def E(expr):
res1 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo), meijerg=True)
res2 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo), meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
def test_issue841():
from sympy import expand_mul
from sympy.abc import k
assert expand_mul(integrate(exp(-x ** 2) * exp(I * k * x), (x, -oo, oo))) == sqrt(pi) * exp(-k ** 2 / 4)
a, d = symbols("a d", positive=True)
assert expand_mul(integrate(exp(-a * x ** 2 + 2 * d * x), (x, -oo, oo))) == sqrt(pi) * exp(d ** 2 / a) / sqrt(a)
def eval(cls, arg):
from sympy.simplify.simplify import signsimp
if hasattr(arg, '_eval_Abs'):
obj = arg._eval_Abs()
if obj is not None:
return obj
# handle what we can
arg = signsimp(arg, evaluate=False)
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if tnew.func is cls:
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known*unk
if arg is S.NaN:
return S.NaN
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if base.is_real:
if exponent.is_integer:
if exponent.is_even:
return arg
if base is S.NegativeOne:
return S.One
if base.func is cls and exponent is S.NegativeOne:
return arg
return Abs(base)**exponent
if base.is_positive == True:
return base**re(exponent)
return (-base)**re(exponent)*C.exp(-S.Pi*im(exponent))
if isinstance(arg, C.exp):
return C.exp(re(arg.args[0]))
if arg.is_zero: # it may be an Expr that is zero
return S.Zero
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -S.ImaginaryUnit * arg
if arg2.is_nonnegative:
return arg2
if arg.is_Add:
if arg.has(S.Infinity, S.NegativeInfinity):
if any(a.is_infinite for a in arg.as_real_imag()):
return S.Infinity
if arg.is_real is None and arg.is_imaginary is None:
if all(a.is_real or a.is_imaginary or (S.ImaginaryUnit*a).is_real for a in arg.args):
from sympy import expand_mul
return sqrt(expand_mul(arg*arg.conjugate()))
if arg.is_real is False and arg.is_imaginary is False:
from sympy import expand_mul
return sqrt(expand_mul(arg*arg.conjugate()))
def test_acsch():
x = Symbol('x')
assert acsch(-x) == acsch(-x)
assert acsch(x) == -acsch(-x)
# values at fixed points
assert acsch(1) == log(1 + sqrt(2))
assert acsch(-1) == - log(1 + sqrt(2))
assert acsch(0) == zoo
assert acsch(2) == log((1+sqrt(5))/2)
assert acsch(-2) == - log((1+sqrt(5))/2)
assert acsch(I) == - I*pi/2
assert acsch(-I) == I*pi/2
assert acsch(-I*(sqrt(6) + sqrt(2))) == I*pi / 12
assert acsch(I*(sqrt(2) + sqrt(6))) == -I*pi / 12
assert acsch(-I*(1 + sqrt(5))) == I*pi / 10
assert acsch(I*(1 + sqrt(5))) == -I*pi / 10
assert acsch(-I*2 / sqrt(2 - sqrt(2))) == I*pi / 8
assert acsch(I*2 / sqrt(2 - sqrt(2))) == -I*pi / 8
assert acsch(-I*2) == I*pi / 6
assert acsch(I*2) == -I*pi / 6
assert acsch(-I*sqrt(2 + 2/sqrt(5))) == I*pi / 5
assert acsch(I*sqrt(2 + 2/sqrt(5))) == -I*pi / 5
assert acsch(-I*sqrt(2)) == I*pi / 4
assert acsch(I*sqrt(2)) == -I*pi / 4
assert acsch(-I*(sqrt(5)-1)) == 3*I*pi / 10
assert acsch(I*(sqrt(5)-1)) == -3*I*pi / 10
assert acsch(-I*2 / sqrt(3)) == I*pi / 3
assert acsch(I*2 / sqrt(3)) == -I*pi / 3
assert acsch(-I*2 / sqrt(2 + sqrt(2))) == 3*I*pi / 8
assert acsch(I*2 / sqrt(2 + sqrt(2))) == -3*I*pi / 8
assert acsch(-I*sqrt(2 - 2/sqrt(5))) == 2*I*pi / 5
assert acsch(I*sqrt(2 - 2/sqrt(5))) == -2*I*pi / 5
assert acsch(-I*(sqrt(6) - sqrt(2))) == 5*I*pi / 12
assert acsch(I*(sqrt(6) - sqrt(2))) == -5*I*pi / 12
# properties
# acsch(x) == asinh(1/x)
assert acsch(-I*sqrt(2)) == asinh(I/sqrt(2))
assert acsch(-I*2 / sqrt(3)) == asinh(I*sqrt(3) / 2)
# acsch(x) == -I*asin(I/x)
assert acsch(-I*sqrt(2)) == -I*asin(-1/sqrt(2))
assert acsch(-I*2 / sqrt(3)) == -I*asin(-sqrt(3)/2)
# csch(acsch(x)) / x == 1
assert expand_mul(csch(acsch(-I*(sqrt(6) + sqrt(2)))) / (-I*(sqrt(6) + sqrt(2)))) == 1
assert expand_mul(csch(acsch(I*(1 + sqrt(5)))) / ((I*(1 + sqrt(5))))) == 1
assert (csch(acsch(I*sqrt(2 - 2/sqrt(5)))) / (I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
assert (csch(acsch(-I*sqrt(2 - 2/sqrt(5)))) / (-I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
# numerical evaluation
assert str(acsch(5*I+1).n(6)) == '0.0391819 - 0.193363*I'
assert str(acsch(-5*I+1).n(6)) == '0.0391819 + 0.193363*I'
def test_asech():
x = Symbol('x')
assert asech(-x) == asech(-x)
# values at fixed points
assert asech(1) == 0
assert asech(-1) == pi*I
assert asech(0) == oo
assert asech(2) == I*pi/3
assert asech(-2) == 2*I*pi / 3
# at infinites
assert asech(oo) == I*pi/2
assert asech(-oo) == I*pi/2
assert asech(zoo) == nan
assert asech(I) == log(1 + sqrt(2)) - I*pi/2
assert asech(-I) == log(1 + sqrt(2)) + I*pi/2
assert asech(sqrt(2) - sqrt(6)) == 11*I*pi / 12
assert asech(sqrt(2 - 2/sqrt(5))) == I*pi / 10
assert asech(-sqrt(2 - 2/sqrt(5))) == 9*I*pi / 10
assert asech(2 / sqrt(2 + sqrt(2))) == I*pi / 8
assert asech(-2 / sqrt(2 + sqrt(2))) == 7*I*pi / 8
assert asech(sqrt(5) - 1) == I*pi / 5
assert asech(1 - sqrt(5)) == 4*I*pi / 5
assert asech(-sqrt(2*(2 + sqrt(2)))) == 5*I*pi / 8
# properties
# asech(x) == acosh(1/x)
assert asech(sqrt(2)) == acosh(1/sqrt(2))
assert asech(2/sqrt(3)) == acosh(sqrt(3)/2)
assert asech(2/sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2))/2)
assert asech(S(2)) == acosh(1/S(2))
# asech(x) == I*acos(1/x)
# (Note: the exact formula is asech(x) == +/- I*acos(1/x))
assert asech(-sqrt(2)) == I*acos(-1/sqrt(2))
assert asech(-2/sqrt(3)) == I*acos(-sqrt(3)/2)
assert asech(-S(2)) == I*acos(-S.Half)
assert asech(-2/sqrt(2)) == I*acos(-sqrt(2)/2)
# sech(asech(x)) / x == 1
assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) / (sqrt(6) - sqrt(2))) == 1
assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) / (sqrt(6) + sqrt(2))) == 1
assert (sech(asech(sqrt(2 + 2/sqrt(5)))) / (sqrt(2 + 2/sqrt(5)))).simplify() == 1
assert (sech(asech(-sqrt(2 + 2/sqrt(5)))) / (-sqrt(2 + 2/sqrt(5)))).simplify() == 1
assert (sech(asech(sqrt(2*(2 + sqrt(2))))) / (sqrt(2*(2 + sqrt(2))))).simplify() == 1
assert expand_mul(sech(asech((1 + sqrt(5)))) / ((1 + sqrt(5)))) == 1
assert expand_mul(sech(asech((-1 - sqrt(5)))) / ((-1 - sqrt(5)))) == 1
assert expand_mul(sech(asech((-sqrt(6) - sqrt(2)))) / ((-sqrt(6) - sqrt(2)))) == 1
# numerical evaluation
assert str(asech(5*I).n(6)) == '0.19869 - 1.5708*I'
assert str(asech(-5*I).n(6)) == '0.19869 + 1.5708*I'
def test_issue_3940():
a, b, c, d = symbols('a:d', positive=True, finite=True)
assert integrate(exp(-x**2 + I*c*x), x) == \
-sqrt(pi)*exp(-c**2/4)*erf(I*c/2 - x)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
sqrt(pi)*exp(c)*exp(-b**2/(4*a))*erfi(sqrt(a)*x + b/(2*sqrt(a)))/(2*sqrt(a))
from sympy import expand_mul
from sympy.abc import k
assert expand_mul(integrate(exp(-x**2)*exp(I*k*x), (x, -oo, oo))) == \
sqrt(pi)*exp(-k**2/4)
