def test_singular_values():
x = Symbol('x', real=True)
A = EigenOnlyMatrix([[0, 1*I], [2, 0]])
# if singular values can be sorted, they should be in decreasing order
assert A.singular_values() == [2, 1]
A = eye(3)
A[1, 1] = x
A[2, 2] = 5
vals = A.singular_values()
# since Abs(x) cannot be sorted, test set equality
assert set(vals) == set([5, 1, Abs(x)])
A = EigenOnlyMatrix([[sin(x), cos(x)], [-cos(x), sin(x)]])
vals = [sv.trigsimp() for sv in A.singular_values()]
assert vals == [S(1), S(1)]
A = EigenOnlyMatrix([
[2, 4],
[1, 3],
[0, 0],
[0, 0]
])
assert A.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221))]
assert A.T.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221)), 0, 0]
def as_real_imag(self, deep=True, **hints):
# TODO: Handle deep and hints
n, m, theta, phi = self.args
re = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
cos(m*phi) * assoc_legendre(n, m, cos(theta)))
im = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
sin(m*phi) * assoc_legendre(n, m, cos(theta)))
return (re, im)
def test_issue_3554():
x = Symbol("x")
assert (1 / sqrt(1 + cos(x) * sin(x ** 2))).series(x, 0, 7) == 1 - x ** 2 / 2 + 5 * x ** 4 / 8 - 5 * x ** 6 / 8 + O(
x ** 7
)
assert (1 / sqrt(1 + cos(x) * sin(x ** 2))).series(x, 0, 8) == 1 - x ** 2 / 2 + 5 * x ** 4 / 8 - 5 * x ** 6 / 8 + O(
x ** 8
)
def arbitrary_point(self, u=None, v=None):
""" Returns an arbitrary point on the Plane. If given two
parameters, the point ranges over the entire plane. If given 1
or no parameters, returns a point with one parameter which,
when varying from 0 to 2*pi, moves the point in a circle of
radius 1 about p1 of the Plane.
Examples
========
>>> from sympy.geometry import Plane, Ray
>>> from sympy.abc import u, v, t, r
>>> p = Plane((1, 1, 1), normal_vector=(1, 0, 0))
>>> p.arbitrary_point(u, v)
Point3D(1, u + 1, v + 1)
>>> p.arbitrary_point(t)
Point3D(1, cos(t) + 1, sin(t) + 1)
While arbitrary values of u and v can move the point anywhere in
the plane, the single-parameter point can be used to construct a
ray whose arbitrary point can be located at angle t and radius
r from p.p1:
>>> Ray(p.p1, _).arbitrary_point(r)
Point3D(1, r*cos(t) + 1, r*sin(t) + 1)
Returns
=======
Point3D
"""
circle = v is None
if circle:
u = _symbol(u or 't', real=True)
else:
u = _symbol(u or 'u', real=True)
v = _symbol(v or 'v', real=True)
x, y, z = self.normal_vector
a, b, c = self.p1.args
# x1, y1, z1 is a nonzero vector parallel to the plane
if x.is_zero and y.is_zero:
x1, y1, z1 = S.One, S.Zero, S.Zero
else:
x1, y1, z1 = -y, x, S.Zero
# x2, y2, z2 is also parallel to the plane, and orthogonal to x1, y1, z1
x2, y2, z2 = tuple(Matrix((x, y, z)).cross(Matrix((x1, y1, z1))))
if circle:
x1, y1, z1 = (w/sqrt(x1**2 + y1**2 + z1**2) for w in (x1, y1, z1))
x2, y2, z2 = (w/sqrt(x2**2 + y2**2 + z2**2) for w in (x2, y2, z2))
p = Point3D(a + x1*cos(u) + x2*sin(u), \
b + y1*cos(u) + y2*sin(u), \
c + z1*cos(u) + z2*sin(u))
else:
p = Point3D(a + x1*u + x2*v, b + y1*u + y2*v, c + z1*u + z2*v)
return p
def test_simplify():
n = Symbol('n')
f = Function('f')
M = OperationsOnlyMatrix([[ 1/x + 1/y, (x + x*y) / x ],
[ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
assert M.simplify() == Matrix([[ (x + y)/(x * y), 1 + y ],
[ 1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
eq = (1 + x)**2
M = OperationsOnlyMatrix([[eq]])
assert M.simplify() == Matrix([[eq]])
assert M.simplify(ratio=oo) == Matrix([[eq.simplify(ratio=oo)]])
def test_jacobian2():
rho, phi = symbols("rho,phi")
X = CalculusOnlyMatrix(3, 1, [rho*cos(phi), rho*sin(phi), rho**2])
Y = CalculusOnlyMatrix(2, 1, [rho, phi])
J = Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)],
[ 2*rho, 0],
])
assert X.jacobian(Y) == J
m = CalculusOnlyMatrix(2, 2, [1, 2, 3, 4])
m2 = CalculusOnlyMatrix(4, 1, [1, 2, 3, 4])
raises(TypeError, lambda: m.jacobian(Matrix([1,2])))
raises(TypeError, lambda: m2.jacobian(m))
def jn(n, z):
"""
Spherical Bessel function of the first kind.
Examples:
>>> from sympy import Symbol, jn, sin, cos
>>> z = Symbol("z")
>>> print jn(0, z)
sin(z)/z
>>> jn(1, z) == sin(z)/z**2 - cos(z)/z
True
>>> jn(3, z) ==(1/z - 15/z**3)*cos(z) + (15/z**4 - 6/z**2)*sin(z)
True
The spherical Bessel functions are calculated using the formula:
jn(n, z) == fn(n, z) * sin(z) + (-1)**(n+1) * fn(-n-1, z) * cos(z)
where fn(n, z) are the coefficients, see fn()'s sourcecode for more
information.
"""
n = sympify(n)
z = sympify(z)
return fn(n, z) * sin(z) + (-1)**(n+1) * fn(-n-1, z) * cos(z)
def finite_check(f, x, L):
def check_fx(exprs, x):
return x not in exprs.free_symbols
def check_sincos(expr, x, L):
if type(expr) == sin or type(expr) == cos:
sincos_args = expr.args[0]
if sincos_args.match(a*(pi/L)*x + b) is not None:
return True
else:
return False
expr = sincos_to_sum(TR2(TR1(f)))
res_expr = S.Zero
add_coeff = expr.as_coeff_add()
res_expr += add_coeff[0]
a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k != S.Zero, ])
b = Wild('b', properties=[lambda k: x not in k.free_symbols or k == S.Zero, ])
for s in add_coeff[1]:
mul_coeffs = s.as_coeff_mul()[1]
for t in mul_coeffs:
if not (check_fx(t, x) or check_sincos(t, x, L)):
return False, f
res_expr += TR10(s)
return True, res_expr.collect([sin(a*(pi/L)*x), cos(a*(pi/L)*x)])
def test_issue_3554s():
x = Symbol("x")
assert (1 / sqrt(1 + cos(x) * sin(x ** 2))).series(
x, 0, 15
) == 1 - x ** 2 / 2 + 5 * x ** 4 / 8 - 5 * x ** 6 / 8 + 4039 * x ** 8 / 5760 - 5393 * x ** 10 / 6720 + 13607537 * x ** 12 / 14515200 - 532056047 * x ** 14 / 479001600 + O(
x ** 15
)
def _eval_expand_func(self, **hints):
from sympy import Sum
n = self.args[0]
m = self.args[1] if len(self.args) == 2 else 1
if m == S.One:
if n.is_Add:
off = n.args[0]
nnew = n - off
if off.is_Integer and off.is_positive:
result = [S.One/(nnew + i) for i in range(off, 0, -1)] + [harmonic(nnew)]
return Add(*result)
elif off.is_Integer and off.is_negative:
result = [-S.One/(nnew + i) for i in range(0, off, -1)] + [harmonic(nnew)]
return Add(*result)
if n.is_Rational:
# Expansions for harmonic numbers at general rational arguments (u + p/q)
# Split n as u + p/q with p < q
p, q = n.as_numer_denom()
u = p // q
p = p - u * q
if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
k = Dummy("k")
t1 = q * Sum(1 / (q * k + p), (k, 0, u))
t2 = 2 * Sum(cos((2 * pi * p * k) / S(q)) *
log(sin((pi * k) / S(q))),
(k, 1, floor((q - 1) / S(2))))
t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
return t1 + t2 - t3
return self
def test_issue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**(S(1)/3)
assert (((1 + I)**(I*(1 + 7*f)))**(S(1)/3)).exp == S(1)/3
r = symbols('r', real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**(5/S(4))
p = symbols('p', positive=True)
assert cbrt(p**2) == p**(2/S(3))
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt(1 - I)/sqrt(2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**-S.Half == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == S.Half
assert sqrt(r**(4/S(3))) != r**(2/S(3))
assert sqrt((p + I)**(4/S(3))) == (p + I)**(2/S(3))
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_literal_evalf_is_number_is_zero_is_comparable():
from sympy.integrals.integrals import Integral
from sympy.core.symbol import symbols
from sympy.core.function import Function
from sympy.functions.elementary.trigonometric import cos, sin
x = symbols('x')
f = Function('f')
