def test_vector_simplify():
A, s, k, m = symbols('A, s, k, m')
test1 = (1 / a + 1 / b) * i
assert (test1 & i) != (a + b) / (a * b)
test1 = simplify(test1)
assert (test1 & i) == (a + b) / (a * b)
assert test1.simplify() == simplify(test1)
test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * i
test2 = simplify(test2)
assert (test2 & i) == (A**2 * s**4 / (4 * pi * k * m**3))
test3 = ((4 + 4 * a - 2 * (2 + 2 * a)) / (2 + 2 * a)) * i
test3 = simplify(test3)
assert (test3 & i) == 0
test4 = ((-4 * a * b**2 - 2 * b**3 - 2 * a**2 * b) / (a + b)**2) * i
test4 = simplify(test4)
assert (test4 & i) == -2 * b
v = (sin(a) + cos(a))**2 * i - j
assert trigsimp(v) == (2 * sin(a + pi / 4)**2) * i + (-1) * j
assert trigsimp(v) == v.trigsimp()
assert simplify(Vector.zero) == Vector.zero
def test_trigsimp2():
assert trigsimp(cos(x)**2*sin(y)**2 + cos(x)**2*cos(y)**2 + sin(x)**2,
recursive=True) == 1
assert trigsimp(sin(x)**2*sin(y)**2 + sin(x)**2*cos(y)**2 + cos(x)**2,
recursive=True) == 1
assert trigsimp(
Subs(x, (x, sin(y)**2 + cos(y)**2))) == Subs(x, (x, 1))
def test_trigsimp2():
assert trigsimp(cos(x)**2*sin(y)**2 + cos(x)**2*cos(y)**2 + sin(x)**2,
recursive=True) == 1
assert trigsimp(sin(x)**2*sin(y)**2 + sin(x)**2*cos(y)**2 + cos(x)**2,
recursive=True) == 1
assert trigsimp(
Subs(x, x, sin(y)**2 + cos(y)**2)) == Subs(x, x, 1)
def test_sympyissue_4280():
assert trigsimp(cos(x)**2 + cos(y)**2 * sin(x)**2 +
sin(y)**2 * sin(x)**2) == 1
assert trigsimp(a**2 * sin(x)**2 + a**2 * cos(y)**2 * cos(x)**2 +
a**2 * cos(x)**2 * sin(y)**2) == a**2
assert trigsimp(a**2 * cos(y)**2 * sin(x)**2 +
a**2 * sin(y)**2 * sin(x)**2) == a**2 * sin(x)**2
def test_trigsimp_groebner():
c = cos(x)
s = sin(x)
ex = (4 * s * c + 12 * s + 5 * c**3 + 21 * c**2 + 23 * c + 15) / (
-s * c**2 + 2 * s * c + 15 * s + 7 * c**3 + 31 * c**2 + 37 * c + 21)
resnum = (5 * s - 5 * c + 1)
resdenom = (8 * s - 6 * c)
results = [resnum / resdenom, (-resnum) / (-resdenom)]
assert trigsimp_groebner(ex) in results
assert trigsimp_groebner(s / c, hints=[tan]) == tan(x)
assert trigsimp((-s + 1) / c + c / (-s + 1), method='groebner') == 2 / c
assert trigsimp((-s + 1) / c + c / (-s + 1),
method='groebner',
polynomial=True) == 2 / c
# Test quick=False works
assert trigsimp_groebner(ex, hints=[2]) in results
# test "I"
assert trigsimp_groebner(sin(I * x) / cos(I * x),
hints=[tanh]) == I * tanh(x)
# test hyperbolic / sums
assert trigsimp_groebner((tanh(x) + tanh(y)) / (1 + tanh(x) * tanh(y)),
hints=[(tanh, x, y)]) == tanh(x + y)
# issue sympy/sympy#11062
trigsimp_groebner(csc(x) * sin(x)) # not raises
def test_trigsimp_groebner():
c = cos(x)
s = sin(x)
ex = (4*s*c + 12*s + 5*c**3 + 21*c**2 + 23*c + 15)/(
-s*c**2 + 2*s*c + 15*s + 7*c**3 + 31*c**2 + 37*c + 21)
resnum = (5*s - 5*c + 1)
resdenom = (8*s - 6*c)
results = [resnum/resdenom, (-resnum)/(-resdenom)]
assert trigsimp_groebner(ex) in results
assert trigsimp_groebner(s/c, hints=[tan]) == tan(x)
assert trigsimp((-s + 1)/c + c/(-s + 1),
method='groebner') == 2/c
assert trigsimp((-s + 1)/c + c/(-s + 1),
method='groebner', polynomial=True) == 2/c
# Test quick=False works
assert trigsimp_groebner(ex, hints=[2]) in results
# test "I"
assert trigsimp_groebner(sin(I*x)/cos(I*x), hints=[tanh]) == I*tanh(x)
# test hyperbolic / sums
assert trigsimp_groebner((tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)),
hints=[(tanh, x, y)]) == tanh(x + y)
# issue sympy/sympy#11062
trigsimp_groebner(csc(x) * sin(x)) # not raises
def test_sympyissue_4661():
eq = -4*sin(x)**4 + 4*cos(x)**4 - 8*cos(x)**2
assert trigsimp(eq) == -4
n = sin(x)**6 + 4*sin(x)**4*cos(x)**2 + 5*sin(x)**2*cos(x)**4 + 2*cos(x)**6
d = -sin(x)**2 - 2*cos(x)**2
assert simplify(n/d) == -1
assert trigsimp(-2*cos(x)**2 + cos(x)**4 - sin(x)**4) == -1
eq = (- sin(x)**3/4)*cos(x) + (cos(x)**3/4)*sin(x) - sin(2*x)*cos(2*x)/8
assert trigsimp(eq) == 0
def test_issue_4661():
eq = -4*sin(x)**4 + 4*cos(x)**4 - 8*cos(x)**2
assert trigsimp(eq) == -4
n = sin(x)**6 + 4*sin(x)**4*cos(x)**2 + 5*sin(x)**2*cos(x)**4 + 2*cos(x)**6
d = -sin(x)**2 - 2*cos(x)**2
assert simplify(n/d) == -1
assert trigsimp(-2*cos(x)**2 + cos(x)**4 - sin(x)**4) == -1
eq = (- sin(x)**3/4)*cos(x) + (cos(x)**3/4)*sin(x) - sin(2*x)*cos(2*x)/8
assert trigsimp(eq) == 0
def test_trigsimp_old(capsys):
e = 2 * sin(x)**2 + 2 * cos(x)**2
assert trigsimp(e, old=True) == 2
e = 3 * tanh(x)**7 - 2 / coth(x)**7
assert trigsimp(e, method='old') == e
e = (-sin(x) + 1) / cos(x) + cos(x) / (-sin(x) + 1)
assert (trigsimp(e, method='old') == (-sin(x) + 1) / cos(x) - cos(x) /
(sin(x) - 1))
e = (-sin(x) + 1) / cos(x) + cos(x) / (-sin(x) + 1)