a, d = symbols('a d', positive=True)
assert expand_mul(integrate(exp(-a*x**2 + 2*d*x), (x, -oo, oo))) == \
sqrt(pi)*exp(d**2/a)/sqrt(a)
def eval(cls, arg):
from sympy.simplify.simplify import signsimp
if hasattr(arg, "_eval_Abs"):
obj = arg._eval_Abs()
if obj is not None:
return obj
# handle what we can
arg = signsimp(arg, evaluate=False)
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if tnew.func is cls:
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known * unk
if arg is S.NaN:
return S.NaN
if arg.is_zero: # it may be an Expr that is zero
return S.Zero
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -S.ImaginaryUnit * arg
if arg2.is_nonnegative:
return arg2
if arg.is_real is False and arg.is_imaginary is False:
from sympy import expand_mul
return sqrt(expand_mul(arg * arg.conjugate()))
if arg.is_real is None and arg.is_imaginary is None and arg.is_Add:
if all(a.is_real or a.is_imaginary or (S.ImaginaryUnit * a).is_real for a in arg.args):
from sympy import expand_mul
return sqrt(expand_mul(arg * arg.conjugate()))
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if exponent.is_even and base.is_real:
return arg
if exponent.is_integer and base is S.NegativeOne:
return S.One
def _eval_expand_trig(self, **hints):
from sympy import expand_mul
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
# TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return sx*cy + sy*cx
else:
n, x = arg.as_coeff_Mul(rational=True)
if n.is_Integer: # n will be positive because of .eval
# canonicalization
# See http://mathworld.wolfram.com/Multiple-AngleFormulas.html
if n.is_odd:
return (-1)**((n - 1)/2)*C.chebyshevt(n, sin(x))
else:
return expand_mul((-1)**(n/2 - 1)*cos(x)*C.chebyshevu(n -
1, sin(x)), deep=False)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return sin(arg)
def _eval_as_leading_term(self, x):
from sympy import expand_mul, factor_terms
old = self
self = expand_mul(self)
if not self.is_Add:
return self.as_leading_term(x)
unbounded = [t for t in self.args if t.is_unbounded]
self = self.func(*[t.as_leading_term(x) for t in self.args]).removeO()
if not self:
# simple leading term analysis gave us 0 but we have to send
# back a term, so compute the leading term (via series)
return old.compute_leading_term(x)
elif self is S.NaN:
return old.func._from_args(unbounded)
elif not self.is_Add:
return self
else:
plain = self.func(*[s for s, _ in self.extract_leading_order(x)])
rv = factor_terms(plain, fraction=False)
rv_simplify = rv.simplify()
# if it simplifies to an x-free expression, return that;
# tests don't fail if we don't but it seems nicer to do this
if x not in rv_simplify.free_symbols:
if rv_simplify.is_zero and plain.is_zero is not True:
return (self - plain)._eval_as_leading_term(x)
return rv_simplify
return rv
def _bell_poly(n, prev):
s = 1
a = 1
for k in xrange(2, n+1):
a = a * (n-k+1) // (k-1)
s += a * prev[k-1]
return expand_mul(_sym * s)
def test_sqrtdenest():
d = {sqrt(5 + 2 * sqrt(6)): sqrt(2) + sqrt(3),
sqrt(sqrt(2)): sqrt(sqrt(2)),
sqrt(5+sqrt(7)): sqrt(5+sqrt(7)),
sqrt(3+sqrt(5+2*sqrt(7))):
sqrt(6+3*sqrt(7))/(sqrt(2)*(5+2*sqrt(7))**Rational(1,4)) +
3*(5+2*sqrt(7))**Rational(1,4)/(sqrt(2)*sqrt(6+3*sqrt(7))),
sqrt(3+2*sqrt(3)): 3**Rational(1,4)/sqrt(2)+3/(sqrt(2)*3**Rational(1,4))}
for i in d:
assert sqrtdenest(i) == d[i] or denester([i])[0] == d[i]
# this test caused a pattern recognition failure in sqrtdenest
# nest = sqrt(2) + sqrt(5) - sqrt(7)
nest = symbols('nest')
x0, x1, x2, x3, x4, x5, x6 = symbols('x:7')
l = sqrt(2) + sqrt(5)
r = sqrt(7) + nest
s = (l**2 - r**2).expand() + nest**2 # == nest**2
ok = solve(nest**4 - s**2, nest)[1] # this will change if results order changes
assert abs((l - r).subs(nest, ok).n()) < 1e-12
x0 = sqrt(3)
x2 = root(45*I*x0 - 28, 3)
x3 = 19/x2
x4 = x2 + x3
x5 = -x4 - 14
x6 = sqrt(-x5)
ans = -x0*x6/3 + x0*sqrt(-x4 + 28 - 6*sqrt(210)*x6/x5)/3
assert expand_mul(radsimp(ok) - ans) == 0
# issue 2554
eq = sqrt(sqrt(sqrt(2) + 2) + 2)
assert sqrtdenest(eq) == eq
def _bell_incomplete_poly(n, k, symbols):
r"""
The second kind of Bell polynomials (incomplete Bell polynomials).
Calculated by recurrence formula:
.. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
\sum_{m=1}^{n-k+1}
\x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})
where
B_{0,0} = 1;
B_{n,0} = 0; for n>=1
B_{0,k} = 0; for k>=1
"""
if (n==0) and (k==0):
return S.One
elif (n==0) or (k==0):
return S.Zero
s = S.Zero
a = S.One
for m in xrange(1, n-k+2):
s += a*bell._bell_incomplete_poly(n-m, k-1, symbols)*symbols[m-1]
a = a*(n-m)/m
return expand_mul(s)
def _eval_as_leading_term(self, x):
from sympy import expand_mul, factor_terms
old = self
self = expand_mul(self)
if not self.is_Add:
return self.as_leading_term(x)
unbounded = [t for t in self.args if t.is_unbounded]
if unbounded:
return Add._from_args(unbounded)
self = Add(*[t.as_leading_term(x) for t in self.args]).removeO()
if not self:
# simple leading term analysis gave us 0 but we have to send
# back a term, so compute the leading term (via series)
return old.compute_leading_term(x)
elif not self.is_Add:
return self
else:
plain = Add(*[s for s, _ in self.extract_leading_order(x)])
rv = factor_terms(plain, fraction=False)
rv_fraction = factor_terms(rv, fraction=True)
# if it simplifies to an x-free expression, return that;
# tests don't fail if we don't but it seems nicer to do this
if x not in rv_fraction.free_symbols:
return rv_fraction
return rv
def eval(cls, arg):
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if tnew.func is cls:
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known*unk
if arg is S.NaN:
return S.NaN
if arg.is_zero:
return arg
if arg.is_positive:
return arg
if arg.is_negative:
return -arg
if arg.is_real is False:
from sympy import expand_mul
return sqrt( expand_mul(arg * arg.conjugate()) )
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if exponent.is_even and base.is_real:
return arg
def eval(cls, arg):
if hasattr(arg, "_eval_Abs"):
obj = arg._eval_Abs()
if obj is not None:
return obj
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if tnew.func is cls:
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known * unk
if arg is S.NaN:
return S.NaN
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -S.ImaginaryUnit * arg
if arg2.is_nonnegative:
return arg2
if arg.is_real is False and arg.is_imaginary is False:
from sympy import expand_mul
return sqrt(expand_mul(arg * arg.conjugate()))
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if exponent.is_even and base.is_real:
return arg
def _combine_inverse(lhs, rhs):
"""
Returns lhs - rhs, but treats arguments like symbols, so things like
oo - oo return 0, instead of a nan.
