def test_besselsimp():
from sympy import besselj, besseli, cosh, cosine_transform, bessely
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)*x**Rational(1, 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
assert besselsimp(x**2*(a*(-2*besselj(5*I, x) + besselj(-2 + 5*I, x) +
besselj(2 + 5*I, x)) + b*(-2*bessely(5*I, x) + bessely(-2 + 5*I, x) +
bessely(2 + 5*I, x)))/4 + x*(a*(besselj(-1 + 5*I, x)/2 - besselj(1 + 5*I, x)/2)
+ b*(bessely(-1 + 5*I, x)/2 - bessely(1 + 5*I, x)/2)) + (x**2 + 25)*(a*besselj(5*I, x)
+ b*bessely(5*I, x))) == 0
assert besselsimp(81*x**2*(a*(besselj(Rational(-5, 3), 9*x) - 2*besselj(Rational(1, 3), 9*x) + besselj(Rational(7, 3), 9*x))
+ b*(bessely(Rational(-5, 3), 9*x) - 2*bessely(Rational(1, 3), 9*x) + bessely(Rational(7, 3), 9*x)))/4 + x*(a*(9*besselj(Rational(-2, 3), 9*x)/2
- 9*besselj(Rational(4, 3), 9*x)/2) + b*(9*bessely(Rational(-2, 3), 9*x)/2 - 9*bessely(Rational(4, 3), 9*x)/2)) +
(81*x**2 - Rational(1, 9))*(a*besselj(Rational(1, 3), 9*x) + b*bessely(Rational(1, 3), 9*x))) == 0
assert besselsimp(besselj(a-1,x) + besselj(a+1, x) - 2*a*besselj(a, x)/x) == 0
assert besselsimp(besselj(a-1,x) + besselj(a+1, x) + besselj(a, x)) == (2*a + x)*besselj(a, x)/x
assert besselsimp(x**2* besselj(a,x) + x**3*besselj(a+1, x) + besselj(a+2, x)) == \
2*a*x*besselj(a + 1, x) + x**3*besselj(a + 1, x) - x**2*besselj(a + 2, x) + 2*x*besselj(a + 1, x) + besselj(a + 2, x)
def test_besselsimp():
from sympy import besselj, besseli, besselk, bessely, jn, yn, exp_polar, cosh
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)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) * \
besseli(-S(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(S(-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)
def test_besselsimp():
from sympy import besselj, besseli, besselk, bessely, jn, yn, exp_polar, cosh
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)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) * \
besseli(-S(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(S(-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)
def test_besselsimp():
from sympy import besselj, besseli, exp_polar, cosh, cosine_transform
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)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-S(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(S(-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_besselsimp():
from sympy import besselj, besseli, exp_polar, cosh, cosine_transform
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)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-S(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(S(-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_besselsimp():
from sympy import besselj, besseli, exp_polar, cosh, cosine_transform, bessely
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)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-S(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(S(-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
assert besselsimp(
x**2 * (a * (-2 * besselj(5 * I, x) + besselj(-2 + 5 * I, x) +
besselj(2 + 5 * I, x)) + b *
(-2 * bessely(5 * I, x) + bessely(-2 + 5 * I, x) +
bessely(2 + 5 * I, x))) / 4 + x *
(a * (besselj(-1 + 5 * I, x) / 2 - besselj(1 + 5 * I, x) / 2) + b *
(bessely(-1 + 5 * I, x) / 2 - bessely(1 + 5 * I, x) / 2)) +
(x**2 + 25) * (a * besselj(5 * I, x) + b * bessely(5 * I, x))) == 0
assert besselsimp(
81 * x**2 *
(a * (besselj(-S(5) / 3, 9 * x) - 2 * besselj(S(1) / 3, 9 * x) +
besselj(S(7) / 3, 9 * x)) + b *
(bessely(-S(5) / 3, 9 * x) - 2 * bessely(S(1) / 3, 9 * x) +
bessely(S(7) / 3, 9 * x))) / 4 + x *
(a * (9 * besselj(-S(2) / 3, 9 * x) / 2 -
9 * besselj(S(4) / 3, 9 * x) / 2) + b *
(9 * bessely(-S(2) / 3, 9 * x) / 2 - 9 * bessely(S(4) / 3, 9 * x) / 2)
) + (81 * x**2 - S(1) / 9) *
(a * besselj(S(1) / 3, 9 * x) + b * bessely(S(1) / 3, 9 * x))) == 0
assert besselsimp(
besselj(a - 1, x) + besselj(a + 1, x) - 2 * a * besselj(a, x) / x) == 0
assert besselsimp(besselj(a - 1, x) + besselj(a + 1, x) +
besselj(a, x)) == (2 * a + x) * besselj(a, x) / x
assert besselsimp(x**2* besselj(a,x) + x**3*besselj(a+1, x) + besselj(a+2, x)) == \
2*a*x*besselj(a + 1, x) + x**3*besselj(a + 1, x) - x**2*besselj(a + 2, x) + 2*x*besselj(a + 1, x) + besselj(a + 2, x)
def test_expand():
from sympy import besselsimp, Symbol, exp, exp_polar, I
assert expand_func(besselj(S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(S(1)/2, z).rewrite(yn)) == \
-sqrt(2)*cos(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(besseli(S(1) / 2,
z)) == sqrt(2) * sinh(z) / (sqrt(pi) * sqrt(z))
def check(eq, ans):
return tn(eq, ans) and eq == ans
rn = randcplx(a=1, b=0, d=0, c=2)
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 + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*exp(-I*pi*(n + S(1)/2)/2)* \
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)
assert expand_func(besselj(n + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
def test_expand():
from sympy import besselsimp, Symbol, exp, exp_polar, I
assert expand_func(besselj(S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(S(1)/2, z).rewrite(yn)) == \
-sqrt(2)*cos(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(besseli(S(1)/2, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