# the following should not be changed without a lot of dicussion
# `foo.is_number` should be equivalent to `not foo.free_symbols`
# it should not attempt anything fancy; see is_zero, is_constant
# and equals for more rigorous tests.
assert f(1).is_number is True
i = Integral(0, (x, x, x))
# expressions that are symbolically 0 can be difficult to prove
# so in case there is some easy way to know if something is 0
# it should appear in the is_zero property for that object;
# if is_zero is true evalf should always be able to compute that
# zero
assert i.n() == 0
assert i.is_zero
assert i.is_number is False
assert i.evalf(2, strict=False) == 0
# issue 10268
n = sin(1)**2 + cos(1)**2 - 1
assert n.is_comparable is False
assert n.n(2).is_comparable is False
assert n.n(2).n(2).is_comparable
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result U_n(x)
# U_n(-x) ---> (-1)**n * U_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevu(n, -x)
# U_{-n}(x) ---> -U_{n-2}(x)
if n.could_extract_minus_sign():
if n == S.NegativeOne:
return S.Zero
else:
return -chebyshevu(-n - 2, x)
# We can evaluate for some special values of x
if x == S.Zero:
return cos(S.Half * S.Pi * n)
if x == S.One:
return S.One + n
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# U_{-n}(x) ---> -U_{n-2}(x)
if n == S.NegativeOne:
return S.Zero
else:
return -cls._eval_at_order(-n - 2, x)
else:
return cls._eval_at_order(n, x)
def _eval_term(self, pt):
if pt == 0:
return self.a0
_term = self.an.get(pt, S.Zero) * cos(pt * (pi / self.L) * self.x) \
+ self.bn.get(pt, S.Zero) * sin(pt * (pi / self.L) * self.x)
return _term
def __new__(cls, f, limits, exprs):
if not (type(exprs) == tuple and len(exprs) == 3): # exprs is not of form (a0, an, bn)
# Converts the expression to fourier form
c, e = exprs.as_coeff_add()
rexpr = c + Add(*[TR10(i) for i in e])
a0, exp_ls = rexpr.expand(trig=False, power_base=False, power_exp=False, log=False).as_coeff_add()
x = limits[0]
L = abs(limits[2] - limits[1]) / 2
a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k is not S.Zero, ])
b = Wild('b', properties=[lambda k: x not in k.free_symbols, ])
an = dict()
bn = dict()
# separates the coefficients of sin and cos terms in dictionaries an, and bn
for p in exp_ls:
t = p.match(b * cos(a * (pi / L) * x))
q = p.match(b * sin(a * (pi / L) * x))
if t:
an[t[a]] = t[b] + an.get(t[a], S.Zero)
elif q:
bn[q[a]] = q[b] + bn.get(q[a], S.Zero)
else:
a0 += p
exprs = (a0, an, bn)
args = map(sympify, (f, limits, exprs))
return Expr.__new__(cls, *args)
def fourier_cos_seq(func, limits, n):
"""Returns the cos sequence in a Fourier series"""
from sympy.integrals import integrate
x, L = limits[0], limits[2] - limits[1]
cos_term = cos(2*n*pi*x / L)
formula = 2 * cos_term * integrate(func * cos_term, limits) / L
a0 = formula.subs(n, S.Zero) / 2
return a0, SeqFormula(2 * cos_term * integrate(func * cos_term, limits)
/ L, (n, 1, oo))
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (
sqrt((2 * n + 1) / (4 * pi) * C.factorial(n - m) / C.factorial(n + m))
* C.exp(I * m * phi)
* assoc_legendre(n, m, C.cos(theta))
)
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta) ** 2 + 1), sin(theta))
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
Examples
========
>>> from sympy import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)
Notes
=====
When creating a random point, one may simply replace the
parameter with a random number. When doing so, however, the
random number should be made a Rational or else the point
may not test as being in the ellipse:
>>> from sympy.abc import t
>>> from sympy import Rational
>>> arb = e1.arbitrary_point(t); arb
Point2D(3*cos(t), 2*sin(t))
>>> arb.subs(t, .1) in e1
False
>>> arb.subs(t, Rational(.1)) in e1
True
>>> arb.subs(t, Rational('.1')) in e1
True
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
"""
from sympy import sin, cos, Rational
t = _symbol('t', real=True)
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
# simplify this now or else the Float will turn s into a Float
r = Rational(rng.random())
c = 2*r - 1
s = sqrt(1 - c**2)
return Point(x.subs(cos(t), c), y.subs(sin(t), s))
def test_expand():
m0 = OperationsOnlyMatrix([[x*(x + y), 2], [((x + y)*y)*x, x*(y + x*(x + y))]])
# Test if expand() returns a matrix
m1 = m0.expand()
assert m1 == Matrix(
[[x*y + x**2, 2], [x*y**2 + y*x**2, x*y + y*x**2 + x**3]])
a = Symbol('a', real=True)
assert OperationsOnlyMatrix(1, 1, [exp(I*a)]).expand(complex=True) == \
Matrix([cos(a) + I*sin(a)])
def taylor_term(n, x, *previous_terms):
if n < 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 1:
p = previous_terms[-1]
return (3**(S(1)/3)*x * Abs(sin(2*pi*(n + S.One)/S(3))) * C.factorial((n - S.One)/S(3)) /
((n + S.One) * Abs(cos(2*pi*(n + S.Half)/S(3))) * C.factorial((n - 2)/S(3))) * p)
else:
return (S.One/(root(3, 6)*pi) * gamma((n + S.One)/S(3)) * Abs(sin(2*pi*(n + S.One)/S(3))) /
C.factorial(n) * (root(3, 3)*x)**n)
def _fourier_transform(seq, dps, inverse=False):
"""Utility function for the Discrete Fourier Transform"""
if not iterable(seq):
raise TypeError("Expected a sequence of numeric coefficients "
"for Fourier Transform")
a = [sympify(arg) for arg in seq]
if any(x.has(Symbol) for x in a):
raise ValueError("Expected non-symbolic coefficients")
n = len(a)
if n < 2:
return a
b = n.bit_length() - 1
if n&(n - 1): # not a power of 2
b += 1
n = 2**b
a += [S.Zero]*(n - len(a))
for i in range(1, n):
j = int(ibin(i, b, str=True)[::-1], 2)
if i < j:
a[i], a[j] = a[j], a[i]
ang = -2*pi/n if inverse else 2*pi/n
if dps is not None:
ang = ang.evalf(dps + 2)
w = [cos(ang*i) + I*sin(ang*i) for i in range(n // 2)]
h = 2
while h <= n:
hf, ut = h // 2, n // h
for i in range(0, n, h):
for j in range(hf):
u, v = a[i + j], expand_mul(a[i + j + hf]*w[ut * j])
a[i + j], a[i + j + hf] = u + v, u - v
h *= 2
if inverse:
a = [(x/n).evalf(dps) for x in a] if dps is not None \
else [x/n for x in a]
return a
def test_Max():
from sympy.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
r = Symbol('r', real=True)
assert Max(5, 4) == 5
# lists
raises(ValueError, lambda: Max())
assert Max(x, y) == Max(y, x)
assert Max(x, y, z) == Max(z, y, x)
assert Max(x, Max(y, z)) == Max(z, y, x)
assert Max(x, Min(y, oo)) == Max(x, y)
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p) == p
assert Max(2, x, p, n, -oo, S.NegativeInfinity, n_, p, 2) == Max(2, x, p)
assert Max(0, x, 1, y) == Max(1, x, y)
assert Max(r, r + 1, r - 1) == 1 + r
assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
assert Max(cos(x), sin(x)).subs(x, 1) == sin(1)
assert Max(cos(x), sin(x)).subs(x, S(1)/2) == cos(S(1)/2)
raises(ValueError, lambda: Max(cos(x), sin(x)).subs(x, I))
raises(ValueError, lambda: Max(I))
raises(ValueError, lambda: Max(I, x))
raises(ValueError, lambda: Max(S.ComplexInfinity, 1))
# interesting:
# Max(n, -oo, n_, p, 2) == Max(p, 2)
# True
# Max(n, -oo, n_, p, 1000) == Max(p, 1000)
# False
assert Max(1, x).diff(x) == Heaviside(x - 1)
assert Max(x, 1).diff(x) == Heaviside(x - 1)
assert Max(x**2, 1 + x, 1).diff(x) == \
2*x*Heaviside(x**2 - Max(1, x + 1)) \
+ Heaviside(x - Max(1, x**2) + 1)
a, b = Symbol('a', real=True), Symbol('b', real=True)
# a and b are both real, Max(a, b) should be real
assert Max(a, b).is_real
# issue 7233
e = Max(0, x)
assert e.evalf == e.n
assert e.n().args == (0, x)
def _eval_expand_func(self, **hints):
n, z = self.args
if n.is_Integer and n.is_nonnegative:
if z.is_Add:
coeff = z.args[0]
if coeff.is_Integer:
e = -(n + 1)
if coeff > 0:
tail = Add(*[Pow(
z - i, e) for i in range(1, int(coeff) + 1)])
else:
tail = -Add(*[Pow(
z + i, e) for i in range(0, int(-coeff))])
return polygamma(n, z - coeff) + (-1)**n*factorial(n)*tail
elif z.is_Mul:
coeff, z = z.as_two_terms()
if coeff.is_Integer and coeff.is_positive:
tail = [ polygamma(n, z + Rational(
i, coeff)) for i in range(0, int(coeff)) ]
if n == 0:
return Add(*tail)/coeff + log(coeff)
else:
return Add(*tail)/coeff**(n + 1)
z *= coeff
if n == 0 and z.is_Rational:
p, q = z.as_numer_denom()
# Reference:
# Values of the polygamma functions at rational arguments, J. Choi, 2007
part_1 = -S.EulerGamma - pi * cot(p * pi / q) / 2 - log(q) + Add(
*[cos(2 * k * pi * p / q) * log(2 * sin(k * pi / q)) for k in range(1, q)])
if z > 0:
n = floor(z)
z0 = z - n
return part_1 + Add(*[1 / (z0 + k) for k in range(n)])
elif z < 0:
n = floor(1 - z)
z0 = z + n
return part_1 - Add(*[1 / (z0 - 1 - k) for k in range(n)])
return polygamma(n, z)
def apothem(self):
"""The inradius of the RegularPolygon.
The apothem/inradius is the radius of the inscribed circle.
Returns
-------
apothem : number or instance of Basic
Examples
--------
>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.apothem
sqrt(2)*r/2
"""
return self.radius * cos(S.Pi/self._n)
def rot_axis3(theta):
"""Returns a rotation matrix for a rotation of theta (in radians) about
the 3-axis.