"""
from sympy import oo, I, expand_mul
if lhs == oo and rhs == oo or lhs == oo*I and rhs == oo*I:
return S.Zero
return expand_mul(lhs - rhs)
def _eval_expand_func(self, **hints):
from sympy import log, expand_mul, Dummy, exp_polar, I
s, z = self.args
if s == 1:
return -log(1 + exp_polar(-I*pi)*z)
if s.is_Integer and s <= 0:
u = Dummy('u')
start = u/(1 - u)
for _ in range(-s):
start = u*start.diff(u)
return expand_mul(start).subs(u, z)
return polylog(s, z)
def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False):
""" A helper for the real inverse_mellin_transform function, this one here
assumes x to be real and positive. """
from sympy import (expand, expand_mul, hyperexpand, meijerg, And, Or,
arg, pi, re, factor, Heaviside, gamma, Add)
x = _dummy('t', 'inverse-mellin-transform', F, positive=True)
# Actually, we won't try integration at all. Instead we use the definition
# of the Meijer G function as a fairly general inverse mellin transform.
F = F.rewrite(gamma)
for g in [factor(F), expand_mul(F), expand(F)]:
if g.is_Add:
# do all terms separately
ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg,
noconds=False) \
for G in g.args]
conds = [p[1] for p in ress]
ress = [p[0] for p in ress]
res = Add(*ress)
if not as_meijerg:
res = factor(res, gens=res.atoms(Heaviside))
return res.subs(x, x_), And(*conds)
try:
a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1])
except IntegralTransformError:
continue
G = meijerg(a, b, C/x**e)
if as_meijerg:
h = G
else:
h = hyperexpand(G)
if h.is_Piecewise and len(h.args) == 3:
# XXX we break modularity here!
h = Heaviside(x - abs(C))*h.args[0].args[0] \
+ Heaviside(abs(C) - x)*h.args[1].args[0]
# We must ensure that the intgral along the line we want converges,
# and return that value.
# See [L], 5.2
cond = [abs(arg(G.argument)) < G.delta*pi]
# Note: we allow ">=" here, this corresponds to convergence if we let
# limits go to oo symetrically. ">" corresponds to absolute convergence.
cond += [And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1),
abs(arg(G.argument)) == G.delta*pi)]
cond = Or(*cond)
if cond is False:
raise IntegralTransformError('Inverse Mellin', F, 'does not converge')
return (h*fac).subs(x, x_), cond
raise IntegralTransformError('Inverse Mellin', F, '')
def solution(pars):
(a2,a1,a0),(t1,t2),(k1,k2),(y0,yd0)=pars
a2,a1,a0 = map(int,[a2,a1,a0])
den = a2*s**2+a1*s+a0
V1 = apart(1/s/den).as_ordered_terms() # const.
V2 = apart(1/s**2/den).as_ordered_terms() # lin.
Iv1,Iv2 = ilt_pfe(V1),ilt_pfe(V2)
LIC = (a2*(s*y0+yd0) + a1*y0)/den
y1 = ILT(LIC, s,t)
y2 = k1*Iv2 # k1, t
y3 = (k2-k1)*Iv2.subs(t,t-t1)*H(t-t1) # k2-k1,(t-t1)*H(t-t1)
y4 = (t1*(k2-k1)-k2*t2)*Iv1.subs(t,t-t1)*H(t-t1) # t1*(k2-k1)-k2*t2,H(t-t1), nulove pre spojite
y5 = -k2*Iv2.subs(t,t-t2)*H(t-t2) # -k2, (t-t2)*H(t-t2)
y = expand_mul(simplify(y1+y2+y3+y4+y5))
return y
def _eval_expand_func(self, **hints):
from sympy import log, expand_mul, Dummy, exp_polar, I
if hints.get('deep', False):
s, z = map(lambda x: x._eval_expand_func(**hints), self.args)
else:
s, z = self.args
if s == 1:
return -log(1 + exp_polar(-I*pi)*z)
if s.is_Integer and s <= 0:
u = Dummy('u')
start = u/(1 - u)
for _ in range(-s):
start = u*start.diff(u)
return expand_mul(start).subs(u, z)
return polylog(s, z)
def eval(cls, arg):
if arg is S.NaN:
return S.NaN
if arg.is_zero: return arg
if arg.is_positive: return arg
if arg.is_negative: return -arg
coeff, terms = arg.as_coeff_mul()
if coeff is not S.One:
return cls(coeff) * cls(Mul(*terms))
if arg.is_real is False:
from sympy import expand_mul
return sqrt( expand_mul(arg * arg.conjugate()) )
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if exponent.is_even and base.is_real:
return arg
return
def eval(cls, nu, z):
from sympy import unpolarify, expand_mul, uppergamma, exp, gamma, factorial
nu2 = unpolarify(nu)
if nu != nu2:
return expint(nu2, z)
if nu.is_Integer and nu <= 0 or (not nu.is_Integer and (2 * nu).is_Integer):
return unpolarify(expand_mul(z ** (nu - 1) * uppergamma(1 - nu, z)))
# Extract branching information. This can be deduced from what is
# explained in lowergamma.eval().
z, n = z.extract_branch_factor()
if n == 0:
return
if nu.is_integer:
if (nu > 0) is not True:
return
return expint(nu, z) - 2 * pi * I * n * (-1) ** (nu - 1) / factorial(nu - 1) * unpolarify(z) ** (nu - 1)
else:
return (exp(2 * I * pi * nu * n) - 1) * z ** (nu - 1) * gamma(1 - nu) + expint(nu, z)
def _eval_expand_mul(self, **hints):
from sympy import fraction, expand_mul
# Handle things like 1/(x*(x + 1)), which are automatically converted
# to 1/x*1/(x + 1)
expr = self
n, d = fraction(expr)
if d.is_Mul:
expr = n/d._eval_expand_mul(**hints)
if not expr.is_Mul:
return expand_mul(expr, deep=False)
plain, sums, rewrite = [], [], False
for factor in expr.args:
if factor.is_Add:
sums.append(factor)
rewrite = True
else:
if factor.is_commutative:
plain.append(factor)
else:
sums.append(Basic(factor)) # Wrapper
if not rewrite:
return expr
else:
plain = Mul(*plain)
if sums:
terms = Mul._expandsums(sums)
args = []
for term in terms:
t = Mul(plain, term)
if t.is_Mul and any(a.is_Add for a in t.args):
t = t._eval_expand_mul()
args.append(t)
return Add(*args)
else:
return plain
def doit(self, **hints):
"""
Try to evaluate the transform in closed form.
This general function handles linearity, but apart from that leaves
pretty much everything to _compute_transform.
Standard hints are the following:
- ``simplify``: whether or not to simplify the result
- ``noconds``: if True, don't return convergence conditions
- ``needeval``: if True, raise IntegralTransformError instead of
returning IntegralTransform objects
The default values of these hints depend on the concrete transform,
usually the default is
``(simplify, noconds, needeval) = (True, False, False)``.