assert trigsimp(e, method='groebner', old=True) == 2 / cos(x)
assert trigsimp(1 / cot(x)**2, compare=True, old=True) == cot(x)**(-2)
assert capsys.readouterr().out == '\tfutrig: tan(x)**2\n'
def test_trigsimp3():
assert trigsimp(sin(x)/cos(x)) == tan(x)
assert trigsimp(sin(x)**2/cos(x)**2) == tan(x)**2
assert trigsimp(sin(x)**3/cos(x)**3) == tan(x)**3
assert trigsimp(sin(x)**10/cos(x)**10) == tan(x)**10
assert trigsimp(cos(x)/sin(x)) == 1/tan(x)
assert trigsimp(cos(x)**2/sin(x)**2) == 1/tan(x)**2
assert trigsimp(cos(x)**10/sin(x)**10) == 1/tan(x)**10
assert trigsimp(tan(x)) == trigsimp(sin(x)/cos(x))
def test_trigsimp3():
assert trigsimp(sin(x) / cos(x)) == tan(x)
assert trigsimp(sin(x)**2 / cos(x)**2) == tan(x)**2
assert trigsimp(sin(x)**3 / cos(x)**3) == tan(x)**3
assert trigsimp(sin(x)**10 / cos(x)**10) == tan(x)**10
assert trigsimp(cos(x) / sin(x)) == 1 / tan(x)
assert trigsimp(cos(x)**2 / sin(x)**2) == 1 / tan(x)**2
assert trigsimp(cos(x)**10 / sin(x)**10) == 1 / tan(x)**10
assert trigsimp(tan(x)) == trigsimp(sin(x) / cos(x))
def test_R2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', extended_real=True)
point_r = R2_r.point([x0, y0])
point_p = R2_p.point([r0, theta0])
# r**2 = x**2 + y**2
assert (R2.r**2 - R2.x**2 - R2.y**2).rcall(point_r) == 0
assert trigsimp((R2.r**2 - R2.x**2 - R2.y**2).rcall(point_p)) == 0
assert trigsimp(R2.e_r(R2.x**2 + R2.y**2).rcall(point_p).doit()) == 2 * r0
# polar->rect->polar == Id
a, b = symbols('a b', positive=True)
m = Matrix([[a], [b]])
# TODO assert m == R2_r.coord_tuple_transform_to(R2_p, R2_p.coord_tuple_transform_to(R2_r, [a, b])).applyfunc(simplify)
assert m == R2_p.coord_tuple_transform_to(
R2_r, R2_r.coord_tuple_transform_to(R2_p, m)).applyfunc(simplify)
def scalar_map(self, other):
"""
Returns a dictionary which expresses the coordinate variables
(base scalars) of this frame in terms of the variables of
otherframe.
Parameters
==========
otherframe : CoordSysCartesian
The other system to map the variables to.
Examples
========
>>> from diofant.vector import CoordSysCartesian
>>> from diofant import Symbol, cos, sin
>>> A = CoordSysCartesian('A')
>>> q = Symbol('q')
>>> B = A.orient_new_axis('B', q, A.k)
>>> A.scalar_map(B) == {A.x: B.x*cos(q) - B.y*sin(q), A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z}
True
"""
relocated_scalars = []
origin_coords = tuple(self.position_wrt(other).to_matrix(other))
for i, x in enumerate(other.base_scalars()):
relocated_scalars.append(x - origin_coords[i])
vars_matrix = (self.rotation_matrix(other) * Matrix(relocated_scalars))
mapping = {}
for i, x in enumerate(self.base_scalars()):
mapping[x] = trigsimp(vars_matrix[i])
return mapping
def test_sympyissue_3210():
eqs = (sin(2)*cos(3) + sin(3)*cos(2),
-sin(2)*sin(3) + cos(2)*cos(3),
sin(2)*cos(3) - sin(3)*cos(2),
sin(2)*sin(3) + cos(2)*cos(3),
sin(2)*sin(3) + cos(2)*cos(3) + cos(2),
sinh(2)*cosh(3) + sinh(3)*cosh(2),
sinh(2)*sinh(3) + cosh(2)*cosh(3))
assert [trigsimp(e) for e in eqs] == [sin(5), cos(5), -sin(1), cos(1),
cos(1) + cos(2), sinh(5), cosh(5)]
def test_sympyissue_2827_trigsimp_methods():
def measure1(expr):
return len(str(expr))
def measure2(expr):
return -count_ops(expr) # Return the most complicated result
expr = (x + 1)/(x + sin(x)**2 + cos(x)**2)
ans = Matrix([1])
M = Matrix([expr])
assert trigsimp(M, method='fu', measure=measure1) == ans
assert trigsimp(M, method='fu', measure=measure2) != ans
# all methods should work with Basic expressions even if they
# aren't Expr
M = Matrix.eye(1)
assert all(trigsimp(M, method=m) == M for m in
'fu matching groebner old'.split())
# watch for E in exptrigsimp, not only exp()
eq = 1/sqrt(E) + E
assert exptrigsimp(eq) == eq
def test_sympyissue_4892b():
# Issues relating to issue sympy/sympy#4596 are making the actual result of this hard
# to test. The answer should be something like
#
# (-sin(y) + sqrt(-72 + 48*cos(y) - 8*cos(y)**2)/2)*log(x + sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/(2*(3 - cos(y)))) + (-sin(y) - sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/2)*log(x - sqrt(-72 + 48*cos(y) -
# 8*cos(y)**2)/(2*(3 - cos(y)))) + x**2*sin(y)/2 + 2*x*cos(y)
expr = (sin(y) * x**3 + 2 * cos(y) * x**2 + 12) / (x**2 + 2)
assert trigsimp(factor(integrate(expr, x).diff(x) - expr)) == 0
def test_sympyissue_4892b():
# Issues relating to issue sympy/sympy#4596 are making the actual result of this hard
# to test. The answer should be something like
#
# (-sin(y) + sqrt(-72 + 48*cos(y) - 8*cos(y)**2)/2)*log(x + sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/(2*(3 - cos(y)))) + (-sin(y) - sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/2)*log(x - sqrt(-72 + 48*cos(y) -
# 8*cos(y)**2)/(2*(3 - cos(y)))) + x**2*sin(y)/2 + 2*x*cos(y)
expr = (sin(y)*x**3 + 2*cos(y)*x**2 + 12)/(x**2 + 2)