Examples
========
>>> from sympy import pi
>>> from sympy.matrices import rot_axis3
A rotation of pi/3 (60 degrees):
>>> theta = pi/3
>>> rot_axis3(theta)
Matrix([
[ 1/2, sqrt(3)/2, 0],
[-sqrt(3)/2, 1/2, 0],
[ 0, 0, 1]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis3(pi/2)
Matrix([
[ 0, 1, 0],
[-1, 0, 0],
[ 0, 0, 1]])
See Also
========
rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
about the 1-axis
rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
about the 2-axis
"""
ct = cos(theta)
st = sin(theta)
lil = ((ct, st, 0),
(-st, ct, 0),
(0, 0, 1))
return Matrix(lil)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point2D(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name))
return Point(self.center.x + self.hradius*cos(t),
self.center.y + self.vradius*sin(t))
def rot_axis1(theta):
"""Returns a rotation matrix for a rotation of theta (in radians) about
the 1-axis.
Examples
========
>>> from sympy import pi
>>> from sympy.matrices import rot_axis1
A rotation of pi/3 (60 degrees):
>>> theta = pi/3
>>> rot_axis1(theta)
[1, 0, 0]
[0, 1/2, sqrt(3)/2]
[0, -sqrt(3)/2, 1/2]
If we rotate by pi/2 (90 degrees):
>>> rot_axis1(pi/2)
[1, 0, 0]
[0, 0, 1]
[0, -1, 0]
See Also
========
rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
about the 2-axis
rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
about the 3-axis
"""
ct = cos(theta)
st = sin(theta)
lil = ((1, 0, 0),
(0, ct, st),
(0, -st, ct))
return Matrix(lil)
def __getitem__(self, k):
"""
>>> from sympy import Polygon, Point
>>> r = Polygon(Point(0, 0), 1, n=3)
>>> r[0]
Point(1, 0)
Note that iteration and indexing do not give the same results.
>>> for ri in r:
... print ri
Point(0, 0)
1
3
0
"""
if k < -self._n or k >= self._n:
raise IndexError('virtual tuple index out of range')
c = self._center
r = self._radius
rot = self._rot
v = 2*S.Pi/self._n
return Point(c[0] + r*cos(k*v + rot), c[1] + r*sin(k*v + rot))
def _minpoly_sin(ex, x):
"""
Returns the minimal polynomial of ``sin(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
n = c.q
q = sympify(n)
if q.is_prime:
# for a = pi*p/q with q odd prime, using chebyshevt
# write sin(q*a) = mp(sin(a))*sin(a);
# the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1
a = dup_chebyshevt(n, ZZ)
return Add(*[x**(n - i - 1)*a[i] for i in range(n)])
if c.p == 1:
if q == 9:
return 64*x**6 - 96*x**4 + 36*x**2 - 3
if n % 2 == 1:
# for a = pi*p/q with q odd, use
# sin(q*a) = 0 to see that the minimal polynomial must be
# a factor of dup_chebyshevt(n, ZZ)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i)*a[i] for i in range(n + 1)]
r = Add(*a)
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
expr = ((1 - cos(2*c*pi))/2)**S.Half
res = _minpoly_compose(expr, x, QQ)
return res
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def test_exp2():
e1 = exp(cos(x)).series(x, 0)
e2 = series(exp(cos(x)), x, 0)
assert e1 == e2
def _eval_rewrite_as_besselj(self, nu, z):
if nu.is_integer is False:
return csc(
pi * nu) * (cos(pi * nu) * besselj(nu, z) - besselj(-nu, z))
def _expand(self):
n = self.order
z = self.argument
return fn(n, z) * sin(z) + (-1)**(n+1) * fn(-n-1, z) * cos(z)
def test_issue_11884():
assert cos(x).series(x, 1, n=1) == cos(1) + O(x - 1, (x, 1))
def test_checksysodesol():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \
Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + Rational(43, 2))) + 4*C2*exp(t*(sqrt(1713)/2 + \
Rational(43, 2)))), Eq(y(t), C1*(-sqrt(1713)/2 + Rational(39, 2))*exp(t*(-sqrt(1713)/2 + \
Rational(43, 2))) + C2*(Rational(39, 2) + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + Rational(43, 2))))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2))), \
Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \
C2/2)*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \
Rational(22, 3)), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \
sqrt(6))*exp(t*(sqrt(6) + 3)) - Rational(5, 3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - Rational(58, 3)), \
Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - Rational(185, 3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*exp(Integral(2, t).doit()) + C2*exp(-(Integral(2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \
C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \
C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
C2*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \
Eq(y(t), (C1*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
C2*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
root0 = -sqrt(-sqrt(47) + 7)
root1 = sqrt(-sqrt(47) + 7)
root2 = -sqrt(sqrt(47) + 7)
root3 = sqrt(sqrt(47) + 7)
sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \
Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \
C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
root0 = -sqrt(-sqrt(109)/2 + Rational(15, 2))
root1 = sqrt(-sqrt(109)/2 + Rational(15, 2))
root2 = -sqrt(sqrt(109)/2 + Rational(15, 2))
root3 = sqrt(sqrt(109)/2 + Rational(15, 2))
sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - Rational(181, 29)), \
Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \
C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + Rational(183, 29))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol = [Eq(x(t), C1*cos(t*(Rational(9, 2) + sqrt(109)/2)) + C2*sin(t*(Rational(9, 2) + sqrt(109)/2)) + \
C3*cos(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*sin(t*(-sqrt(109)/2 + Rational(9, 2)))), Eq(y(t), -C1*sin(t*(Rational(9, 2) + sqrt(109)/2)) \
+ C2*cos(t*(Rational(9, 2) + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*cos(t*(-sqrt(109)/2 + Rational(9, 2))))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
I1 = sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erfi(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(3*sqrt(7)*t**2/2)/t
I2 = -sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erf(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(-3*sqrt(7)*t**2/2)/t
sol = [Eq(x(t), C3*t + t*(9*C1*I1 + 9*C2*I2)), Eq(y(t), C4*t + t*(3*sqrt(7)*C1*I1 - 3*sqrt(7)*C2*I2))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t)))
sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \
Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol = [Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol = [Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)}
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_deltaintegrate():
assert deltaintegrate(x, x) is None
assert deltaintegrate(x + DiracDelta(x), x) is None
assert deltaintegrate(DiracDelta(x, 0), x) == Heaviside(x)
for n in range(10):
assert deltaintegrate(DiracDelta(x, n + 1), x) == DiracDelta(x, n)
assert deltaintegrate(DiracDelta(x), x) == Heaviside(x)
assert deltaintegrate(DiracDelta(-x), x) == Heaviside(x)
assert deltaintegrate(DiracDelta(x - y), x) == Heaviside(x - y)
assert deltaintegrate(DiracDelta(y - x), x) == Heaviside(x - y)
assert deltaintegrate(x * DiracDelta(x), x) == 0
assert deltaintegrate((x - y) * DiracDelta(x - y), x) == 0
assert deltaintegrate(DiracDelta(x)**2, x) == DiracDelta(0) * Heaviside(x)
assert deltaintegrate(y*DiracDelta(x)**2, x) == \
y*DiracDelta(0)*Heaviside(x)
assert deltaintegrate(DiracDelta(x, 1), x) == DiracDelta(x, 0)
assert deltaintegrate(y * DiracDelta(x, 1), x) == y * DiracDelta(x, 0)
assert deltaintegrate(DiracDelta(x, 1)**2,
x) == -DiracDelta(0, 2) * Heaviside(x)
assert deltaintegrate(y * DiracDelta(x, 1)**2,
x) == -y * DiracDelta(0, 2) * Heaviside(x)
assert deltaintegrate(DiracDelta(x) * f(x), x) == f(0) * Heaviside(x)
assert deltaintegrate(DiracDelta(-x) * f(x), x) == f(0) * Heaviside(x)
assert deltaintegrate(DiracDelta(x - 1) * f(x),
x) == f(1) * Heaviside(x - 1)
assert deltaintegrate(DiracDelta(1 - x) * f(x),
x) == f(1) * Heaviside(x - 1)
assert deltaintegrate(DiracDelta(x**2 + x - 2), x) == \
Heaviside(x - 1)/3 + Heaviside(x + 2)/3
p = cos(x) * (DiracDelta(x) + DiracDelta(x**2 - 1)) * sin(x) * (x - pi)
assert deltaintegrate(p, x) - (-pi*(cos(1)*Heaviside(-1 + x)*sin(1)/2 - \
cos(1)*Heaviside(1 + x)*sin(1)/2) + \
cos(1)*Heaviside(1 + x)*sin(1)/2 + \
cos(1)*Heaviside(-1 + x)*sin(1)/2) == 0
p = x_2 * DiracDelta(x - x_2) * DiracDelta(x_2 - x_1)
assert deltaintegrate(p,
x_2) == x * DiracDelta(x - x_1) * Heaviside(x_2 - x)
