) == \
-besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
assert expand_func(besselk(n + 2, z)) == \
besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
assert expand_func(bessely(n + 2, z)) == \
-bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z
assert expand_func(besseli(n + Rational(1, 2), z).rewrite(jn)) == \
(sqrt(2)*sqrt(z)*exp(-I*pi*(n + Rational(1, 2))/2) *
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
assert expand_func(besselj(n + Rational(1, 2), z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
r = Symbol('r', extended_real=True)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
for besselx in [besselj, bessely, besseli, besselk]:
assert besselx(i, p).is_extended_real
assert besselx(i, x).is_extended_real is None
assert besselx(x, z).is_extended_real is None
for besselx in [besselj, besseli]:
assert besselx(i, r).is_extended_real
for besselx in [bessely, besselk]:
assert besselx(i, r).is_extended_real is None
def test_expand():
assert expand_func(besselj(Rational(1, 2), z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(Rational(1, 2), z).rewrite(yn)) == \
-sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert expand_func(besselj(I, z)) == besselj(I, z)
# Test simplify helper
assert simplify(besselj(Rational(1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
# XXX: teach sin/cos to work around arguments like
# x*exp_polar(I*pi*n/2). Then change besselsimp -> expand_func
assert besselsimp(besselj(Rational(1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(Rational(-1, 2), z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(Rational(5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselj(-Rational(5, 2), z)) == \
-sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(1, 2), z)) == \
-(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(-1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(5, 2), z)) == \
sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(-5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(1, 2), z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(-1, 2), z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(5, 2), z)) == \
sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(-5, 2), z)) == \
sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselk(Rational(1, 2), z)) == \
besselsimp(besselk(Rational(-1, 2), z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
assert besselsimp(besselk(Rational(5, 2), z)) == \
besselsimp(besselk(Rational(-5, 2), z)) == \
sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**Rational(5, 2))
def check(eq, ans):
return tn(eq, ans) and eq == ans
rn = randcplx(a=1, b=0, d=0, c=2)
for besselx in [besselj, bessely, besseli, besselk]:
ri = Rational(2*randint(-11, 10) + 1, 2) # half integer in [-21/2, 21/2]
assert tn(besselsimp(besselx(ri, z)), besselx(ri, z))
assert check(expand_func(besseli(rn, x)),
besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x)
assert check(expand_func(besseli(-rn, x)),
besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x)
assert check(expand_func(besselj(rn, x)),
-besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x)
assert check(expand_func(besselj(-rn, x)),
-besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x)
assert check(expand_func(besselk(rn, x)),
besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x)
assert check(expand_func(besselk(-rn, x)),
besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x)
assert check(expand_func(bessely(rn, x)),
-bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x)
assert check(expand_func(bessely(-rn, x)),
-bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x)
n = Symbol('n', integer=True, positive=True)
assert expand_func(besseli(n + 2, z)) == \
besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
assert expand_func(besselj(n + 2, z)) == \
-besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
assert expand_func(besselk(n + 2, z)) == \
besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
assert expand_func(bessely(n + 2, z)) == \
-bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z
assert expand_func(besseli(n + Rational(1, 2), z).rewrite(jn)) == \
(sqrt(2)*sqrt(z)*exp(-I*pi*(n + Rational(1, 2))/2) *
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
assert expand_func(besselj(n + Rational(1, 2), z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
r = Symbol('r', extended_real=True)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
for besselx in [besselj, bessely, besseli, besselk]:
assert besselx(i, p).is_extended_real
assert besselx(i, x).is_extended_real is None
assert besselx(x, z).is_extended_real is None
for besselx in [besselj, besseli]:
assert besselx(i, r).is_extended_real
for besselx in [bessely, besselk]:
assert besselx(i, r).is_extended_real is None