assert trigsimp(factor(integrate(expr, x).diff(x) - expr)) == 0
def test_sympyissue_2827_trigsimp_methods():
def measure1(expr):
return len(str(expr))
def measure2(expr):
return -count_ops(expr) # Return the most complicated result
expr = (x + 1) / (x + sin(x)**2 + cos(x)**2)
ans = Matrix([1])
M = Matrix([expr])
assert trigsimp(M, method='fu', measure=measure1) == ans
assert trigsimp(M, method='fu', measure=measure2) != ans
# all methods should work with Basic expressions even if they
# aren't Expr
M = Matrix.eye(1)
assert all(
trigsimp(M, method=m) == M for m in 'fu matching groebner old'.split())
# watch for E in exptrigsimp, not only exp()
eq = 1 / sqrt(E) + E
assert exptrigsimp(eq) == eq
def test_functional_diffgeom_ch4():
x0, y0, theta0 = symbols('x0, y0, theta0', extended_real=True)
x, y, r, theta = symbols('x, y, r, theta', extended_real=True)
r0 = symbols('r0', positive=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
f_field = b1(R2.x, R2.y) * R2.dx + b2(R2.x, R2.y) * R2.dy
assert f_field.rcall(R2.e_x).rcall(p_r) == b1(x0, y0)
assert f_field.rcall(R2.e_y).rcall(p_r) == b2(x0, y0)
s_field_r = f(R2.x, R2.y)
df = Differential(s_field_r)
assert df(R2.e_x).rcall(p_r).doit() == Derivative(f(x0, y0), x0)
assert df(R2.e_y).rcall(p_r).doit() == Derivative(f(x0, y0), y0)
s_field_p = f(R2.r, R2.theta)
df = Differential(s_field_p)
assert trigsimp(df(R2.e_x).rcall(p_p).doit()) == (
cos(theta0) * Derivative(f(r0, theta0), r0) -
sin(theta0) * Derivative(f(r0, theta0), theta0) / r0)
assert trigsimp(df(R2.e_y).rcall(p_p).doit()) == (
sin(theta0) * Derivative(f(r0, theta0), r0) +
cos(theta0) * Derivative(f(r0, theta0), theta0) / r0)
assert R2.dx(R2.e_x).rcall(p_r) == 1
assert R2.dx(R2.e_x) == 1
assert R2.dx(R2.e_y).rcall(p_r) == 0
assert R2.dx(R2.e_y) == 0
circ = -R2.y * R2.e_x + R2.x * R2.e_y
assert R2.dx(circ).rcall(p_r).doit() == -y0
assert R2.dy(circ).rcall(p_r) == x0
assert R2.dr(circ).rcall(p_r) == 0
assert simplify(R2.dtheta(circ).rcall(p_r)) == 1
assert (circ - R2.e_theta).rcall(s_field_r).rcall(p_r) == 0
def test_Piecewise():
e1 = x*(x + y) - y*(x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y)*y/x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
# trigsimp tries not to touch non-trig containing args
assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
Piecewise((e1, e3 < s2), (e3, True))
def test_Piecewise():
e1 = x * (x + y) - y * (x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y) * y / x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
# trigsimp tries not to touch non-trig containing args
assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
Piecewise((e1, e3 < s2), (e3, True))
def test_sympyissue_3210():
eqs = (sin(2) * cos(3) + sin(3) * cos(2), -sin(2) * sin(3) +
cos(2) * cos(3), sin(2) * cos(3) - sin(3) * cos(2),
sin(2) * sin(3) + cos(2) * cos(3), sin(2) * sin(3) +
cos(2) * cos(3) + cos(2), sinh(2) * cosh(3) + sinh(3) * cosh(2),
sinh(2) * sinh(3) + cosh(2) * cosh(3))
assert [trigsimp(e) for e in eqs] == [
sin(5),
cos(5), -sin(1),
cos(1),
cos(1) + cos(2),
sinh(5),
cosh(5)
]
def test_trigsimp_sympyissue_6925():
assert trigsimp(tan(2*x).expand(trig=True)) == tan(2*x)
def test_sympyissue_5948():
assert trigsimp(diff(integrate(cos(x)/sin(x)**7, x), x)) == cos(x)/sin(x)**7
def test_trigsimp_noncommutative():
A, B = symbols('A,B', commutative=False)
assert trigsimp(A - A * sin(x)**2) == A * cos(x)**2
assert trigsimp(A - A * cos(x)**2) == A * sin(x)**2
assert trigsimp(A * sin(x)**2 + A * cos(x)**2) == A
assert trigsimp(A + A * tan(x)**2) == A / cos(x)**2
assert trigsimp(A / cos(x)**2 - A) == A * tan(x)**2
assert trigsimp(A / cos(x)**2 - A * tan(x)**2) == A
assert trigsimp(A + A * cot(x)**2) == A / sin(x)**2
assert trigsimp(A / sin(x)**2 - A) == A / tan(x)**2
assert trigsimp(A / sin(x)**2 - A * cot(x)**2) == A
assert trigsimp(y * A * cos(x)**2 + y * A * sin(x)**2) == y * A
assert trigsimp(A * sin(x) / cos(x)) == A * tan(x)
assert trigsimp(A * tan(x) * cos(x)) == A * sin(x)
assert trigsimp(A * cot(x)**3 * sin(x)**3) == A * cos(x)**3
assert trigsimp(y * A * tan(x)**2 / sin(x)**2) == y * A / cos(x)**2
assert trigsimp(A * cot(x) / cos(x)) == A / sin(x)
assert trigsimp(A * sin(x + y) + A * sin(x - y)) == 2 * A * sin(x) * cos(y)
assert trigsimp(A * sin(x + y) - A * sin(x - y)) == 2 * A * sin(y) * cos(x)
assert trigsimp(A * cos(x + y) + A * cos(x - y)) == 2 * A * cos(x) * cos(y)