p = x * y**2 * z * DiracDelta(y - x) * DiracDelta(y - z) * DiracDelta(x -
z)
assert deltaintegrate(
p, y) == x**3 * z * DiracDelta(x - z)**2 * Heaviside(y - x)
assert deltaintegrate((x + 1) * DiracDelta(2 * x),
x) == S.Half * Heaviside(x)
assert deltaintegrate((x + 1)*DiracDelta(x*Rational(2, 3) + Rational(4, 9)), x) == \
S.Half * Heaviside(x + Rational(2, 3))
a, b, c = symbols('a b c', commutative=False)
assert deltaintegrate(DiracDelta(x - y)*f(x - b)*f(x - a), x) == \
f(y - b)*f(y - a)*Heaviside(x - y)
p = f(x - a) * DiracDelta(x - y) * f(x - c) * f(x - b)
assert deltaintegrate(
p, x) == f(y - a) * f(y - c) * f(y - b) * Heaviside(x - y)
p = DiracDelta(x - z) * f(x - b) * f(x - a) * DiracDelta(x - y)
assert deltaintegrate(p, x) == DiracDelta(y - z)*f(y - b)*f(y - a) * \
Heaviside(x - y)
def expr(self):
from sympy.functions.elementary.trigonometric import sin, cos
r, theta = self.variables
return r * (cos(theta) + S.ImaginaryUnit * sin(theta))
def test_issue_6653():
x = Symbol('x')
assert (1 / sqrt(1 + cos(x) * sin(x**2))).series(x, 0, 7) == \
1 - x**2/2 + 5*x**4/8 - 5*x**6/8 + O(x**7)
assert (1 / sqrt(1 + cos(x) * sin(x**2))).series(x, 0, 8) == \
1 - x**2/2 + 5*x**4/8 - 5*x**6/8 + O(x**8)
def test_inverse_mellin_transform():
from sympy.core.function import expand
from sympy.functions.elementary.miscellaneous import (Max, Min)
from sympy.functions.elementary.trigonometric import cot
from sympy.simplify.powsimp import powsimp
from sympy.simplify.simplify import simplify
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x
# test factorisation of polys
r = symbols('r', real=True)
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
).subs(x, r).rewrite(sin).simplify() \
== sin(r)*Heaviside(1 - exp(-r))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a)
assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', real=True)
assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
*gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
*gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi/(s*sin(2*pi*s)), s, x, (Rational(-1, 2), 0)) == log(sqrt(x) + 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x)
# TODO
def mysimp(expr):
from sympy.core.function import expand
from sympy.simplify.powsimp import powsimp
from sympy.simplify.simplify import logcombine
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [
log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1),
log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1)]
# test passing cot
assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [
log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1),
-log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1), ]
# 8.4.14
assert IMT(-gamma(s + S.Half)/(sqrt(pi)*s), s, x, (Rational(-1, 2), 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(S.Half - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S.Half - 2*s)
/ (gamma(S.Half - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, Rational(1, 4)))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(S.Half - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), S.Half))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(S.Half - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, S.Half))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
*gamma(a/2 + b/2 - s + 1)),
s, x, (-(re(a) + re(b))/2, S.Half))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
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)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S.Half))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi/cos(pi*s), s, x, (0, S.Half)) == sqrt(x)/(x + 1)
def test_mellin_transform_bessel():
from sympy.functions.elementary.miscellaneous 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 + S.Half)*gamma(a/2 + s + S.Half)/(
gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
-re(a)/2 - S.Half, Rational(1, 4)), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(-2*s + S.Half)/(
gamma(-a/2 - s + S.Half)*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(S.Half - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), S.Half), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(S.Half - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, S.Half), 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 - S.Half)
/ (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + S.Half)),
(S.Half - re(a), S.Half), 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, S.Half), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), S.Half), 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(S.Half - 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(S.Half - 2*s)
/ (sqrt(pi)*gamma(S.Half - s - a/2)*gamma(S.Half - 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(S.Half - s)
/ (pi**S('3/2')*gamma(1 + a - s)),
(Max(-re(a), 0), S.Half), 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), S.Half), 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)), S.Half), 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 + S.Half)/(2*sqrt(pi)*gamma(a - s + 1)),
(Max(-re(a), 0), S.Half), 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), S.Half), True)
# TODO products of besselk are a mess
mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
mt0 = gammasimp(trigsimp(gammasimp(mt[0].expand(func=True))))
assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(S.Half - s)/(
(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_laplace_transform():
from sympy import lowergamma
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import (fresnelc, fresnels)
LT = laplace_transform
a, b, c, = symbols('a, b, c', positive=True)
t, w, x = symbols('t, w, x')
f = Function("f")
g = Function("g")
# Test rule-base evaluation according to
# http://eqworld.ipmnet.ru/en/auxiliary/inttrans/
# Power-law functions (laplace2.pdf)
assert LT(a*t+t**2+t**(S(5)/2), t, s) ==\
(a/s**2 + 2/s**3 + 15*sqrt(pi)/(8*s**(S(7)/2)), 0, True)
assert LT(b/(t+a), t, s) == (-b*exp(-a*s)*Ei(-a*s), 0, True)
assert LT(1/sqrt(t+a), t, s) ==\
(sqrt(pi)*sqrt(1/s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True)
assert LT(sqrt(t)/(t+a), t, s) ==\
(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
0, True)
assert LT((t+a)**(-S(3)/2), t, s) ==\
(-2*sqrt(pi)*sqrt(s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + 2/sqrt(a),
0, True)
assert LT(t**(S(1)/2)*(t+a)**(-1), t, s) ==\
(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
0, True)
assert LT(1/(a*sqrt(t) + t**(3/2)), t, s) ==\
(pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True)
assert LT((t+a)**b, t, s) ==\
(s**(-b - 1)*exp(-a*s)*lowergamma(b + 1, a*s), 0, True)
assert LT(t**5/(t+a), t, s) == (120*a**5*lowergamma(-5, a*s), 0, True)
# Exponential functions (laplace3.pdf)
assert LT(exp(t), t, s) == (1/(s - 1), 1, True)
assert LT(exp(2*t), t, s) == (1/(s - 2), 2, True)
assert LT(exp(a*t), t, s) == (1/(s - a), a, True)
assert LT(exp(a*(t-b)), t, s) == (exp(-a*b)/(-a + s), a, True)
assert LT(t*exp(-a*(t)), t, s) == ((a + s)**(-2), -a, True)
assert LT(t*exp(-a*(t-b)), t, s) == (exp(a*b)/(a + s)**2, -a, True)
assert LT(b*t*exp(-a*t), t, s) == (b/(a + s)**2, -a, True)
assert LT(t**(S(7)/4)*exp(-8*t)/gamma(S(11)/4), t, s) ==\
((s + 8)**(-S(11)/4), -8, True)
assert LT(t**(S(3)/2)*exp(-8*t), t, s) ==\
(3*sqrt(pi)/(4*(s + 8)**(S(5)/2)), -8, True)
assert LT(t**a*exp(-a*t), t, s) == ((a+s)**(-a-1)*gamma(a+1), -a, True)
assert LT(b*exp(-a*t**2), t, s) ==\
(sqrt(pi)*b*exp(s**2/(4*a))*erfc(s/(2*sqrt(a)))/(2*sqrt(a)), 0, True)
assert LT(exp(-2*t**2), t, s) ==\
(sqrt(2)*sqrt(pi)*exp(s**2/8)*erfc(sqrt(2)*s/4)/4, 0, True)
assert LT(b*exp(2*t**2), t, s) == b*LaplaceTransform(exp(2*t**2), t, s)
assert LT(t*exp(-a*t**2), t, s) ==\
(1/(2*a) - s*erfc(s/(2*sqrt(a)))/(4*sqrt(pi)*a**(S(3)/2)), 0, True)
assert LT(exp(-a/t), t, s) ==\
(2*sqrt(a)*sqrt(1/s)*besselk(1, 2*sqrt(a)*sqrt(s)), 0, True)
assert LT(sqrt(t)*exp(-a/t), t, s) ==\
(sqrt(pi)*(2*sqrt(a)*sqrt(s) + 1)*sqrt(s**(-3))*exp(-2*sqrt(a)*\
sqrt(s))/2, 0, True)
assert LT(exp(-a/t)/sqrt(t), t, s) ==\
(sqrt(pi)*sqrt(1/s)*exp(-2*sqrt(a)*sqrt(s)), 0, True)
assert LT( exp(-a/t)/(t*sqrt(t)), t, s) ==\
(sqrt(pi)*sqrt(1/a)*exp(-2*sqrt(a)*sqrt(s)), 0, True)
assert LT(exp(-2*sqrt(a*t)), t, s) ==\
( 1/s -sqrt(pi)*sqrt(a) * exp(a/s)*erfc(sqrt(a)*sqrt(1/s))/\