"""
from sympy import Add, expand_mul, Mul
from sympy.core.function import AppliedUndef
needeval = hints.pop('needeval', False)
try_directly = not any(func.has(self.function_variable) \
for func in self.function.atoms(AppliedUndef))
if try_directly:
try:
return self._compute_transform(self.function,
self.function_variable, self.transform_variable, **hints)
except IntegralTransformError:
pass
fn = self.function
if not fn.is_Add:
fn = expand_mul(fn)
if fn.is_Add:
hints['needeval'] = needeval
res = [self.__class__(*([x] + list(self.args[1:]))).doit(**hints)
for x in fn.args]
extra = []
ress = []
for x in res:
if not isinstance(x, tuple):
x = [x]
ress.append(x[0])
if len(x) > 1:
extra += [x[1:]]
res = Add(*ress)
if not extra:
return res
try:
extra = self._collapse_extra(extra)
return tuple([res]) + tuple(extra)
except IntegralTransformError:
pass
if needeval:
raise IntegralTransformError(self.__class__._name, self.function, 'needeval')
# TODO handle derivatives etc
# pull out constant coefficients
coeff, rest = fn.as_coeff_mul(self.function_variable)
return coeff*self.__class__(*([Mul(*rest)] + list(self.args[1:])))
>>> _rewrite_sin((2*pi, 0), s, 0, S(1)/2)
(gamma(2*s), gamma(-2*s + 1), pi)
>>> _rewrite_sin((2*pi, 0), s, S(1)/2, 1)
(gamma(2*s - 1), gamma(-2*s + 2), -pi)
"""
# (This is a separate function because it is moderately complicated,
# and I want to doctest it.)
# We want to use pi/sin(pi*x) = gamma(x)*gamma(1-x).
# But there is one comlication: the gamma functions determine the
# inegration contour in the definition of the G-function. Usually
# it would not matter if this is slightly shifted, unless this way
# we create an undefined function!
# So we try to write this in such a way that the gammas are
# eminently on the right side of the strip.
from sympy import expand_mul, pi, ceiling, gamma, re
m = expand_mul(m/pi)
n = expand_mul(n/pi)
r = ceiling(-m*a - n.as_real_imag()[0]) # Don't use re(n), does not expand
return gamma(m*s + n + r), gamma(1 - n - r - m*s), (-1)**r*pi
class MellinTransformStripError(ValueError):
"""
Exception raised by _rewrite_gamma. Mainly for internal use.
"""
pass
def _rewrite_gamma(f, s, a, b):
"""
Try to rewrite the product f(s) as a product of gamma functions,
so that the inverse mellin transform of f can be expressed as a meijer
G function.
def test_pinjoint_arbitrary_axis():
theta, omega = dynamicsymbols('theta_J, omega_J')
# When the bodies are attached though masscenters but axess are opposite.
N, A, P, C = _generate_body()
PinJoint('J', P, C, child_axis=-A.x)
assert (-A.x).angle_between(N.x) == 0
assert -A.x.express(N) == N.x
assert A.dcm(N) == Matrix([[-1, 0, 0],
[0, -cos(theta), -sin(theta)],
[0, -sin(theta), cos(theta)]])
assert A.ang_vel_in(N) == omega*N.x
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
assert C.masscenter.pos_from(P.masscenter) == 0
assert C.masscenter.pos_from(P.masscenter).express(N).simplify() == 0
assert C.masscenter.vel(N) == 0
# When axes are different and parent joint is at masscenter but child joint
# is at a unit vector from child masscenter.
N, A, P, C = _generate_body()
PinJoint('J', P, C, child_axis=A.y, child_joint_pos=A.x)
assert A.y.angle_between(N.x) == 0 # Axis are aligned
assert A.y.express(N) == N.x
assert A.dcm(N) == Matrix([[0, -cos(theta), -sin(theta)],
[1, 0, 0],
[0, -sin(theta), cos(theta)]])
assert A.ang_vel_in(N) == omega*N.x
assert A.ang_vel_in(N).express(A) == omega * A.y
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.y)
assert angle.xreplace({omega: 1}) == 0
assert C.masscenter.vel(N) == omega*A.z
assert C.masscenter.pos_from(P.masscenter) == -A.x
assert (C.masscenter.pos_from(P.masscenter).express(N).simplify() ==
cos(theta)*N.y + sin(theta)*N.z)
assert C.masscenter.vel(N).angle_between(A.x) == pi/2
# Similar to previous case but wrt parent body
N, A, P, C = _generate_body()
PinJoint('J', P, C, parent_axis=N.y, parent_joint_pos=N.x)
assert N.y.angle_between(A.x) == 0 # Axis are aligned
assert N.y.express(A) == A.x
assert A.dcm(N) == Matrix([[0, 1, 0],
[-cos(theta), 0, sin(theta)],
[sin(theta), 0, cos(theta)]])
assert A.ang_vel_in(N) == omega*N.y
assert A.ang_vel_in(N).express(A) == omega*A.x
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x)
assert angle.xreplace({omega: 1}) == 0
assert C.masscenter.vel(N).simplify() == - omega*N.z
assert C.masscenter.pos_from(P.masscenter) == N.x
# Both joint pos id defined but different axes
N, A, P, C = _generate_body()
PinJoint('J', P, C, parent_joint_pos=N.x, child_joint_pos=A.x,
child_axis=A.x+A.y)
assert expand_mul(N.x.angle_between(A.x + A.y)) == 0 # Axis are aligned
assert (A.x + A.y).express(N).simplify() == sqrt(2)*N.x
assert _simplify_matrix(A.dcm(N)) == Matrix([
[sqrt(2)/2, -sqrt(2)*cos(theta)/2, -sqrt(2)*sin(theta)/2],
[sqrt(2)/2, sqrt(2)*cos(theta)/2, sqrt(2)*sin(theta)/2],
[0, -sin(theta), cos(theta)]])
assert A.ang_vel_in(N) == omega*N.x
assert (A.ang_vel_in(N).express(A).simplify() ==
(omega*A.x + omega*A.y)/sqrt(2))
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x + A.y)
assert angle.xreplace({omega: 1}) == 0
assert C.masscenter.vel(N).simplify() == (omega * A.z)/sqrt(2)
assert C.masscenter.pos_from(P.masscenter) == N.x - A.x
assert (C.masscenter.pos_from(P.masscenter).express(N).simplify() ==
(1 - sqrt(2)/2)*N.x + sqrt(2)*cos(theta)/2*N.y +
sqrt(2)*sin(theta)/2*N.z)
assert (C.masscenter.vel(N).express(N).simplify() ==
-sqrt(2)*omega*sin(theta)/2*N.y + sqrt(2)*omega*cos(theta)/2*N.z)
assert C.masscenter.vel(N).angle_between(A.x) == pi/2
N, A, P, C = _generate_body()
PinJoint('J', P, C, parent_joint_pos=N.x, child_joint_pos=A.x,
child_axis=A.x+A.y-A.z)
assert expand_mul(N.x.angle_between(A.x + A.y - A.z)) == 0 # Axis aligned
assert (A.x + A.y - A.z).express(N).simplify() == sqrt(3)*N.x
assert _simplify_matrix(A.dcm(N)) == Matrix([
[sqrt(3)/3, -sqrt(6)*sin(theta + pi/4)/3,
sqrt(6)*cos(theta + pi/4)/3],
[sqrt(3)/3, sqrt(6)*cos(theta + pi/12)/3,
sqrt(6)*sin(theta + pi/12)/3],
[-sqrt(3)/3, sqrt(6)*cos(theta + 5*pi/12)/3,
sqrt(6)*sin(theta + 5*pi/12)/3]])
assert A.ang_vel_in(N) == omega*N.x
assert A.ang_vel_in(N).express(A).simplify() == (omega*A.x + omega*A.y -
omega*A.z)/sqrt(3)
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x + A.y-A.z)
assert angle.xreplace({omega: 1}) == 0
assert C.masscenter.vel(N).simplify() == (omega*A.y + omega*A.z)/sqrt(3)
assert C.masscenter.pos_from(P.masscenter) == N.x - A.x
assert (C.masscenter.pos_from(P.masscenter).express(N).simplify() ==