assert trigsimp(A * cos(x + y) -
A * cos(x - y)) == -2 * A * sin(x) * sin(y)
assert trigsimp(A * sinh(x + y) +
A * sinh(x - y)) == 2 * A * sinh(x) * cosh(y)
assert trigsimp(A * sinh(x + y) -
A * sinh(x - y)) == 2 * A * sinh(y) * cosh(x)
assert trigsimp(A * cosh(x + y) +
A * cosh(x - y)) == 2 * A * cosh(x) * cosh(y)
assert trigsimp(A * cosh(x + y) -
A * cosh(x - y)) == 2 * A * sinh(x) * sinh(y)
assert trigsimp(A * cos(0.12345)**2 + A * sin(0.12345)**2) == 1.0 * A
def test_trigsimp_sympyissue_7131():
n = Symbol('n', integer=True, positive=True)
assert trigsimp(2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2) == \
2**(n/2)*cos(pi*n/4)/2 + 2**n/4
def test_trigsimp_sympyissue_5614():
assert trigsimp(x * cos(x) * tan(x)) == x * sin(x)
assert trigsimp(-sin(x) + cos(x) * tan(x)) == 0
def test_sympyissue_4494():
eq = sin(a)**2*sin(b)**2 + cos(a)**2*cos(b)**2*tan(a)**2 + cos(a)**2
assert trigsimp(eq) == 1
def test_trigsimp1():
assert trigsimp(1 - sin(x)**2) == cos(x)**2
assert trigsimp(1 - cos(x)**2) == sin(x)**2
assert trigsimp(sin(x)**2 + cos(x)**2) == 1
assert trigsimp(1 + tan(x)**2) == 1 / cos(x)**2
assert trigsimp(1 / cos(x)**2 - 1) == tan(x)**2
assert trigsimp(1 / cos(x)**2 - tan(x)**2) == 1
assert trigsimp(1 + cot(x)**2) == 1 / sin(x)**2
assert trigsimp(1 / sin(x)**2 - 1) == 1 / tan(x)**2
assert trigsimp(1 / sin(x)**2 - cot(x)**2) == 1
assert trigsimp(5 * cos(x)**2 + 5 * sin(x)**2) == 5
assert trigsimp(5 * cos(x / 2)**2 +
2 * sin(x / 2)**2) == 3 * cos(x) / 2 + Rational(7, 2)
assert trigsimp(sin(x) / cos(x)) == tan(x)
assert trigsimp(2 * tan(x) * cos(x)) == 2 * sin(x)
assert trigsimp(cot(x)**3 * sin(x)**3) == cos(x)**3
assert trigsimp(y * tan(x)**2 / sin(x)**2) == y / cos(x)**2
assert trigsimp(cot(x) / cos(x)) == 1 / sin(x)
assert trigsimp(sin(x + y) + sin(x - y)) == 2 * sin(x) * cos(y)
assert trigsimp(sin(x + y) - sin(x - y)) == 2 * sin(y) * cos(x)
assert trigsimp(cos(x + y) + cos(x - y)) == 2 * cos(x) * cos(y)
assert trigsimp(cos(x + y) - cos(x - y)) == -2 * sin(x) * sin(y)
assert trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y))) == \
sin(y)/(-sin(y)*tan(x) + cos(y)) # -tan(y)/(tan(x)*tan(y) - 1)
assert trigsimp(sinh(x + y) + sinh(x - y)) == 2 * sinh(x) * cosh(y)
assert trigsimp(sinh(x + y) - sinh(x - y)) == 2 * sinh(y) * cosh(x)
assert trigsimp(cosh(x + y) + cosh(x - y)) == 2 * cosh(x) * cosh(y)
assert trigsimp(cosh(x + y) - cosh(x - y)) == 2 * sinh(x) * sinh(y)
assert trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y))) == \
sinh(y)/(sinh(y)*tanh(x) + cosh(y))
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1
e = 2 * sin(x)**2 + 2 * cos(x)**2
assert trigsimp(log(e)) == log(2)
def test_sympyissue_6811_fail():
xp = Symbol('xp')
eq = 4*(-19*sin(x)*y + 5*sin(3*x)*y + 15*cos(2*x)*z - 21*z)*xp/(9*cos(x) - 5*cos(3*x))
assert trigsimp(eq) == -2*(2*cos(x)*tan(x)*y + 3*z)*xp/cos(x)
def test_trigsimp1():
assert trigsimp(1 - sin(x)**2) == cos(x)**2
assert trigsimp(1 - cos(x)**2) == sin(x)**2
assert trigsimp(sin(x)**2 + cos(x)**2) == 1
assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
assert trigsimp(1/sin(x)**2 - 1) == 1/tan(x)**2
assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) == 3*cos(x)/2 + Rational(7, 2)
assert trigsimp(sin(x)/cos(x)) == tan(x)
assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x)
assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3
assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2
assert trigsimp(cot(x)/cos(x)) == 1/sin(x)
assert trigsimp(sin(x + y) + sin(x - y)) == 2*sin(x)*cos(y)
assert trigsimp(sin(x + y) - sin(x - y)) == 2*sin(y)*cos(x)
assert trigsimp(cos(x + y) + cos(x - y)) == 2*cos(x)*cos(y)
assert trigsimp(cos(x + y) - cos(x - y)) == -2*sin(x)*sin(y)
assert trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y))) == \
sin(y)/(-sin(y)*tan(x) + cos(y)) # -tan(y)/(tan(x)*tan(y) - 1)
assert trigsimp(sinh(x + y) + sinh(x - y)) == 2*sinh(x)*cosh(y)
assert trigsimp(sinh(x + y) - sinh(x - y)) == 2*sinh(y)*cosh(x)
assert trigsimp(cosh(x + y) + cosh(x - y)) == 2*cosh(x)*cosh(y)
assert trigsimp(cosh(x + y) - cosh(x - y)) == 2*sinh(x)*sinh(y)
assert trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y))) == \
sinh(y)/(sinh(y)*tanh(x) + cosh(y))
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1
e = 2*sin(x)**2 + 2*cos(x)**2
assert trigsimp(log(e)) == log(2)
def test_trigsimp1a():
assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2)
assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2)
assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2)
assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2)
assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2)
assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2)
assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2)
assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2)
assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2)
assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2)
assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2)
assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2)
def test_trigsimp_sympyissue_7131():
n = Symbol('n', integer=True, positive=True)
assert trigsimp(2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2) == \
2**(n/2)*cos(pi*n/4)/2 + 2**n/4
def test_trigsimp_sympyissue_5614():
assert trigsimp(x*cos(x)*tan(x)) == x*sin(x)
assert trigsimp(-sin(x) + cos(x)*tan(x)) == 0
def test_sympyissue_5948():
assert trigsimp(diff(integrate(cos(x) / sin(x)**7, x),
x)) == cos(x) / sin(x)**7
def test_trigsimp_issues():
# issue sympy/sympy#4625 - factor_terms works, too
assert trigsimp(sin(x)**3 + cos(x)**2*sin(x)) == sin(x)
# issue sympy/sympy#5948
assert trigsimp(diff(integrate(cos(x)/sin(x)**3, x), x)) == \
cos(x)/sin(x)**3
assert trigsimp(diff(integrate(sin(x)/cos(x)**3, x), x)) == \
sin(x)/cos(x)**3
# check integer exponents
e = sin(x)**y/cos(x)**y
assert trigsimp(e) == e
assert trigsimp(e.subs({y: 2})) == tan(x)**2
assert trigsimp(e.subs({x: 1})) == tan(1)**y
# check for multiple patterns
assert (cos(x)**2/sin(x)**2*cos(y)**2/sin(y)**2).trigsimp() == \
1/tan(x)**2/tan(y)**2
assert trigsimp(cos(x)/sin(x)*cos(x+y)/sin(x+y)) == \
1/(tan(x)*tan(x + y))
eq = cos(2)*(cos(3) + 1)**2/(cos(3) - 1)**2
assert trigsimp(eq) == eq.factor() # factor makes denom (-1 + cos(3))**2
assert trigsimp(cos(2)*(cos(3) + 1)**2*(cos(3) - 1)**2) == \
cos(2)*sin(3)**4
# issue sympy/sympy#6789; this generates an expression that formerly caused
# trigsimp to hang
assert cot(x).equals(tan(x)) is False
# nan or the unchanged expression is ok, but not sin(1)
z = cos(x)**2 + sin(x)**2 - 1
z1 = tan(x)**2 - 1/cot(x)**2
n = (1 + z1/z)
assert trigsimp(sin(n)) != sin(1)
eq = x*(n - 1) - x*n
assert trigsimp(eq) is nan
assert trigsimp(eq, recursive=True) is nan
assert trigsimp(1).is_Integer
assert trigsimp(-sin(x)**4 - 2*sin(x)**2*cos(x)**2 - cos(x)**4) == -1
def test_sympyissue_4775():
assert trigsimp(sin(x) * cos(y) + cos(x) * sin(y)) == sin(x + y)
assert trigsimp(sin(x) * cos(y) + cos(x) * sin(y) + 3) == sin(x + y) + 3
def test_trigsimp_old():
e = 2*sin(x)**2 + 2*cos(x)**2
assert trigsimp(e, old=True) == 2
def test_sympyissue_4775():
assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)) == sin(x + y)
assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)+3) == sin(x + y) + 3
def test_sympyissue_6811_fail():
xp = Symbol('xp')
eq = 4 * (-19 * sin(x) * y + 5 * sin(3 * x) * y + 15 * cos(2 * x) * z -
21 * z) * xp / (9 * cos(x) - 5 * cos(3 * x))
assert trigsimp(eq) == -2 * (2 * cos(x) * tan(x) * y + 3 * z) * xp / cos(x)
def test_trigsimp_issues():
# issue sympy/sympy#4625 - factor_terms works, too
assert trigsimp(sin(x)**3 + cos(x)**2 * sin(x)) == sin(x)
# issue sympy/sympy#5948
assert trigsimp(diff(integrate(cos(x)/sin(x)**3, x), x)) == \
cos(x)/sin(x)**3
assert trigsimp(diff(integrate(sin(x)/cos(x)**3, x), x)) == \
sin(x)/cos(x)**3
# check integer exponents
e = sin(x)**y / cos(x)**y
assert trigsimp(e) == e
assert trigsimp(e.subs(y, 2)) == tan(x)**2
assert trigsimp(e.subs(x, 1)) == tan(1)**y
# check for multiple patterns
assert (cos(x)**2/sin(x)**2*cos(y)**2/sin(y)**2).trigsimp() == \
1/tan(x)**2/tan(y)**2
assert trigsimp(cos(x)/sin(x)*cos(x+y)/sin(x+y)) == \
1/(tan(x)*tan(x + y))
eq = cos(2) * (cos(3) + 1)**2 / (cos(3) - 1)**2
assert trigsimp(eq) == eq.factor() # factor makes denom (-1 + cos(3))**2
assert trigsimp(cos(2)*(cos(3) + 1)**2*(cos(3) - 1)**2) == \
cos(2)*sin(3)**4
# issue sympy/sympy#6789; this generates an expression that formerly caused
# trigsimp to hang
assert cot(x).equals(tan(x)) is False
# nan or the unchanged expression is ok, but not sin(1)
z = cos(x)**2 + sin(x)**2 - 1
z1 = tan(x)**2 - 1 / cot(x)**2
n = (1 + z1 / z)
assert trigsimp(sin(n)) != sin(1)
eq = x * (n - 1) - x * n
assert trigsimp(eq) is nan
assert trigsimp(eq, recursive=True) is nan
assert trigsimp(1).is_Integer
assert trigsimp(-sin(x)**4 - 2 * sin(x)**2 * cos(x)**2 - cos(x)**4) == -1
def test_trigsimp1a():
assert trigsimp(sin(2)**2 * cos(3) * exp(2) /
cos(2)**2) == tan(2)**2 * cos(3) * exp(2)
assert trigsimp(tan(2)**2 * cos(3) * exp(2) *
cos(2)**2) == sin(2)**2 * cos(3) * exp(2)
assert trigsimp(cot(2) * cos(3) * exp(2) *