s**(S(3)/2), 0, True)
assert LT(exp(-2*sqrt(a*t))/sqrt(t), t, s) == (exp(a/s)*erfc(sqrt(a)*\
sqrt(1/s))*(sqrt(pi)*sqrt(1/s)), 0, True)
assert LT(t**4*exp(-2/t), t, s) ==\
(8*sqrt(2)*(1/s)**(S(5)/2)*besselk(5, 2*sqrt(2)*sqrt(s)), 0, True)
# Hyperbolic functions (laplace4.pdf)
assert LT(sinh(a*t), t, s) == (a/(-a**2 + s**2), a, True)
assert LT(b*sinh(a*t)**2, t, s) == (2*a**2*b/(-4*a**2*s**2 + s**3),
2*a, True)
# The following line confirms that issue #21202 is solved
assert LT(cosh(2*t), t, s) == (s/(-4 + s**2), 2, True)
assert LT(cosh(a*t), t, s) == (s/(-a**2 + s**2), a, True)
assert LT(cosh(a*t)**2, t, s) == ((-2*a**2 + s**2)/(-4*a**2*s**2 + s**3),
2*a, True)
assert LT(sinh(x + 3), x, s) == (
(-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 0, Abs(s) > 1)
# The following line replaces the old test test_issue_7173()
assert LT(sinh(a*t)*cosh(a*t), t, s) == (a/(-4*a**2 + s**2), 2*a, True)
assert LT(sinh(a*t)/t, t, s) == (log((a + s)/(-a + s))/2, a, True)
assert LT(t**(-S(3)/2)*sinh(a*t), t, s) ==\
(-sqrt(pi)*(sqrt(-a + s) - sqrt(a + s)), a, True)
assert LT(sinh(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*sqrt(a)*exp(a/s)/s**(S(3)/2), 0, True)
assert LT(sqrt(t)*sinh(2*sqrt(a*t)), t, s) ==\
(-sqrt(a)/s**2 + sqrt(pi)*(a + s/2)*exp(a/s)*erf(sqrt(a)*\
sqrt(1/s))/s**(S(5)/2), 0, True)
assert LT(sinh(2*sqrt(a*t))/sqrt(t), t, s) ==\
(sqrt(pi)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/sqrt(s), 0, True)
assert LT(sinh(sqrt(a*t))**2/sqrt(t), t, s) ==\
(sqrt(pi)*(exp(a/s) - 1)/(2*sqrt(s)), 0, True)
assert LT(t**(S(3)/7)*cosh(a*t), t, s) ==\
(((a + s)**(-S(10)/7) + (-a+s)**(-S(10)/7))*gamma(S(10)/7)/2, a, True)
assert LT(cosh(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*sqrt(a)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/s**(S(3)/2) + 1/s,
0, True)
assert LT(sqrt(t)*cosh(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*(a + s/2)*exp(a/s)/s**(S(5)/2), 0, True)
assert LT(cosh(2*sqrt(a*t))/sqrt(t), t, s) ==\
(sqrt(pi)*exp(a/s)/sqrt(s), 0, True)
assert LT(cosh(sqrt(a*t))**2/sqrt(t), t, s) ==\
(sqrt(pi)*(exp(a/s) + 1)/(2*sqrt(s)), 0, True)
# logarithmic functions (laplace5.pdf)
assert LT(log(t), t, s) == (-log(s+S.EulerGamma)/s, 0, True)
assert LT(log(t/a), t, s) == (-log(a*s + S.EulerGamma)/s, 0, True)
assert LT(log(1+a*t), t, s) == (-exp(s/a)*Ei(-s/a)/s, 0, True)
assert LT(log(t+a), t, s) == ((log(a) - exp(s/a)*Ei(-s/a)/s)/s, 0, True)
assert LT(log(t)/sqrt(t), t, s) ==\
(sqrt(pi)*(-log(s) - 2*log(2) - S.EulerGamma)/sqrt(s), 0, True)
assert LT(t**(S(5)/2)*log(t), t, s) ==\
(15*sqrt(pi)*(-log(s)-2*log(2)-S.EulerGamma+S(46)/15)/(8*s**(S(7)/2)),
0, True)
assert (LT(t**3*log(t), t, s, noconds=True)-6*(-log(s) - S.EulerGamma\
+ S(11)/6)/s**4).simplify() == S.Zero
assert LT(log(t)**2, t, s) ==\
(((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True)
assert LT(exp(-a*t)*log(t), t, s) ==\
((-log(a + s) - S.EulerGamma)/(a + s), -a, True)
# Trigonometric functions (laplace6.pdf)
assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
assert LT(Abs(sin(a*t)), t, s) ==\
(a*coth(pi*s/(2*a))/(a**2 + s**2), 0, True)
assert LT(sin(a*t)/t, t, s) == (atan(a/s), 0, True)
assert LT(sin(a*t)**2/t, t, s) == (log(4*a**2/s**2 + 1)/4, 0, True)
assert LT(sin(a*t)**2/t**2, t, s) ==\
(a*atan(2*a/s) - s*log(4*a**2/s**2 + 1)/4, 0, True)
assert LT(sin(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*sqrt(a)*exp(-a/s)/s**(S(3)/2), 0, True)
assert LT(sin(2*sqrt(a*t))/t, t, s) == (pi*erf(sqrt(a)*sqrt(1/s)), 0, True)
assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
assert LT(cos(a*t)**2, t, s) ==\
((2*a**2 + s**2)/(s*(4*a**2 + s**2)), 0, True)
assert LT(sqrt(t)*cos(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*(-2*a + s)*exp(-a/s)/(2*s**(S(5)/2)), 0, True)
assert LT(cos(2*sqrt(a*t))/sqrt(t), t, s) ==\
(sqrt(pi)*sqrt(1/s)*exp(-a/s), 0, True)
assert LT(sin(a*t)*sin(b*t), t, s) ==\
(2*a*b*s/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True)
assert LT(cos(a*t)*sin(b*t), t, s) ==\
(b*(-a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
0, True)
assert LT(cos(a*t)*cos(b*t), t, s) ==\
(s*(a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
0, True)
assert LT(c*exp(-b*t)*sin(a*t), t, s) == (a*c/(a**2 + (b + s)**2),
-b, True)
assert LT(c*exp(-b*t)*cos(a*t), t, s) == ((b + s)*c/(a**2 + (b + s)**2),
-b, True)
assert LT(cos(x + 3), x, s) == ((s*cos(3) - sin(3))/(s**2 + 1), 0, True)
# Error functions (laplace7.pdf)
assert LT(erf(a*t), t, s) == (exp(s**2/(4*a**2))*erfc(s/(2*a))/s, 0, True)
assert LT(erf(sqrt(a*t)), t, s) == (sqrt(a)/(s*sqrt(a + s)), 0, True)
assert LT(exp(a*t)*erf(sqrt(a*t)), t, s) ==\
(sqrt(a)/(sqrt(s)*(-a + s)), a, True)
assert LT(erf(sqrt(a/t)/2), t, s) == ((1-exp(-sqrt(a)*sqrt(s)))/s, 0, True)
assert LT(erfc(sqrt(a*t)), t, s) ==\
((-sqrt(a) + sqrt(a + s))/(s*sqrt(a + s)), 0, True)
assert LT(exp(a*t)*erfc(sqrt(a*t)), t, s) ==\
(1/(sqrt(a)*sqrt(s) + s), 0, True)
assert LT(erfc(sqrt(a/t)/2), t, s) == (exp(-sqrt(a)*sqrt(s))/s, 0, True)
# Bessel functions (laplace8.pdf)
assert LT(besselj(0, a*t), t, s) == (1/sqrt(a**2 + s**2), 0, True)
assert LT(besselj(1, a*t), t, s) ==\
(a/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))), 0, True)
assert LT(besselj(2, a*t), t, s) ==\
(a**2/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))**2), 0, True)
assert LT(t*besselj(0, a*t), t, s) ==\
(s/(a**2 + s**2)**(S(3)/2), 0, True)
assert LT(t*besselj(1, a*t), t, s) ==\
(a/(a**2 + s**2)**(S(3)/2), 0, True)
assert LT(t**2*besselj(2, a*t), t, s) ==\
(3*a**2/(a**2 + s**2)**(S(5)/2), 0, True)
assert LT(besselj(0, 2*sqrt(a*t)), t, s) == (exp(-a/s)/s, 0, True)
assert LT(t**(S(3)/2)*besselj(3, 2*sqrt(a*t)), t, s) ==\
(a**(S(3)/2)*exp(-a/s)/s**4, 0, True)
assert LT(besselj(0, a*sqrt(t**2+b*t)), t, s) ==\
(exp(b*s - b*sqrt(a**2 + s**2))/sqrt(a**2 + s**2), 0, True)
assert LT(besseli(0, a*t), t, s) == (1/sqrt(-a**2 + s**2), a, True)
assert LT(besseli(1, a*t), t, s) ==\
(a/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))), a, True)
assert LT(besseli(2, a*t), t, s) ==\
(a**2/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))**2), a, True)
assert LT(t*besseli(0, a*t), t, s) == (s/(-a**2 + s**2)**(S(3)/2), a, True)
assert LT(t*besseli(1, a*t), t, s) == (a/(-a**2 + s**2)**(S(3)/2), a, True)
assert LT(t**2*besseli(2, a*t), t, s) ==\
(3*a**2/(-a**2 + s**2)**(S(5)/2), a, True)
assert LT(t**(S(3)/2)*besseli(3, 2*sqrt(a*t)), t, s) ==\
(a**(S(3)/2)*exp(a/s)/s**4, 0, True)
assert LT(bessely(0, a*t), t, s) ==\
(-2*asinh(s/a)/(pi*sqrt(a**2 + s**2)), 0, True)
assert LT(besselk(0, a*t), t, s) ==\
(log(s + sqrt(-a**2 + s**2))/sqrt(-a**2 + s**2), a, True)
assert LT(sin(a*t)**8, t, s) ==\
(40320*a**8/(s*(147456*a**8 + 52480*a**6*s**2 + 4368*a**4*s**4 +\
120*a**2*s**6 + s**8)), 0, True)
# Test general rules and unevaluated forms
# These all also test whether issue #7219 is solved.
assert LT(Heaviside(t-1)*cos(t-1), t, s) == (s*exp(-s)/(s**2 + 1), 0, True)
assert LT(a*f(t), t, w) == a*LaplaceTransform(f(t), t, w)
assert LT(a*Heaviside(t+1)*f(t+1), t, s) ==\
a*LaplaceTransform(f(t + 1)*Heaviside(t + 1), t, s)
assert LT(a*Heaviside(t-1)*f(t-1), t, s) ==\
a*LaplaceTransform(f(t), t, s)*exp(-s)
assert LT(b*f(t/a), t, s) == a*b*LaplaceTransform(f(t), t, a*s)
assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -f(x), True)
assert LT(exp(-a*t)*f(t), t, s) == LaplaceTransform(f(t), t, a + s)
assert LT(exp(-a*t)*erfc(sqrt(b/t)/2), t, s) ==\
(exp(-sqrt(b)*sqrt(a + s))/(a + s), -a, True)
assert LT(sinh(a*t)*f(t), t, s) ==\
LaplaceTransform(f(t), t, -a+s)/2 - LaplaceTransform(f(t), t, a+s)/2
assert LT(sinh(a*t)*t, t, s) ==\
(-1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True)
assert LT(cosh(a*t)*f(t), t, s) ==\
LaplaceTransform(f(t), t, -a+s)/2 + LaplaceTransform(f(t), t, a+s)/2
assert LT(cosh(a*t)*t, t, s) ==\
(1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True)
assert LT(sin(a*t)*f(t), t, s) ==\
I*(-LaplaceTransform(f(t), t, -I*a + s) +\
LaplaceTransform(f(t), t, I*a + s))/2
assert LT(sin(a*t)*t, t, s) ==\
(2*a*s/(a**4 + 2*a**2*s**2 + s**4), 0, True)
assert LT(cos(a*t)*f(t), t, s) ==\
LaplaceTransform(f(t), t, -I*a + s)/2 +\
LaplaceTransform(f(t), t, I*a + s)/2
assert LT(cos(a*t)*t, t, s) ==\
((-a**2 + s**2)/(a**4 + 2*a**2*s**2 + s**4), 0, True)