def check(eq, ans):
return tn(eq, ans) and eq == ans
rn = randcplx(a=1, b=0, d=0, c=2)
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 + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*exp(-I*pi*(n + S(1)/2)/2)* \
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)
assert expand_func(besselj(n + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
def test_expand():
from sympy import besselsimp, Symbol, exp, exp_polar, I
assert expand_func(besselj(S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(S(1)/2, z).rewrite(yn)) == \
-sqrt(2)*cos(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(S(1)/2, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(S(-1)/2, z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(S(5)/2, z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besselj(-S(5)/2, z)) == \
-sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(bessely(S(1)/2, z)) == \
-(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(S(-1)/2, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(S(5)/2, z)) == \
sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(bessely(S(-5)/2, z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besseli(S(1)/2, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(S(-1)/2, z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(S(5)/2, z)) == \
sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besseli(S(-5)/2, z)) == \
sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besselk(S(1)/2, z)) == \
besselsimp(besselk(S(-1)/2, z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
assert besselsimp(besselk(S(5)/2, z)) == \
besselsimp(besselk(S(-5)/2, z)) == \
sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**(S(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 = S(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 + S(1)/2, z).rewrite(jn)) == \
(sqrt(2)*sqrt(z)*exp(-I*pi*(n + S(1)/2)/2) *
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
assert expand_func(besselj(n + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
for besselx in [besselj, bessely, besseli, besselk]:
assert besselx(i, p).is_real
assert besselx(i, x).is_real is None
assert besselx(x, z).is_real is None
for besselx in [besselj, besseli]:
assert besselx(i, r).is_real
for besselx in [bessely, besselk]:
assert besselx(i, r).is_real is None
def test_expand():
from sympy import besselsimp, Symbol, exp, exp_polar, I
assert expand_func(besselj(
S.Half, z).rewrite(jn)) == sqrt(2) * sin(z) / (sqrt(pi) * sqrt(z))
assert expand_func(bessely(
S.Half, z).rewrite(yn)) == -sqrt(2) * cos(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(S.Half,
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(S.Half,
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(S.Half,
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(S.Half, 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 = S(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 + S.Half,
z).rewrite(jn)) == (sqrt(2) * sqrt(z) * exp(-I * pi *
(n + S.Half) / 2) *
exp_polar(I * pi / 4) *
jn(n, z * exp_polar(I * pi / 2)) / sqrt(pi))
assert expand_func(besselj(
n + S.Half, z).rewrite(jn)) == sqrt(2) * sqrt(z) * jn(n, z) / sqrt(pi)
r = Symbol("r", 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 is True
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 is True
for besselx in [bessely, besselk]:
assert besselx(i, r).is_extended_real is None
def test_expand():
from sympy import besselsimp, Symbol, exp, exp_polar, I
assert expand_func(besselj(S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(S(1)/2, z).rewrite(yn)) == \
-sqrt(2)*cos(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(S(1) / 2,
z)) == sqrt(2) * sin(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(besselj(S(-1) / 2,
z)) == sqrt(2) * cos(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(besselj(S(5)/2, z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besselj(-S(5)/2, z)) == \
-sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(bessely(S(1)/2, z)) == \
-(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(S(-1) / 2,
z)) == sqrt(2) * sin(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(bessely(S(5)/2, z)) == \
sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(bessely(S(-5)/2, z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besseli(S(1) / 2,
z)) == sqrt(2) * sinh(z) / (sqrt(pi) * sqrt(z))
assert besselsimp(besseli(S(-1)/2, z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(S(5)/2, z)) == \
sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besseli(S(-5)/2, z)) == \
sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**(S(5)/2))
assert besselsimp(besselk(S(1)/2, z)) == \
besselsimp(besselk(S(-1)/2, z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
assert besselsimp(besselk(S(5)/2, z)) == \
besselsimp(besselk(S(-5)/2, z)) == \
sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**(S(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 = S(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 + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*exp(-I*pi*(n + S(1)/2)/2)* \
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)
assert expand_func(besselj(n + S(1)/2, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