(1 - sqrt(3)/3)*N.x + sqrt(6)*sin(theta + pi/4)/3*N.y -
sqrt(6)*cos(theta + pi/4)/3*N.z)
assert (C.masscenter.vel(N).express(N).simplify() ==
sqrt(6)*omega*cos(theta + pi/4)/3*N.y +
sqrt(6)*omega*sin(theta + pi/4)/3*N.z)
assert C.masscenter.vel(N).angle_between(A.x) == pi/2
N, A, P, C = _generate_body()
m, n = symbols('m n')
PinJoint('J', P, C, parent_joint_pos=m*N.x, child_joint_pos=n*A.x,
child_axis=A.x+A.y-A.z, parent_axis=N.x-N.y+N.z)
angle = (N.x-N.y+N.z).angle_between(A.x+A.y-A.z)
assert expand_mul(angle) == 0 # Axis are aligned
assert ((A.x-A.y+A.z).express(N).simplify() ==
(-4*cos(theta)/3 - S(1)/3)*N.x + (S(1)/3 - 4*sin(theta + pi/6)/3)*N.y +
(4*cos(theta + pi/3)/3 - S(1)/3)*N.z)
assert _simplify_matrix(A.dcm(N)) == Matrix([
[S(1)/3 - 2*cos(theta)/3, -2*sin(theta + pi/6)/3 - S(1)/3,
2*cos(theta + pi/3)/3 + S(1)/3],
[2*cos(theta + pi/3)/3 + S(1)/3, 2*cos(theta)/3 - S(1)/3,
2*sin(theta + pi/6)/3 + S(1)/3],
[-2*sin(theta + pi/6)/3 - S(1)/3, 2*cos(theta + pi/3)/3 + S(1)/3,
2*cos(theta)/3 - S(1)/3]])
assert A.ang_vel_in(N) == (omega*N.x - omega*N.y + omega*N.z)/sqrt(3)
assert A.ang_vel_in(N).express(A).simplify() == (omega*A.x + omega*A.y -
omega*A.z)/sqrt(3)
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x+A.y-A.z)
assert angle.xreplace({omega: 1}) == 0
assert (C.masscenter.vel(N).simplify() ==
(m*omega*N.y + m*omega*N.z + n*omega*A.y + n*omega*A.z)/sqrt(3))
assert C.masscenter.pos_from(P.masscenter) == m*N.x - n*A.x
assert (C.masscenter.pos_from(P.masscenter).express(N).simplify() ==
(m + n*(2*cos(theta) - 1)/3)*N.x + n*(2*sin(theta + pi/6) +
1)/3*N.y - n*(2*cos(theta + pi/3) + 1)/3*N.z)
assert (C.masscenter.vel(N).express(N).simplify() ==
-2*n*omega*sin(theta)/3*N.x + (sqrt(3)*m +
2*n*cos(theta + pi/6))*omega/3*N.y
+ (sqrt(3)*m + 2*n*sin(theta + pi/3))*omega/3*N.z)
assert expand_mul(C.masscenter.vel(N).angle_between(m*N.x - n*A.x)) == pi/2
def test_acsch():
x = Symbol('x')
assert unchanged(acsch, x)
assert acsch(-x) == -acsch(x)
# values at fixed points
assert acsch(1) == log(1 + sqrt(2))
assert acsch(-1) == -log(1 + sqrt(2))
assert acsch(0) is zoo
assert acsch(2) == log((1 + sqrt(5)) / 2)
assert acsch(-2) == -log((1 + sqrt(5)) / 2)
assert acsch(I) == -I * pi / 2
assert acsch(-I) == I * pi / 2
assert acsch(-I * (sqrt(6) + sqrt(2))) == I * pi / 12
assert acsch(I * (sqrt(2) + sqrt(6))) == -I * pi / 12
assert acsch(-I * (1 + sqrt(5))) == I * pi / 10
assert acsch(I * (1 + sqrt(5))) == -I * pi / 10
assert acsch(-I * 2 / sqrt(2 - sqrt(2))) == I * pi / 8
assert acsch(I * 2 / sqrt(2 - sqrt(2))) == -I * pi / 8
assert acsch(-I * 2) == I * pi / 6
assert acsch(I * 2) == -I * pi / 6
assert acsch(-I * sqrt(2 + 2 / sqrt(5))) == I * pi / 5
assert acsch(I * sqrt(2 + 2 / sqrt(5))) == -I * pi / 5
assert acsch(-I * sqrt(2)) == I * pi / 4
assert acsch(I * sqrt(2)) == -I * pi / 4
assert acsch(-I * (sqrt(5) - 1)) == 3 * I * pi / 10
assert acsch(I * (sqrt(5) - 1)) == -3 * I * pi / 10
assert acsch(-I * 2 / sqrt(3)) == I * pi / 3
assert acsch(I * 2 / sqrt(3)) == -I * pi / 3
assert acsch(-I * 2 / sqrt(2 + sqrt(2))) == 3 * I * pi / 8
assert acsch(I * 2 / sqrt(2 + sqrt(2))) == -3 * I * pi / 8
assert acsch(-I * sqrt(2 - 2 / sqrt(5))) == 2 * I * pi / 5
assert acsch(I * sqrt(2 - 2 / sqrt(5))) == -2 * I * pi / 5
assert acsch(-I * (sqrt(6) - sqrt(2))) == 5 * I * pi / 12
assert acsch(I * (sqrt(6) - sqrt(2))) == -5 * I * pi / 12
assert acsch(nan) is nan
# properties
# acsch(x) == asinh(1/x)
assert acsch(-I * sqrt(2)) == asinh(I / sqrt(2))
assert acsch(-I * 2 / sqrt(3)) == asinh(I * sqrt(3) / 2)
# acsch(x) == -I*asin(I/x)
assert acsch(-I * sqrt(2)) == -I * asin(-1 / sqrt(2))
assert acsch(-I * 2 / sqrt(3)) == -I * asin(-sqrt(3) / 2)
# csch(acsch(x)) / x == 1
assert expand_mul(
csch(acsch(-I * (sqrt(6) + sqrt(2)))) / (-I *
(sqrt(6) + sqrt(2)))) == 1
assert expand_mul(csch(acsch(I * (1 + sqrt(5)))) / (I *
(1 + sqrt(5)))) == 1
assert (csch(acsch(I * sqrt(2 - 2 / sqrt(5)))) /
(I * sqrt(2 - 2 / sqrt(5)))).simplify() == 1
assert (csch(acsch(-I * sqrt(2 - 2 / sqrt(5)))) /
(-I * sqrt(2 - 2 / sqrt(5)))).simplify() == 1
# numerical evaluation
assert str(acsch(5 * I + 1).n(6)) == '0.0391819 - 0.193363*I'
assert str(acsch(-5 * I + 1).n(6)) == '0.0391819 + 0.193363*I'
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp, sin
mu1, mu2 = symbols("mu1 mu2", real=True, nonzero=True, finite=True)
sigma1, sigma2 = symbols("sigma1 sigma2", real=True, nonzero=True, finite=True, positive=True)
rate = Symbol("lambda", real=True, positive=True, finite=True)
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma ** 2) * exp(-(x - mu) ** 2 / 2 / sigma ** 2)
def exponential(x, rate):
return rate * exp(-rate * x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1
assert integrate(x ** 2 * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1 ** 2 + sigma1 ** 2
assert integrate(x ** 3 * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1 ** 3 + 3 * mu1 * sigma1 ** 2
assert integrate(normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True) == 1
assert (
integrate(x * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1
)
assert (
integrate(y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu2
)
assert (
integrate(x * y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True)
== mu1 * mu2
)
assert (
integrate(
(x + y + 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True
)
== 1 + mu1 + mu2
)
assert (
integrate(
(x + y - 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True
)
== -1 + mu1 + mu2
)
i = integrate(x ** 2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1 ** 2 + sigma1 ** 2
assert (
integrate(y ** 2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True)
== sigma2 ** 2 + mu2 ** 2
)
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x * exponential(x, rate), (x, 0, oo), meijerg=True) == 1 / rate
assert integrate(x ** 2 * exponential(x, rate), (x, 0, oo), meijerg=True) == 2 / rate ** 2
def E(expr):
res1 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1), (x, 0, oo), (y, -oo, oo), meijerg=True)
res2 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1), (y, -oo, oo), (x, 0, oo), meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x * y) == mu1 / rate
assert E(x * y ** 2) == mu1 ** 2 / rate + sigma1 ** 2 / rate
ans = sigma1 ** 2 + 1 / rate ** 2
assert simplify(E((x + y + 1) ** 2) - E(x + y + 1) ** 2) == ans
assert simplify(E((x + y - 1) ** 2) - E(x + y - 1) ** 2) == ans
assert simplify(E((x + y) ** 2) - E(x + y) ** 2) == ans
# Beta' distribution
alpha, beta = symbols("alpha beta", positive=True)