# The following two lines test whether issues #5813 and #7176 are solved.
assert LT(diff(f(t), (t, 1)), t, s) == s*LaplaceTransform(f(t), t, s)\
- f(0)
assert LT(diff(f(t), (t, 3)), t, s) == s**3*LaplaceTransform(f(t), t, s)\
- s**2*f(0) - s*Subs(Derivative(f(t), t), t, 0)\
- Subs(Derivative(f(t), (t, 2)), t, 0)
assert LT(a*f(b*t)+g(c*t), t, s) == a*LaplaceTransform(f(t), t, s/b)/b +\
LaplaceTransform(g(t), t, s/c)/c
assert inverse_laplace_transform(
f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)
assert LT(f(t)*g(t), t, s) == LaplaceTransform(f(t)*g(t), t, s)
# additional basic tests from wikipedia
assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \
((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True)
assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
# DiracDelta function: standard cases
assert LT(DiracDelta(t), t, s) == (1, 0, True)
assert LT(DiracDelta(a*t), t, s) == (1/a, 0, True)
assert LT(DiracDelta(t/42), t, s) == (42, 0, True)
assert LT(DiracDelta(t+42), t, s) == (0, 0, True)
assert LT(DiracDelta(t)+DiracDelta(t-42), t, s) == \
(1 + exp(-42*s), 0, True)
assert LT(DiracDelta(t)-a*exp(-a*t), t, s) == (s/(a + s), 0, True)
assert LT(exp(-t)*(DiracDelta(t)+DiracDelta(t-42)), t, s) == \
(exp(-42*s - 42) + 1, -oo, True)
# Collection of cases that cannot be fully evaluated and/or would catch
# some common implementation errors
assert LT(DiracDelta(t**2), t, s) == LaplaceTransform(DiracDelta(t**2), t, s)
assert LT(DiracDelta(t**2 - 1), t, s) == (exp(-s)/2, -oo, True)
assert LT(DiracDelta(t*(1 - t)), t, s) == \
LaplaceTransform(DiracDelta(-t**2 + t), t, s)
assert LT((DiracDelta(t) + 1)*(DiracDelta(t - 1) + 1), t, s) == \
(LaplaceTransform(DiracDelta(t)*DiracDelta(t - 1), t, s) + \
1 + exp(-s) + 1/s, 0, True)
assert LT(DiracDelta(2*t-2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, True)
assert LT(DiracDelta(-2*t+2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, True)
# Heaviside tests
assert LT(Heaviside(t), t, s) == (1/s, 0, True)
assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True)
assert LT(Heaviside(t-1), t, s) == (exp(-s)/s, 0, True)
assert LT(Heaviside(2*t-4), t, s) == (exp(-2*s)/s, 0, True)
assert LT(Heaviside(-2*t+4), t, s) == ((1 - exp(-2*s))/s, 0, True)
assert LT(Heaviside(2*t+4), t, s) == (1/s, 0, True)
assert LT(Heaviside(-2*t+4), t, s) == ((1 - exp(-2*s))/s, 0, True)
# Fresnel functions
assert laplace_transform(fresnels(t), t, s) == \
((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 -
cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True)
assert laplace_transform(fresnelc(t), t, s) == (
((2*sin(s**2/(2*pi))*fresnelc(s/pi) - 2*cos(s**2/(2*pi))*fresnels(s/pi)
+ sqrt(2)*cos(s**2/(2*pi) + pi/4))/(2*s), 0, True))
# Matrix tests
Mt = Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]])
Ms = Matrix([[ 1/(s - 1), (s + 1)**(-2)],
[(s + 1)**(-2), 1/(s - 1)]])
# The default behaviour for Laplace tranform of a Matrix returns a Matrix
# of Tuples and is deprecated:
with warns_deprecated_sympy():
Ms_conds = Matrix([[(1/(s - 1), 1, True), ((s + 1)**(-2),
-1, True)], [((s + 1)**(-2), -1, True), (1/(s - 1), 1, True)]])
with warns_deprecated_sympy():
assert LT(Mt, t, s) == Ms_conds
# The new behavior is to return a tuple of a Matrix and the convergence
# conditions for the matrix as a whole:
assert LT(Mt, t, s, legacy_matrix=False) == (Ms, 1, True)
# With noconds=True the transformed matrix is returned without conditions
# either way:
assert LT(Mt, t, s, noconds=True) == Ms
assert LT(Mt, t, s, legacy_matrix=False, noconds=True) == Ms
def test_mellin_transform():
from sympy.functions.elementary.miscellaneous import (Max, Min)
MT = mellin_transform
bpos = symbols('b', positive=True)
# 8.4.2
assert MT(x**nu*Heaviside(x - 1), x, s) == \
(-1/(nu + s), (-oo, -re(nu)), True)
assert MT(x**nu*Heaviside(1 - x), x, s) == \
(1/(nu + s), (-re(nu), oo), True)
assert MT((1 - x)**(beta - 1)*Heaviside(1 - x), x, s) == \
(gamma(beta)*gamma(s)/gamma(beta + s), (0, oo), re(beta) > 0)
assert MT((x - 1)**(beta - 1)*Heaviside(x - 1), x, s) == \
(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
(-oo, 1 - re(beta)), re(beta) > 0)
assert MT((1 + x)**(-rho), x, s) == \
(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
assert MT(abs(1 - x)**(-rho), x, s) == (
2*sin(pi*rho/2)*gamma(1 - rho)*
cos(pi*(s - rho/2))*gamma(s)*gamma(rho-s)/pi,
(0, re(rho)), re(rho) < 1)
mt = MT((1 - x)**(beta - 1)*Heaviside(1 - x)
+ a*(x - 1)**(beta - 1)*Heaviside(x - 1), x, s)
assert mt[1], mt[2] == ((0, -re(beta) + 1), re(beta) > 0)
assert MT((x**a - b**a)/(x - b), x, s)[0] == \
pi*b**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s)))
assert MT((x**a - bpos**a)/(x - bpos), x, s) == \
(pi*bpos**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s))),
(Max(0, -re(a)), Min(1, 1 - re(a))), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, bpos), x, s) == \
(-a*(2*bpos)**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
(0, -re(a)/2), True)
expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
assert MT(expr.subs(b, bpos), x, s) == \
(2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(s)
*gamma(1 - a - 2*s)/gamma(1 - a - s),
(0, -re(a)/2 + S.Half), True)
# 8.4.2
assert MT(exp(-x), x, s) == (gamma(s), (0, oo), True)
assert MT(exp(-1/x), x, s) == (gamma(-s), (-oo, 0), True)
# 8.4.5
assert MT(log(x)**4*Heaviside(1 - x), x, s) == (24/s**5, (0, oo), True)
assert MT(log(x)**3*Heaviside(x - 1), x, s) == (6/s**4, (-oo, 0), True)
assert MT(log(x + 1), x, s) == (pi/(s*sin(pi*s)), (-1, 0), True)
assert MT(log(1/x + 1), x, s) == (pi/(s*sin(pi*s)), (0, 1), True)
assert MT(log(abs(1 - x)), x, s) == (pi/(s*tan(pi*s)), (-1, 0), True)
assert MT(log(abs(1 - 1/x)), x, s) == (pi/(s*tan(pi*s)), (0, 1), True)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S.Half)/(sqrt(pi)*s), (Rational(-1, 2), 0), True)
def gauss_chebyshev_t(n, n_digits):
r"""
Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
the first kind.