sin(2)) == cos(3) * exp(2) * cos(2)
assert trigsimp(tan(2) * cos(3) * exp(2) /
sin(2)) == cos(3) * exp(2) / cos(2)
assert trigsimp(cot(2) * cos(3) * exp(2) /
cos(2)) == cos(3) * exp(2) / sin(2)
assert trigsimp(cot(2) * cos(3) * exp(2) * tan(2)) == cos(3) * exp(2)
assert trigsimp(sinh(2) * cos(3) * exp(2) /
cosh(2)) == tanh(2) * cos(3) * exp(2)
assert trigsimp(tanh(2) * cos(3) * exp(2) *
cosh(2)) == sinh(2) * cos(3) * exp(2)
assert trigsimp(coth(2) * cos(3) * exp(2) *
sinh(2)) == cosh(2) * cos(3) * exp(2)
assert trigsimp(tanh(2) * cos(3) * exp(2) /
sinh(2)) == cos(3) * exp(2) / cosh(2)
assert trigsimp(coth(2) * cos(3) * exp(2) /
cosh(2)) == cos(3) * exp(2) / sinh(2)
assert trigsimp(coth(2) * cos(3) * exp(2) * tanh(2)) == cos(3) * exp(2)
def test_trigsimp_sympyissue_6925():
assert trigsimp(tan(2 * x).expand(trig=True)) == tan(2 * x)
def test_hyperbolic_simp():
assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
assert trigsimp(1 - tanh(x)**2) == 1/cosh(x)**2
assert trigsimp(1 - 1/cosh(x)**2) == tanh(x)**2
assert trigsimp(tanh(x)**2 + 1/cosh(x)**2) == 1
assert trigsimp(coth(x)**2 - 1) == 1/sinh(x)**2
assert trigsimp(1/sinh(x)**2 + 1) == 1/tanh(x)**2
assert trigsimp(coth(x)**2 - 1/sinh(x)**2) == 1
assert trigsimp(5*cosh(x)**2 - 5*sinh(x)**2) == 5
assert trigsimp(5*cosh(x/2)**2 - 2*sinh(x/2)**2) == 3*cosh(x)/2 + Rational(7, 2)
assert trigsimp(sinh(x)/cosh(x)) == tanh(x)
assert trigsimp(tanh(x)) == trigsimp(sinh(x)/cosh(x))
assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
assert trigsimp(2*tanh(x)*cosh(x)) == 2*sinh(x)
assert trigsimp(coth(x)**3*sinh(x)**3) == cosh(x)**3
assert trigsimp(y*tanh(x)**2/sinh(x)**2) == y/cosh(x)**2
assert trigsimp(coth(x)/cosh(x)) == 1/sinh(x)
e = 2*cosh(x)**2 - 2*sinh(x)**2
assert trigsimp(log(e)) == log(2)
assert trigsimp(cosh(x)**2*cosh(y)**2 - cosh(x)**2*sinh(y)**2 - sinh(x)**2,
recursive=True) == 1
assert trigsimp(sinh(x)**2*sinh(y)**2 - sinh(x)**2*cosh(y)**2 + cosh(x)**2,
recursive=True) == 1
assert abs(trigsimp(2.0*cosh(x)**2 - 2.0*sinh(x)**2) - 2.0) < 1e-10
assert trigsimp(sinh(x)**2/cosh(x)**2) == tanh(x)**2
assert trigsimp(sinh(x)**3/cosh(x)**3) == tanh(x)**3
assert trigsimp(sinh(x)**10/cosh(x)**10) == tanh(x)**10
assert trigsimp(cosh(x)**3/sinh(x)**3) == 1/tanh(x)**3
assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
assert trigsimp(cosh(x)**2/sinh(x)**2) == 1/tanh(x)**2
assert trigsimp(cosh(x)**10/sinh(x)**10) == 1/tanh(x)**10
assert trigsimp(x*cosh(x)*tanh(x)) == x*sinh(x)
assert trigsimp(-sinh(x) + cosh(x)*tanh(x)) == 0
assert tan(x) != 1/cot(x) # cot doesn't auto-simplify
assert trigsimp(tan(x) - 1/cot(x)) == 0
assert trigsimp(3*tanh(x)**7 - 2/coth(x)**7) == tanh(x)**7
def test_trigsimp_sympyissue_7761():
assert trigsimp(cosh(pi / 4)) == cosh(pi / 4)
def test_trigsimp_sympyissue_7761():
assert trigsimp(cosh(pi/4)) == cosh(pi/4)
def test_hyperbolic_simp():
assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
assert trigsimp(1 - tanh(x)**2) == 1 / cosh(x)**2
assert trigsimp(1 - 1 / cosh(x)**2) == tanh(x)**2
assert trigsimp(tanh(x)**2 + 1 / cosh(x)**2) == 1
assert trigsimp(coth(x)**2 - 1) == 1 / sinh(x)**2
assert trigsimp(1 / sinh(x)**2 + 1) == 1 / tanh(x)**2
assert trigsimp(coth(x)**2 - 1 / sinh(x)**2) == 1
assert trigsimp(5 * cosh(x)**2 - 5 * sinh(x)**2) == 5
assert trigsimp(5 * cosh(x / 2)**2 -
2 * sinh(x / 2)**2) == 3 * cosh(x) / 2 + Rational(7, 2)
assert trigsimp(sinh(x) / cosh(x)) == tanh(x)
assert trigsimp(tanh(x)) == trigsimp(sinh(x) / cosh(x))
assert trigsimp(cosh(x) / sinh(x)) == 1 / tanh(x)
assert trigsimp(2 * tanh(x) * cosh(x)) == 2 * sinh(x)
assert trigsimp(coth(x)**3 * sinh(x)**3) == cosh(x)**3
assert trigsimp(y * tanh(x)**2 / sinh(x)**2) == y / cosh(x)**2
assert trigsimp(coth(x) / cosh(x)) == 1 / sinh(x)
e = 2 * cosh(x)**2 - 2 * sinh(x)**2
assert trigsimp(log(e)) == log(2)
assert trigsimp(cosh(x)**2 * cosh(y)**2 - cosh(x)**2 * sinh(y)**2 -
sinh(x)**2,
recursive=True) == 1
assert trigsimp(sinh(x)**2 * sinh(y)**2 - sinh(x)**2 * cosh(y)**2 +
cosh(x)**2,
recursive=True) == 1
assert abs(trigsimp(2.0 * cosh(x)**2 - 2.0 * sinh(x)**2) - 2.0) < 1e-10
assert trigsimp(sinh(x)**2 / cosh(x)**2) == tanh(x)**2
assert trigsimp(sinh(x)**3 / cosh(x)**3) == tanh(x)**3
assert trigsimp(sinh(x)**10 / cosh(x)**10) == tanh(x)**10
assert trigsimp(cosh(x)**3 / sinh(x)**3) == 1 / tanh(x)**3
assert trigsimp(cosh(x) / sinh(x)) == 1 / tanh(x)
assert trigsimp(cosh(x)**2 / sinh(x)**2) == 1 / tanh(x)**2
assert trigsimp(cosh(x)**10 / sinh(x)**10) == 1 / tanh(x)**10
assert trigsimp(x * cosh(x) * tanh(x)) == x * sinh(x)
assert trigsimp(-sinh(x) + cosh(x) * tanh(x)) == 0