betadist = x ** (alpha - 1) * (1 + x) ** (-alpha - beta) * gamma(alpha + beta) / gamma(alpha) / gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x * betadist, (x, 0, oo), meijerg=True, conds="separate")
assert (combsimp(i[0]), i[1]) == (alpha / (beta - 1), 1 < beta)
j = integrate(x ** 2 * betadist, (x, 0, oo), meijerg=True, conds="separate")
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0] ** 2) == (alpha + beta - 1) * alpha / (beta - 2) / (beta - 1) ** 2
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols("a b", positive=True)
betadist = x ** (a - 1) * (-x + 1) ** (b - 1) * gamma(a + b) / (gamma(a) * gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x * betadist, (x, 0, 1), meijerg=True)) == a / (a + b)
assert simplify(integrate(x ** 2 * betadist, (x, 0, 1), meijerg=True)) == a * (a + 1) / (a + b) / (a + b + 1)
assert simplify(integrate(x ** y * betadist, (x, 0, 1), meijerg=True)) == gamma(a + b) * gamma(a + y) / gamma(
a
) / gamma(a + b + y)
# Chi distribution
k = Symbol("k", integer=True, positive=True)
chi = 2 ** (1 - k / 2) * x ** (k - 1) * exp(-x ** 2 / 2) / gamma(k / 2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chi, (x, 0, oo), meijerg=True)) == sqrt(2) * gamma((k + 1) / 2) / gamma(k / 2)
assert simplify(integrate(x ** 2 * chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2 ** (-k / 2) / gamma(k / 2) * x ** (k / 2 - 1) * exp(-x / 2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x ** 2 * chisquared, (x, 0, oo), meijerg=True)) == k * (k + 2)
assert combsimp(integrate(((x - k) / sqrt(2 * k)) ** 3 * chisquared, (x, 0, oo), meijerg=True)) == 2 * sqrt(
2
) / sqrt(k)
# Dagum distribution
a, b, p = symbols("a b p", positive=True)
# XXX (x/b)**a does not work
dagum = a * p / x * (x / b) ** (a * p) / (1 + x ** a / b ** a) ** (p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x * dagum
assert simplify(integrate(arg, (x, 0, oo), meijerg=True, conds="none")) == a * b * gamma(1 - 1 / a) * gamma(
p + 1 + 1 / a
) / ((a * p + 1) * gamma(p))
assert simplify(integrate(x * arg, (x, 0, oo), meijerg=True, conds="none")) == a * b ** 2 * gamma(
1 - 2 / a
) * gamma(p + 1 + 2 / a) / ((a * p + 2) * gamma(p))
# F-distribution
d1, d2 = symbols("d1 d2", positive=True)
f = (
sqrt(((d1 * x) ** d1 * d2 ** d2) / (d1 * x + d2) ** (d1 + d2))
/ x
/ gamma(d1 / 2)
/ gamma(d2 / 2)
* gamma((d1 + d2) / 2)
)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x * f, (x, 0, oo), meijerg=True, conds="none")) == d2 / (d2 - 2)
assert simplify(integrate(x ** 2 * f, (x, 0, oo), meijerg=True, conds="none")) == d2 ** 2 * (d1 + 2) / d1 / (
d2 - 4
) / (d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols("lamda mu", positive=True)
dist = sqrt(lamda / 2 / pi) * x ** (-S(3) / 2) * exp(-lamda * (x - mu) ** 2 / x / 2 / mu ** 2)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x * dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu) ** 2 * dist, (x, 0, oo))) == mu ** 3 / lamda
assert mysimp(integrate((x - mu) ** 3 * dist, (x, 0, oo))) == 3 * mu ** 5 / lamda ** 2
# Levi
c = Symbol("c", positive=True)
assert integrate(sqrt(c / 2 / pi) * exp(-c / 2 / (x - mu)) / (x - mu) ** S("3/2"), (x, mu, oo)) == 1
# higher moments oo
# log-logistic
distn = (beta / alpha) * x ** (beta - 1) / alpha ** (beta - 1) / (1 + x ** beta / alpha ** beta) ** 2
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(integrate(x * distn, (x, 0, oo), conds="none")) == pi * alpha / beta / sin(pi / beta)
# (similar comment for conditions applies)
assert simplify(integrate(x ** y * distn, (x, 0, oo), conds="none")) == pi * alpha ** y * y / beta / sin(
pi * y / beta
)
# weibull
k = Symbol("k", positive=True)
n = Symbol("n", positive=True)
distn = k / lamda * (x / lamda) ** (k - 1) * exp(-(x / lamda) ** k)
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x ** n * distn, (x, 0, oo))) == lamda ** n * gamma(1 + n / k)
# rice distribution
from sympy import besseli
nu, sigma = symbols("nu sigma", positive=True)
rice = x / sigma ** 2 * exp(-(x ** 2 + nu ** 2) / 2 / sigma ** 2) * besseli(0, x * nu / sigma ** 2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol("mu", real=True)
b = Symbol("b", positive=True)
laplace = exp(-abs(x - mu) / b) / 2 / b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x * laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x ** 2 * laplace, (x, -oo, oo), meijerg=True) == 2 * b ** 2 + mu ** 2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol("k", positive=True)
assert combsimp(expand_mul(integrate(log(x) * x ** (k - 1) * exp(-x) / gamma(k), (x, 0, oo)))) == polygamma(0, k)
def simp_pows(expr): return expand_mul(simplify(powsimp(expr, force=True)), deep=True).replace(exp_polar, exp) # XXX ?
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
def _eval_Abs(self):
from sympy import expand_mul
return sqrt( expand_mul(self * self.conjugate()) )
def simp_hyp(expr):
return factor_terms(expand_mul(expr)).rewrite(sin)
def solve_linear(lhs, rhs=0, x=[], exclude=[]):
""" Return a tuple containing derived from f = lhs - rhs that is either:
(numerator, denominator) of f; if this comes back as (0, 1) it means
that f was actually zero even though it may have had symbols:
e.g. y*cos(x)**2 + y*sin(x)**2 - y = y*(0) = 0 If the numerator
is not zero then the function is guaranteed not to be zero.
or
(symbol, solution) where symbol appears linearly in the numerator of f,
is in x (if given) and is not in exclude (if given).
No simplification is done to f other than and mul=True expansion, so
the solution will correspond strictly to a unique solution.
Examples:
>>> from sympy.solvers.solvers import solve_linear
>>> from sympy.abc import x, y, z
These are linear in x and 1/x:
>>> solve_linear(x + y**2)
(x, -y**2)
>>> solve_linear(1/x - y**2)
(x, y**(-2))
When not linear in x or y then the numerator and denominator are returned.
>>> solve_linear(x**2/y**2 - 3)
(x**2 - 3*y**2, y**2)
If x is allowed to cancel, then this appears linear, but this sort of
cancellation is not done so the solultion will always satisfy the original
expression without causing a division by zero error.
>>> solve_linear(x**2*(1/x - z**2/x))
(x**2*(x - x*z**2), x**2)
You can give a list of what you prefer for x candidates:
>>> solve_linear(x + y + z, x=[y])
(y, -x - z)
You can also indicate what variables you don't want to consider:
>>> solve_linear(x + y + z, exclude=[x, z])
(y, -x - z)
If only x was excluded then a solution for y or z might be obtained.