The Gauss-Chebyshev quadrature of the first kind approximates the integral:
.. math::
\int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx
\sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `T_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{\pi}{n}
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_chebyshev_t
>>> x, w = gauss_chebyshev_t(3, 5)
>>> x
[0.86602, 0, -0.86602]
>>> w
[1.0472, 1.0472, 1.0472]
>>> x, w = gauss_chebyshev_t(6, 5)
>>> x
[0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593]
>>> w
[0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236]
See Also
========
gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html
"""
xi = []
w = []
for i in range(1, n + 1):
xi.append((cos((2 * i - S.One) / (2 * n) * S.Pi)).n(n_digits))
w.append((S.Pi / n).n(n_digits))
return xi, w
def test_meijer():
raises(TypeError, lambda: meijerg(1, z))
raises(TypeError, lambda: meijerg(((1,), (2,)), (3,), (4,), z))
assert meijerg(((1, 2), (3,)), ((4,), (5,)), z) == \
meijerg(Tuple(1, 2), Tuple(3), Tuple(4), Tuple(5), z)
g = meijerg((1, 2), (3, 4, 5), (6, 7, 8, 9), (10, 11, 12, 13, 14), z)
assert g.an == Tuple(1, 2)
assert g.ap == Tuple(1, 2, 3, 4, 5)
assert g.aother == Tuple(3, 4, 5)
assert g.bm == Tuple(6, 7, 8, 9)
assert g.bq == Tuple(6, 7, 8, 9, 10, 11, 12, 13, 14)
assert g.bother == Tuple(10, 11, 12, 13, 14)
assert g.argument == z
assert g.nu == 75
assert g.delta == -1
assert g.is_commutative is True
assert g.is_number is False
#issue 13071
assert meijerg([[],[]], [[S.Half],[0]], 1).is_number is True
assert meijerg([1, 2], [3], [4], [5], z).delta == S.Half
# just a few checks to make sure that all arguments go where they should
assert tn(meijerg(Tuple(), Tuple(), Tuple(0), Tuple(), -z), exp(z), z)
assert tn(sqrt(pi)*meijerg(Tuple(), Tuple(),
Tuple(0), Tuple(S.Half), z**2/4), cos(z), z)
assert tn(meijerg(Tuple(1, 1), Tuple(), Tuple(1), Tuple(0), z),
log(1 + z), z)
# test exceptions
raises(ValueError, lambda: meijerg(((3, 1), (2,)), ((oo,), (2, 0)), x))
raises(ValueError, lambda: meijerg(((3, 1), (2,)), ((1,), (2, 0)), x))
# differentiation
g = meijerg((randcplx(),), (randcplx() + 2*I,), Tuple(),
(randcplx(), randcplx()), z)
assert td(g, z)
g = meijerg(Tuple(), (randcplx(),), Tuple(),
(randcplx(), randcplx()), z)
assert td(g, z)
g = meijerg(Tuple(), Tuple(), Tuple(randcplx()),
Tuple(randcplx(), randcplx()), z)
assert td(g, z)
a1, a2, b1, b2, c1, c2, d1, d2 = symbols('a1:3, b1:3, c1:3, d1:3')
assert meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z).diff(z) == \
(meijerg((a1 - 1, a2), (b1, b2), (c1, c2), (d1, d2), z)
+ (a1 - 1)*meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z))/z
assert meijerg([z, z], [], [], [], z).diff(z) == \
Derivative(meijerg([z, z], [], [], [], z), z)
# meijerg is unbranched wrt parameters
from sympy.functions.elementary.complexes import polar_lift as pl
assert meijerg([pl(a1)], [pl(a2)], [pl(b1)], [pl(b2)], pl(z)) == \
meijerg([a1], [a2], [b1], [b2], pl(z))
# integrand
from sympy.abc import a, b, c, d, s
assert meijerg([a], [b], [c], [d], z).integrand(s) == \
z**s*gamma(c - s)*gamma(-a + s + 1)/(gamma(b - s)*gamma(-d + s + 1))
def _yn(n, z):
# (-1)**(n + 1) * _jn(-n - 1, z)
return (-1)**(n + 1) * fn(-n - 1, z) * sin(z) - fn(n, z) * cos(z)
def test_issue_6653s():
x = Symbol('x')
assert (1 / sqrt(1 + cos(x) * sin(x**2))).series(x, 0, 15) == \
1 - x**2/2 + 5*x**4/8 - 5*x**6/8 + 4039*x**8/5760 - 5393*x**10/6720 + \
13607537*x**12/14515200 - 532056047*x**14/479001600 + O(x**15)
def _jn(n, z):
return fn(n, z) * sin(z) + (-1)**(n + 1) * fn(-n - 1, z) * cos(z)
def test_floats():
a, b = map(Wild, 'ab')
e = cos(0.12345, evaluate=False)**2
r = e.match(a * cos(b)**2)
assert r == {a: 1, b: Float(0.12345)}
def test_trigsimp():
assert trigsimp(A*sin(x)**2 + A*cos(x)**2) == A
def test_derivatives_elementwise_applyfunc():
expr = x.applyfunc(tan)
assert expr.diff(x).dummy_eq(
DiagMatrix(x.applyfunc(lambda x: tan(x)**2 + 1)))
assert expr[i,
0].diff(x[m,
0]).doit() == (tan(x[i, 0])**2 + 1) * KDelta(i, m)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (i**2 * x).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct((2 * i) * x, (i**2 * x).applyfunc(cos)))
assert expr[i, 0].diff(i).doit() == 2 * i * x[i, 0] * cos(i**2 * x[i, 0])
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = (log(i) * A * B).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct(A * B / i, (log(i) * A * B).applyfunc(cos)))
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = A * x.applyfunc(exp)
# TODO: restore this result (currently returning the transpose):
# assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(exp))*A.T)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.T * A * x + k * y.applyfunc(sin).T * x
assert expr.diff(x).dummy_eq(A.T * x + A * x + k * y.applyfunc(sin))
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.applyfunc(sin).T * y
# TODO: restore (currently returning the traspose):
# assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(cos))*y)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (a.T * X * b).applyfunc(sin)
assert expr.diff(X).dummy_eq(a * (a.T * X * b).applyfunc(cos) * b.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * X.applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
DiagMatrix(a) * X.applyfunc(cos) * DiagMatrix(b))
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A * X * B).applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
A.T * DiagMatrix(a) * (A * X * B).applyfunc(cos) * DiagMatrix(b) * B.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A * X * b).applyfunc(sin) * b.T
# TODO: not implemented
#assert expr.diff(X) == ...
#_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * A * X.applyfunc(sin) * B * b
assert expr.diff(X).dummy_eq(
HadamardProduct(A.T * a * b.T * B.T, X.applyfunc(cos)))
expr = a.T * (A * X.applyfunc(sin) * B).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == A.T*DiagMatrix(a)*(A*X.applyfunc(sin)*B).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)*B.T
expr = a.T * (X.applyfunc(sin)).applyfunc(log) * b
def _expand(self, **hints):
n = self.order
z = self.argument
return (-1)**(n + 1) * \
(fn(-n - 1, z) * sin(z) + (-1)**(-n) * fn(n, z) * cos(z))
def wigner_d_small(J, beta):
"""Return the small Wigner d matrix for angular momentum J.
Explanation
===========
J : An integer, half-integer, or SymPy symbol for the total angular
momentum of the angular momentum space being rotated.
beta : A real number representing the Euler angle of rotation about
the so-called line of nodes. See [Edmonds74]_.
Returns
=======
A matrix representing the corresponding Euler angle rotation( in the basis
of eigenvectors of `J_z`).
.. math ::
\\mathcal{d}_{\\beta} = \\exp\\big( \\frac{i\\beta}{\\hbar} J_y\\big)
The components are calculated using the general form [Edmonds74]_,
equation 4.1.15.
Examples
========
>>> from sympy import Integer, symbols, pi, pprint
>>> from sympy.physics.wigner import wigner_d_small
>>> half = 1/Integer(2)
>>> beta = symbols("beta", real=True)
>>> pprint(wigner_d_small(half, beta), use_unicode=True)
⎡ ⎛β⎞ ⎛β⎞⎤
⎢cos⎜─⎟ sin⎜─⎟⎥
⎢ ⎝2⎠ ⎝2⎠⎥
⎢ ⎥
⎢ ⎛β⎞ ⎛β⎞⎥
⎢-sin⎜─⎟ cos⎜─⎟⎥
⎣ ⎝2⎠ ⎝2⎠⎦
>>> pprint(wigner_d_small(2*half, beta), use_unicode=True)
⎡ 2⎛β⎞ ⎛β⎞ ⎛β⎞ 2⎛β⎞ ⎤
⎢ cos ⎜─⎟ √2⋅sin⎜─⎟⋅cos⎜─⎟ sin ⎜─⎟ ⎥
⎢ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎥
⎢ ⎥
⎢ ⎛β⎞ ⎛β⎞ 2⎛β⎞ 2⎛β⎞ ⎛β⎞ ⎛β⎞⎥
⎢-√2⋅sin⎜─⎟⋅cos⎜─⎟ - sin ⎜─⎟ + cos ⎜─⎟ √2⋅sin⎜─⎟⋅cos⎜─⎟⎥
⎢ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠⎥
⎢ ⎥
⎢ 2⎛β⎞ ⎛β⎞ ⎛β⎞ 2⎛β⎞ ⎥
⎢ sin ⎜─⎟ -√2⋅sin⎜─⎟⋅cos⎜─⎟ cos ⎜─⎟ ⎥
⎣ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎦
From table 4 in [Edmonds74]_
>>> pprint(wigner_d_small(half, beta).subs({beta:pi/2}), use_unicode=True)
⎡ √2 √2⎤
⎢ ── ──⎥
⎢ 2 2 ⎥
⎢ ⎥
⎢-√2 √2⎥
⎢──── ──⎥
⎣ 2 2 ⎦
>>> pprint(wigner_d_small(2*half, beta).subs({beta:pi/2}),
... use_unicode=True)
⎡ √2 ⎤
⎢1/2 ── 1/2⎥
⎢ 2 ⎥
⎢ ⎥
⎢-√2 √2 ⎥
⎢──── 0 ── ⎥
⎢ 2 2 ⎥
⎢ ⎥
⎢ -√2 ⎥
⎢1/2 ──── 1/2⎥
⎣ 2 ⎦
>>> pprint(wigner_d_small(3*half, beta).subs({beta:pi/2}),
... use_unicode=True)
⎡ √2 √6 √6 √2⎤
⎢ ── ── ── ──⎥
⎢ 4 4 4 4 ⎥
⎢ ⎥
⎢-√6 -√2 √2 √6⎥
⎢──── ──── ── ──⎥
⎢ 4 4 4 4 ⎥
⎢ ⎥
⎢ √6 -√2 -√2 √6⎥
⎢ ── ──── ──── ──⎥
⎢ 4 4 4 4 ⎥
⎢ ⎥
⎢-√2 √6 -√6 √2⎥
⎢──── ── ──── ──⎥
⎣ 4 4 4 4 ⎦
>>> pprint(wigner_d_small(4*half, beta).subs({beta:pi/2}),
... use_unicode=True)
⎡ √6 ⎤
⎢1/4 1/2 ── 1/2 1/4⎥
⎢ 4 ⎥
⎢ ⎥
⎢-1/2 -1/2 0 1/2 1/2⎥
⎢ ⎥
⎢ √6 √6 ⎥
⎢ ── 0 -1/2 0 ── ⎥
⎢ 4 4 ⎥
⎢ ⎥
⎢-1/2 1/2 0 -1/2 1/2⎥
⎢ ⎥
⎢ √6 ⎥
⎢1/4 -1/2 ── -1/2 1/4⎥
⎣ 4 ⎦
"""
M = [J - i for i in range(2 * J + 1)]
d = zeros(2 * J + 1)
for i, Mi in enumerate(M):
for j, Mj in enumerate(M):