assert tan(x) != 1 / cot(x) # cot doesn't auto-simplify
assert trigsimp(tan(x) - 1 / cot(x)) == 0
assert trigsimp(3 * tanh(x)**7 - 2 / coth(x)**7) == tanh(x)**7
def test_sympyissue_4494():
eq = sin(a)**2 * sin(b)**2 + cos(a)**2 * cos(b)**2 * tan(a)**2 + cos(a)**2
assert trigsimp(eq) == 1
def test_trigsimp():
assert trigsimp(A*sin(x)**2 + A*cos(x)**2) == A
def test_sympyissue_4373():
assert abs(trigsimp(2.0*sin(x)**2 + 2.0*cos(x)**2) - 2.0) < 1e-10
def test_trigsimp():
assert trigsimp(A * sin(x)**2 + A * cos(x)**2) == A
def test_sympyissue_4280():
assert trigsimp(cos(x)**2 + cos(y)**2*sin(x)**2 + sin(y)**2*sin(x)**2) == 1
assert trigsimp(a**2*sin(x)**2 + a**2*cos(y)**2*cos(x)**2 + a**2*cos(x)**2*sin(y)**2) == a**2
assert trigsimp(a**2*cos(y)**2*sin(x)**2 + a**2*sin(y)**2*sin(x)**2) == a**2*sin(x)**2
def test_sympyissue_4373():
assert abs(trigsimp(2.0 * sin(x)**2 + 2.0 * cos(x)**2) - 2.0) < 1e-10
def test_trigsimp_noncommutative():
A, B = symbols('A,B', commutative=False)
assert trigsimp(A - A*sin(x)**2) == A*cos(x)**2
assert trigsimp(A - A*cos(x)**2) == A*sin(x)**2
assert trigsimp(A*sin(x)**2 + A*cos(x)**2) == A
assert trigsimp(A + A*tan(x)**2) == A/cos(x)**2
assert trigsimp(A/cos(x)**2 - A) == A*tan(x)**2
assert trigsimp(A/cos(x)**2 - A*tan(x)**2) == A
assert trigsimp(A + A*cot(x)**2) == A/sin(x)**2
assert trigsimp(A/sin(x)**2 - A) == A/tan(x)**2
assert trigsimp(A/sin(x)**2 - A*cot(x)**2) == A
assert trigsimp(y*A*cos(x)**2 + y*A*sin(x)**2) == y*A
assert trigsimp(A*sin(x)/cos(x)) == A*tan(x)
assert trigsimp(A*tan(x)*cos(x)) == A*sin(x)
assert trigsimp(A*cot(x)**3*sin(x)**3) == A*cos(x)**3
assert trigsimp(y*A*tan(x)**2/sin(x)**2) == y*A/cos(x)**2
assert trigsimp(A*cot(x)/cos(x)) == A/sin(x)
assert trigsimp(A*sin(x + y) + A*sin(x - y)) == 2*A*sin(x)*cos(y)
assert trigsimp(A*sin(x + y) - A*sin(x - y)) == 2*A*sin(y)*cos(x)
assert trigsimp(A*cos(x + y) + A*cos(x - y)) == 2*A*cos(x)*cos(y)
assert trigsimp(A*cos(x + y) - A*cos(x - y)) == -2*A*sin(x)*sin(y)
assert trigsimp(A*sinh(x + y) + A*sinh(x - y)) == 2*A*sinh(x)*cosh(y)
assert trigsimp(A*sinh(x + y) - A*sinh(x - y)) == 2*A*sinh(y)*cosh(x)
assert trigsimp(A*cosh(x + y) + A*cosh(x - y)) == 2*A*cosh(x)*cosh(y)
assert trigsimp(A*cosh(x + y) - A*cosh(x - y)) == 2*A*sinh(x)*sinh(y)
assert trigsimp(A*cos(0.12345)**2 + A*sin(0.12345)**2) == 1.0*A
def test_mellin_transform_bessel():
from diofant import Max
MT = mellin_transform
# 8.4.19
assert MT(besselj(a, 2*sqrt(x)), x, s) == \
(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True)
assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(-2*s + Rational(1, 2))*gamma(a/2 + s + Rational(1, 2))/(
gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
-re(a)/2 - Rational(1, 2), Rational(1, 4)), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(-2*s + Rational(1, 2))/(
gamma(-a/2 - s + Rational(1, 2))*gamma(a - 2*s + 1)), (
-re(a)/2, Rational(1, 4)), True)
assert MT(besselj(a, sqrt(x))**2, x, s) == \
(gamma(a + s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, Rational(1, 2)), True)
# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(gamma(1 - s)*gamma(a + s - Rational(1, 2))
/ (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + Rational(1, 2))),
(Rational(1, 2) - re(a), Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
(4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s)
/ (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2)
* gamma( 1 - s + (a + b)/2)),
(-(re(a) + re(b))/2, Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), Rational(1, 2)), True)
# Section 8.4.20
assert MT(bessely(a, 2*sqrt(x)), x, s) == \
(-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi,
(Max(-re(a)/2, re(a)/2), Rational(3, 4)), True)
assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*sin(pi*(a/2 - s))*gamma(Rational(1, 2) - 2*s)
* gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s)
/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
(Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True)
assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(Rational(1, 2) - 2*s)
/ (sqrt(pi)*gamma(Rational(1, 2) - s - a/2)*gamma(Rational(1, 2) - s + a/2)),
(Max(-re(a)/2, re(a)/2), Rational(1, 4)), True)
assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(Rational(1, 2) - s)
/ (pi**Rational(3, 2)*gamma(1 + a - s)),
(Max(-re(a), 0), Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s)
* gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s)
/ (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
(Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), Rational(1, 2)), True)
# NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x))
# are a mess (no matter what way you look at it ...)
assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \
((Max(-re(a), 0, re(a)), Rational(1, 2)), True)
# Section 8.4.22
# TODO we can't do any of these (delicate cancellation)
# Section 8.4.23
assert MT(besselk(a, 2*sqrt(x)), x, s) == \
(gamma(
s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True)
assert MT(
besselj(a, 2 * sqrt(2 * sqrt(x))) * besselk(a, 2 * sqrt(2 * sqrt(x))),
x, s) == (4**(-s) * gamma(2 * s) * gamma(a / 2 + s) /
(2 * gamma(a / 2 - s + 1)), (Max(0, -re(a) / 2), oo), True)
# TODO bessely(a, x)*besselk(a, x) is a mess
assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(gamma(s)*gamma(
a + s)*gamma(-s + Rational(1, 2))/(2*sqrt(pi)*gamma(a - s + 1)),
(Max(-re(a), 0), Rational(1, 2)), True)
assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s) *
gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1) *
gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2,
re(a)/2 - re(b)/2), Rational(1, 2)), True)
# TODO products of besselk are a mess
mt = MT(exp(-x / 2) * besselk(a, x / 2), x, s)
mt0 = combsimp((trigsimp(combsimp(mt[0].expand(func=True)))))
assert mt0 == 2 * pi**Rational(3, 2) * cos(
pi * s) * gamma(-s + Rational(1, 2)) / (
(cos(2 * pi * a) - cos(2 * pi * s)) * gamma(-a - s + 1) *
gamma(a - s + 1))
assert mt[1:] == ((Max(-re(a), re(a)), oo), True)
def test_mellin_transform_bessel():
MT = mellin_transform
# 8.4.19
assert MT(besselj(a, 2*sqrt(x)), x, s) == \
(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True)
assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(-2*s + Rational(1, 2))*gamma(a/2 + s + Rational(1, 2))/(
gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
-re(a)/2 - Rational(1, 2), Rational(1, 4)), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(-2*s + Rational(1, 2))/(
gamma(-a/2 - s + Rational(1, 2))*gamma(a - 2*s + 1)), (
-re(a)/2, Rational(1, 4)), True)
assert MT(besselj(a, sqrt(x))**2, x, s) == \
(gamma(a + s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, Rational(1, 2)), True)
# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(gamma(1 - s)*gamma(a + s - Rational(1, 2))
/ (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + Rational(1, 2))),
(Rational(1, 2) - re(a), Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
(4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s)
/ (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2)
* gamma( 1 - s + (a + b)/2)),
(-(re(a) + re(b))/2, Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), Rational(1, 2)), True)
# Section 8.4.20
assert MT(bessely(a, 2*sqrt(x)), x, s) == \
(-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi,
(Max(-re(a)/2, re(a)/2), Rational(3, 4)), True)
assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*sin(pi*(a/2 - s))*gamma(Rational(1, 2) - 2*s)
* gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s)
/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
(Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True)
assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(Rational(1, 2) - 2*s)
/ (sqrt(pi)*gamma(Rational(1, 2) - s - a/2)*gamma(Rational(1, 2) - s + a/2)),
(Max(-re(a)/2, re(a)/2), Rational(1, 4)), True)
assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(Rational(1, 2) - s)
/ (pi**Rational(3, 2)*gamma(1 + a - s)),
(Max(-re(a), 0), Rational(1, 2)), True)
assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s)
* gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s)
/ (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
(Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), Rational(1, 2)), True)
# NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x))
# are a mess (no matter what way you look at it ...)
assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \
((Max(-re(a), 0, re(a)), Rational(1, 2)), True)
# Section 8.4.22
# TODO we can't do any of these (delicate cancellation)
# Section 8.4.23
assert MT(besselk(a, 2*sqrt(x)), x, s) == \
(gamma(
s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True)
assert MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(
a, 2*sqrt(2*sqrt(x))), x, s) == (4**(-s)*gamma(2*s) *
gamma(a/2 + s)/(2*gamma(a/2 - s + 1)), (Max(0, -re(a)/2), oo), True)
# TODO bessely(a, x)*besselk(a, x) is a mess
assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(gamma(s)*gamma(
a + s)*gamma(-s + Rational(1, 2))/(2*sqrt(pi)*gamma(a - s + 1)),
(Max(-re(a), 0), Rational(1, 2)), True)
assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s) *
gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1) *
gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2,
re(a)/2 - re(b)/2), Rational(1, 2)), True)
# TODO products of besselk are a mess
mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
mt0 = combsimp((trigsimp(combsimp(mt[0].expand(func=True)))))
assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(-s + Rational(1, 2))/(
(cos(2*pi*a) - cos(2*pi*s))*gamma(-a - s + 1)*gamma(a - s + 1))
assert mt[1:] == ((Max(-re(a), re(a)), oo), True)