"""
from sympy import expand_mul, Equality
if isinstance(lhs, Equality):
rhs += lhs.rhs
lhs = lhs.lhs
n, d = (lhs - rhs).as_numer_denom()
ex = expand_mul(n)
if not ex:
return ex, d
exclude = set(exclude)
syms = ex.free_symbols
if not x:
x = syms
else:
x = syms.intersection(x)
x = x.difference(exclude)
for xi in x:
dn = n.diff(xi)
# if not dn then this is a pseudo-function of xi
if dn and not dn.has(xi):
return xi, -(n.subs(xi, 0)) / dn
return n, d
def _minpoly_groebner(ex, x, cls):
"""
Computes the minimal polynomial of an algebraic number
using Groebner bases
Examples
========
>>> from sympy import minimal_polynomial, sqrt, Rational
>>> from sympy.abc import x
>>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False)
x**2 - 2*x - 1
"""
from sympy.polys.polytools import degree
from sympy.core.function import expand_multinomial
generator = numbered_symbols('a', cls=Dummy)
mapping, symbols, replace = {}, {}, []
def update_mapping(ex, exp, base=None):
a = next(generator)
symbols[ex] = a
if base is not None:
mapping[ex] = a**exp + base
else:
mapping[ex] = exp.as_expr(a)
return a
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0 and ex.base.is_Add:
coeff, terms = ex.base.as_coeff_add()
elt, _ = primitive_element(terms, polys=True)
alg = ex.base - coeff
# XXX: turn this into eval()
inverse = invert(elt.gen + coeff, elt).as_expr()
base = inverse.subs(elt.gen, alg).expand()
if ex.exp == -1:
return bottom_up_scan(base)
else:
ex = base**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
def simpler_inverse(ex):
"""
Returns True if it is more likely that the minimal polynomial
algorithm works better with the inverse
"""
if ex.is_Pow:
if (1 / ex.exp).is_integer and ex.exp < 0:
if ex.base.is_Add:
return True
if ex.is_Mul:
hit = True
a = []
for p in ex.args:
if p.is_Add:
return False
if p.is_Pow:
if p.base.is_Add and p.exp > 0:
return False
if hit:
return True
return False
inverted = False
ex = expand_multinomial(ex)
if ex.is_AlgebraicNumber:
return ex.minpoly.as_expr(x)
elif ex.is_Rational:
result = ex.q * x - ex.p
else:
inverted = simpler_inverse(ex)
if inverted:
ex = ex**-1
res = None
if ex.is_Pow and (1 / ex.exp).is_Integer:
n = 1 / ex.exp
res = _minimal_polynomial_sq(ex.base, n, x)
elif _is_sum_surds(ex):
res = _minimal_polynomial_sq(ex, S.One, x)
if res is not None:
result = res
if res is None:
bus = bottom_up_scan(ex)
F = [x - bus] + list(mapping.values())
G = groebner(F, list(symbols.values()) + [x], order='lex')
_, factors = factor_list(G[-1])
# by construction G[-1] has root `ex`
result = _choose_factor(factors, x, ex)
if inverted:
result = _invertx(result, x)
if result.coeff(x**degree(result, x)) < 0:
result = expand_mul(-result)
return result
def doit(self, **hints):
"""
Try to evaluate the transform in closed form.
This general function handles linearity, but apart from that leaves
pretty much everything to _compute_transform.
Standard hints are the following:
- ``simplify``: whether or not to simplify the result
- ``noconds``: if True, don't return convergence conditions
- ``needeval``: if True, raise IntegralTransformError instead of
returning IntegralTransform objects
The default values of these hints depend on the concrete transform,
usually the default is
``(simplify, noconds, needeval) = (True, False, False)``.
"""
from sympy import Add, expand_mul, Mul
from sympy.core.function import AppliedUndef
needeval = hints.pop('needeval', False)
try_directly = not any(func.has(self.function_variable) \
for func in self.function.atoms(AppliedUndef))
if try_directly:
try:
return self._compute_transform(self.function,
self.function_variable,
self.transform_variable,
**hints)
except IntegralTransformError:
pass
fn = self.function
if not fn.is_Add:
fn = expand_mul(fn)
if fn.is_Add:
hints['needeval'] = needeval
res = [
self.__class__(*([x] + list(self.args[1:]))).doit(**hints)
for x in fn.args
]
extra = []
ress = []
for x in res:
if not isinstance(x, tuple):
x = [x]
ress.append(x[0])
if len(x) > 1:
extra += [x[1:]]
res = Add(*ress)
if not extra:
return res
try:
extra = self._collapse_extra(extra)
return tuple([res]) + tuple(extra)
except IntegralTransformError:
pass
if needeval:
raise IntegralTransformError(self.__class__._name, self.function,
'needeval')
# TODO handle derivatives etc
# pull out constant coefficients
coeff, rest = fn.as_coeff_mul(self.function_variable)
return coeff * self.__class__(*([Mul(*rest)] + list(self.args[1:])))
def simp_hyp(expr):
return factor_terms(expand_mul(expr)).rewrite(sin)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp, sin
mu1, mu2 = symbols('mu1 mu2', real=True, nonzero=True, finite=True)
sigma1, sigma2 = symbols('sigma1 sigma2',
real=True,
nonzero=True,
finite=True,
positive=True)
rate = Symbol('lambda', real=True, positive=True, finite=True)
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma**2) * exp(-(x - mu)**2 / 2 / sigma**2)
def exponential(x, rate):
return rate * exp(-rate * x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == \
mu1
assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**2 + sigma1**2
assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**3 + 3*mu1*sigma1**2
assert integrate(normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == 1
assert integrate(x * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu1
assert integrate(y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu2
assert integrate(x * y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu1 * mu2
assert integrate(
(x + y + 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == 1 + mu1 + mu2
assert integrate((x + y - 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
-1 + mu1 + mu2
i = integrate(x**2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1**2 + sigma1**2
assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
sigma2**2 + mu2**2
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == \
1/rate
assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) == \
2/rate**2
def E(expr):
res1 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo),
meijerg=True)
res2 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo),
meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x * y) == mu1 / rate
assert E(x * y**2) == mu1**2 / rate + sigma1**2 / rate
ans = sigma1**2 + 1 / rate**2
assert simplify(E((x + y + 1)**2) - E(x + y + 1)**2) == ans
assert simplify(E((x + y - 1)**2) - E(x + y - 1)**2) == ans
assert simplify(E((x + y)**2) - E(x + y)**2) == ans
# Beta' distribution
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha - 1)*(1 + x)**(-alpha - beta)*gamma(alpha + beta) \
/gamma(alpha)/gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x * betadist, (x, 0, oo), meijerg=True, conds='separate')
assert (combsimp(i[0]), i[1]) == (alpha / (beta - 1), 1 < beta)
j = integrate(x**2 * betadist, (x, 0, oo), meijerg=True, conds='separate')
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
/(beta - 2)/(beta - 1)**2
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols('a b', positive=True)
betadist = x**(a - 1) * (-x + 1)**(b - 1) * gamma(a + b) / (gamma(a) *
gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x*betadist, (x, 0, 1), meijerg=True)) == \
a/(a + b)
assert simplify(integrate(x**2*betadist, (x, 0, 1), meijerg=True)) == \
a*(a + 1)/(a + b)/(a + b + 1)
assert simplify(integrate(x**y*betadist, (x, 0, 1), meijerg=True)) == \
gamma(a + b)*gamma(a + y)/gamma(a)/gamma(a + b + y)
# Chi distribution
k = Symbol('k', integer=True, positive=True)
chi = 2**(1 - k / 2) * x**(k - 1) * exp(-x**2 / 2) / gamma(k / 2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
assert simplify(integrate(x**2 * chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2**(-k / 2) / gamma(k / 2) * x**(k / 2 - 1) * exp(-x / 2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
k*(k + 2)
assert combsimp(
integrate(((x - k) / sqrt(2 * k))**3 * chisquared, (x, 0, oo),
meijerg=True)) == 2 * sqrt(2) / sqrt(k)
# Dagum distribution
a, b, p = symbols('a b p', positive=True)
# XXX (x/b)**a does not work
dagum = a * p / x * (x / b)**(a * p) / (1 + x**a / b**a)**(p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x * dagum
assert simplify(integrate(
arg, (x, 0, oo), meijerg=True,
conds='none')) == a * b * gamma(1 - 1 / a) * gamma(p + 1 + 1 / a) / (
(a * p + 1) * gamma(p))
assert simplify(integrate(
x * arg, (x, 0, oo), meijerg=True,
conds='none')) == a * b**2 * gamma(1 -
2 / a) * gamma(p + 1 + 2 / a) / (
(a * p + 2) * gamma(p))
# F-distribution
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1 + d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x * f, (x, 0, oo), meijerg=True,
conds='none')) == d2 / (d2 - 2)
assert simplify(
integrate(x**2 * f, (x, 0, oo), meijerg=True,
conds='none')) == d2**2 * (d1 + 2) / d1 / (d2 - 4) / (d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols('lamda mu', positive=True)
dist = sqrt(lamda / 2 / pi) * x**(-S(3) / 2) * exp(
-lamda * (x - mu)**2 / x / 2 / mu**2)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x * dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu)**2 * dist, (x, 0, oo))) == mu**3 / lamda
assert mysimp(integrate((x - mu)**3 * dist,
(x, 0, oo))) == 3 * mu**5 / lamda**2
# Levi
c = Symbol('c', positive=True)
assert integrate(
sqrt(c / 2 / pi) * exp(-c / 2 / (x - mu)) / (x - mu)**S('3/2'),
(x, mu, oo)) == 1
# higher moments oo
# log-logistic
distn = (beta/alpha)*x**(beta - 1)/alpha**(beta - 1)/ \
(1 + x**beta/alpha**beta)**2
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(integrate(x*distn, (x, 0, oo), conds='none')) == \
pi*alpha/beta/sin(pi/beta)
# (similar comment for conditions applies)
assert simplify(integrate(x**y*distn, (x, 0, oo), conds='none')) == \
pi*alpha**y*y/beta/sin(pi*y/beta)
# weibull
k = Symbol('k', positive=True)
n = Symbol('n', positive=True)
distn = k / lamda * (x / lamda)**(k - 1) * exp(-(x / lamda)**k)
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x**n*distn, (x, 0, oo))) == \
lamda**n*gamma(1 + n/k)
# rice distribution
from sympy import besseli
nu, sigma = symbols('nu sigma', positive=True)
rice = x / sigma**2 * exp(-(x**2 + nu**2) / 2 / sigma**2) * besseli(
0, x * nu / sigma**2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol('mu', real=True)
b = Symbol('b', positive=True)
laplace = exp(-abs(x - mu) / b) / 2 / b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x * laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == \
2*b**2 + mu**2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol('k', positive=True)
assert combsimp(
expand_mul(
integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k),
(x, 0, oo)))) == polygamma(0, k)
def simp_pows(expr):
return expand_mul(simplify(powsimp(expr, force=True)),
deep=True).replace(exp_polar, exp) # XXX ?