# We get the maximum and minimum value of sigma.
sigmamax = max([-Mi - Mj, J - Mj])
sigmamin = min([0, J - Mi])
dij = sqrt(
factorial(J + Mi) * factorial(J - Mi) / factorial(J + Mj) /
factorial(J - Mj))
terms = [(-1)**(J - Mi - s) * binomial(J + Mj, J - Mi - s) *
binomial(J - Mj, s) * cos(beta / 2)**(2 * s + Mi + Mj) *
sin(beta / 2)**(2 * J - 2 * s - Mj - Mi)
for s in range(sigmamin, sigmamax + 1)]
d[i, j] = dij * Add(*terms)
return ImmutableMatrix(d)
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return cos(S.Pi*(k + 1)/(n + 1))
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
exp(I*m*phi) * assoc_legendre(n, m, cos(theta)))
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
def test_Max():
from sympy.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
r = Symbol('r', real=True)
assert Max(5, 4) == 5
# lists
assert Max() is S.NegativeInfinity
assert Max(x) == x
assert Max(x, y) == Max(y, x)
assert Max(x, y, z) == Max(z, y, x)
assert Max(x, Max(y, z)) == Max(z, y, x)
assert Max(x, Min(y, oo)) == Max(x, y)
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p) == p
assert Max(2, x, p, n, -oo, S.NegativeInfinity, n_, p, 2) == Max(2, x, p)
assert Max(0, x, 1, y) == Max(1, x, y)
assert Max(r, r + 1, r - 1) == 1 + r
assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
assert Max(cos(x), sin(x)).subs(x, 1) == sin(1)
assert Max(cos(x), sin(x)).subs(x, S.Half) == cos(S.Half)
raises(ValueError, lambda: Max(cos(x), sin(x)).subs(x, I))
raises(ValueError, lambda: Max(I))
raises(ValueError, lambda: Max(I, x))
raises(ValueError, lambda: Max(S.ComplexInfinity, 1))
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p, 1000) == Max(p, 1000)
assert Max(1, x).diff(x) == Heaviside(x - 1)
assert Max(x, 1).diff(x) == Heaviside(x - 1)
assert Max(x**2, 1 + x, 1).diff(x) == \
2*x*Heaviside(x**2 - Max(1, x + 1)) \
+ Heaviside(x - Max(1, x**2) + 1)
e = Max(0, x)
assert e.n().args == (0, x)
# issue 8643
m = Max(p, p_, n, r)
assert m.is_positive == True
assert m.is_nonnegative == True
assert m.is_negative == False
m = Max(n, n_)
assert m.is_positive == False
assert m.is_nonnegative == False
assert m.is_negative == True
m = Max(n, n_, r)
assert m.is_positive is None
assert m.is_nonnegative is None
assert m.is_negative is None
m = Max(n, nn, r)
assert m.is_positive is None
assert m.is_nonnegative == True
assert m.is_negative == False
def test_issue_18751():
r = Symbol('r', positive=True)
theta = Symbol('theta', real=True)
f = y(n) - 2 * r * cos(theta) * y(n - 1) + r**2 * y(n - 2)
assert rsolve(f, y(n)) == \
C0*(r*(cos(theta) - I*Abs(sin(theta))))**n + C1*(r*(cos(theta) + I*Abs(sin(theta))))**n
def test_issue_18762():
e, p = symbols('e p')
g0 = sqrt(1 + e**2 - 2 * e * cos(p))
assert len(g0.series(e, 1, 3).args) == 4
def _eval_rewrite_as_cos(self, *args, **kwargs):
return self.amplitude * cos(self.wavenumber * Symbol('x') -
self.angular_velocity * Symbol('t') +
self.phase)
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
Notes
-----
A random point may not appear to be on the ellipse, ie, `p in e` may
return False. This is because the coordinates of the point will be
floating point values, and when these values are substituted into the
equation for the ellipse the result may not be zero because of floating
point rounding error.
Examples
========
>>> from sympy import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)
The random_point method assures that the point will test as being
in the ellipse:
>>> p1 in e1
True
Notes
=====
An arbitrary_point with a random value of t substituted into it may
not test as being on the ellipse because the expression tested that
a point is on the ellipse doesn't simplify to zero and doesn't evaluate
exactly to zero:
>>> from sympy.abc import t
>>> e1.arbitrary_point(t)
Point2D(3*cos(t), 2*sin(t))
>>> p2 = _.subs(t, 0.1)
>>> p2 in e1
False
Note that arbitrary_point routine does not take this approach. A value
for cos(t) and sin(t) (not t) is substituted into the arbitrary point.
There is a small chance that this will give a point that will not
test as being in the ellipse, so the process is repeated (up to 10
times) until a valid point is obtained.
"""
from sympy import sin, cos, Rational
t = _symbol('t')
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
for i in range(10): # should be enough?
# simplify this now or else the Float will turn s into a Float
c = 2*Rational(rng.random()) - 1
s = sqrt(1 - c**2)
p1 = Point(x.subs(cos(t), c), y.subs(sin(t), s))
if p1 in self:
return p1
raise GeometryError(
'Having problems generating a point in the ellipse.')
def test_Min():
from sympy.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
r = Symbol('r', real=True)
assert Min(5, 4) == 4
assert Min(-oo, -oo) is -oo
assert Min(-oo, n) is -oo
assert Min(n, -oo) is -oo
assert Min(-oo, np) is -oo
assert Min(np, -oo) is -oo
assert Min(-oo, 0) is -oo
assert Min(0, -oo) is -oo
assert Min(-oo, nn) is -oo
assert Min(nn, -oo) is -oo
assert Min(-oo, p) is -oo
assert Min(p, -oo) is -oo
assert Min(-oo, oo) is -oo
assert Min(oo, -oo) is -oo
assert Min(n, n) == n
assert unchanged(Min, n, np)
assert Min(np, n) == Min(n, np)
assert Min(n, 0) == n
assert Min(0, n) == n
assert Min(n, nn) == n
assert Min(nn, n) == n
assert Min(n, p) == n
assert Min(p, n) == n
assert Min(n, oo) == n
assert Min(oo, n) == n
assert Min(np, np) == np
assert Min(np, 0) == np
assert Min(0, np) == np
assert Min(np, nn) == np
assert Min(nn, np) == np
assert Min(np, p) == np
assert Min(p, np) == np
assert Min(np, oo) == np
assert Min(oo, np) == np
assert Min(0, 0) == 0
assert Min(0, nn) == 0
assert Min(nn, 0) == 0
assert Min(0, p) == 0
assert Min(p, 0) == 0
assert Min(0, oo) == 0
assert Min(oo, 0) == 0
assert Min(nn, nn) == nn
assert unchanged(Min, nn, p)
assert Min(p, nn) == Min(nn, p)
assert Min(nn, oo) == nn
assert Min(oo, nn) == nn
assert Min(p, p) == p
assert Min(p, oo) == p
assert Min(oo, p) == p
assert Min(oo, oo) is oo
assert Min(n, n_).func is Min
assert Min(nn, nn_).func is Min
assert Min(np, np_).func is Min
assert Min(p, p_).func is Min
# lists
assert Min() is S.Infinity
assert Min(x) == x
assert Min(x, y) == Min(y, x)
assert Min(x, y, z) == Min(z, y, x)
assert Min(x, Min(y, z)) == Min(z, y, x)
assert Min(x, Max(y, -oo)) == Min(x, y)
assert Min(p, oo, n, p, p, p_) == n
assert Min(p_, n_, p) == n_
assert Min(n, oo, -7, p, p, 2) == Min(n, -7)
assert Min(2, x, p, n, oo, n_, p, 2, -2, -2) == Min(-2, x, n, n_)
assert Min(0, x, 1, y) == Min(0, x, y)
assert Min(1000, 100, -100, x, p, n) == Min(n, x, -100)
assert unchanged(Min, sin(x), cos(x))
assert Min(sin(x), cos(x)) == Min(cos(x), sin(x))
assert Min(cos(x), sin(x)).subs(x, 1) == cos(1)
assert Min(cos(x), sin(x)).subs(x, S.Half) == sin(S.Half)
raises(ValueError, lambda: Min(cos(x), sin(x)).subs(x, I))
raises(ValueError, lambda: Min(I))
raises(ValueError, lambda: Min(I, x))
raises(ValueError, lambda: Min(S.ComplexInfinity, x))
assert Min(1, x).diff(x) == Heaviside(1 - x)
assert Min(x, 1).diff(x) == Heaviside(1 - x)
assert Min(0, -x, 1 - 2*x).diff(x) == -Heaviside(x + Min(0, -2*x + 1)) \
- 2*Heaviside(2*x + Min(0, -x) - 1)
# issue 7619
f = Function('f')
assert Min(1, 2 * Min(f(1), 2)) # doesn't fail
# issue 7233
e = Min(0, x)
assert e.n().args == (0, x)
# issue 8643
m = Min(n, p_, n_, r)
assert m.is_positive == False
assert m.is_nonnegative == False
assert m.is_negative == True
m = Min(p, p_)
assert m.is_positive == True
assert m.is_nonnegative == True
assert m.is_negative == False
m = Min(p, nn_, p_)
assert m.is_positive is None
assert m.is_nonnegative == True
assert m.is_negative == False
m = Min(nn, p, r)
assert m.is_positive is None
assert m.is_nonnegative is None
assert m.is_negative is None
def gauss_chebyshev_u(n, n_digits):
r"""
Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
the second kind.
The Gauss-Chebyshev quadrature of the second kind approximates the
integral:
.. math::
\int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `U_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right)
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_chebyshev_u
>>> x, w = gauss_chebyshev_u(3, 5)
>>> x
[0.70711, 0, -0.70711]
>>> w
[0.3927, 0.7854, 0.3927]
>>> x, w = gauss_chebyshev_u(6, 5)
>>> x
[0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097]
>>> w
[0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489]
See Also
========
gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_t, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html
"""
xi = []
w = []
for i in range(1, n + 1):
xi.append((cos(i / (n + S.One) * S.Pi)).n(n_digits))
w.append(
(S.Pi / (n + S.One) * sin(i * S.Pi / (n + S.One))**2).n(n_digits))
return xi, w