def eval(cls, arg):
from sympy.simplify.simplify import signsimp
from sympy import Atom
if hasattr(arg, '_eval_Abs'):
obj = arg._eval_Abs()
if obj is not None:
return obj
if not isinstance(arg, Expr):
raise TypeError("Bad argument type for Abs(): %s" % type(arg))
# handle what we can
arg = signsimp(arg, evaluate=False)
if arg.is_Mul:
known = []
unk = []
for t in Mul.make_args(arg):
tnew = cls(t)
if tnew.func is cls:
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known * unk
if arg is S.NaN:
return S.NaN
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if base.is_real:
if exponent.is_integer:
if exponent.is_even:
return arg
if base is S.NegativeOne:
return S.One
if base.func is cls and exponent is S.NegativeOne:
return arg
return Abs(base)**exponent
if base.is_positive == True:
return base**re(exponent)
return (-base)**re(exponent) * exp(-S.Pi * im(exponent))
if isinstance(arg, exp):
return exp(re(arg.args[0]))
if arg.is_number or isinstance(arg, (cls, Atom)):
if arg.is_zero:
return S.Zero
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -S.ImaginaryUnit * arg
if arg2.is_nonnegative:
return arg2
if arg.is_Add:
if arg.has(S.Infinity, S.NegativeInfinity):
if any(a.is_infinite for a in arg.as_real_imag()):
return S.Infinity
if arg.is_real is None and arg.is_imaginary is None:
if all(a.is_real or a.is_imaginary or (S.ImaginaryUnit *
a).is_real
for a in arg.args):
from sympy import expand_mul
return sqrt(expand_mul(arg * arg.conjugate()))
if arg.is_real is False and arg.is_imaginary is False:
from sympy import expand_mul
return sqrt(expand_mul(arg * arg.conjugate()))
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
def minimal_polynomial(ex, x=None, **args):
"""
Computes the minimal polynomial of an algebraic element.
Parameters
==========
ex : algebraic element expression
x : independent variable of the minimal polynomial
Options
=======
compose : if ``True`` ``_minpoly_compose`` is used, if ``False`` the ``groebner`` algorithm
polys : if ``True`` returns a ``Poly`` object
domain : ground domain
Notes
=====
By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
are computed, then the arithmetic operations on them are performed using the resultant
and factorization.
If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
The default algorithm stalls less frequently.
If no ground domain is given, it will be generated automatically from the expression.
Examples
========
>>> from sympy import minimal_polynomial, sqrt, solve, QQ
>>> from sympy.abc import x, y
>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y
"""
from sympy.polys.polytools import degree
from sympy.polys.domains import FractionField
from sympy.core.basic import preorder_traversal
compose = args.get('compose', True)
polys = args.get('polys', False)
dom = args.get('domain', None)
ex = sympify(ex)
for expr in preorder_traversal(ex):
if expr.is_AlgebraicNumber:
compose = False
break
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not dom:
dom = FractionField(QQ, list(
ex.free_symbols)) if ex.free_symbols else QQ
if hasattr(dom, 'symbols') and x in dom.symbols:
raise GeneratorsError(
"the variable %s is an element of the ground domain %s" % (x, dom))
if compose:
result = _minpoly_compose(ex, x, dom)
result = result.primitive()[1]
c = result.coeff(x**degree(result, x))
if c.is_negative:
result = expand_mul(-result)
return cls(result, x, field=True) if polys else result.collect(x)
if not dom.is_QQ:
raise NotImplementedError("groebner method only works for QQ")
result = _minpoly_groebner(ex, x, cls)
return cls(result, x, field=True) if polys else result.collect(x)
>>> _rewrite_sin((2*pi, 0), s, 0, S(1)/2)
(gamma(2*s), gamma(-2*s + 1), pi)
>>> _rewrite_sin((2*pi, 0), s, S(1)/2, 1)
(gamma(2*s - 1), gamma(-2*s + 2), -pi)
"""
# (This is a separate function because it is moderately complicated,
# and I want to doctest it.)
# We want to use pi/sin(pi*x) = gamma(x)*gamma(1-x).
# But there is one comlication: the gamma functions determine the
# inegration contour in the definition of the G-function. Usually
# it would not matter if this is slightly shifted, unless this way
# we create an undefined function!
# So we try to write this in such a way that the gammas are
# eminently on the right side of the strip.
from sympy import expand_mul, pi, ceiling, gamma, re
m = expand_mul(m / pi)
n = expand_mul(n / pi)
r = ceiling(-m * a -
n.as_real_imag()[0]) # Don't use re(n), does not expand
return gamma(m * s + n + r), gamma(1 - n - r - m * s), (-1)**r * pi
class MellinTransformStripError(ValueError):
"""
Exception raised by _rewrite_gamma. Mainly for internal use.
"""
pass
def _rewrite_gamma(f, s, a, b):
"""
def minimal_polynomial(ex, x=None, **args):
"""
Computes the minimal polynomial of an algebraic number.
Parameters
==========
ex : algebraic number expression
x : indipendent variable of the minimal polynomial
Options
=======
compose : if ``True`` _minpoly1`` is used, else the ``groebner`` algorithm
polys : if ``True`` returns a ``Poly`` object
Notes
=====
By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
are computed, then the arithmetic operations on them are performed using the resultant
and factorization.
If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
The default algorithm stalls less frequently.
Examples
========
>>> from sympy import minimal_polynomial, sqrt, solve
>>> from sympy.abc import x
>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
"""
from sympy.polys.polytools import degree
from sympy.core.function import expand_multinomial
from sympy.core.basic import preorder_traversal
compose = args.get('compose', True)
polys = args.get('polys', False)
ex = sympify(ex)
for expr in preorder_traversal(ex):
if expr.is_AlgebraicNumber:
compose = False
break
if ex.is_AlgebraicNumber:
compose = False
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if compose:
result = _minpoly1(ex, x)
result = result.primitive()[1]
c = result.coeff(x**degree(result, x))
if c < 0:
result = expand_mul(-result)
c = -c
return cls(result, x, field=True) if polys else result
generator = numbered_symbols('a', cls=Dummy)
mapping, symbols, replace = {}, {}, []
def update_mapping(ex, exp, base=None):
a = generator.next()
symbols[ex] = a
if base is not None:
mapping[ex] = a**exp + base
else:
mapping[ex] = exp.as_expr(a)
return a
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0 and ex.base.is_Add:
coeff, terms = ex.base.as_coeff_add()
elt, _ = primitive_element(terms, polys=True)
alg = ex.base - coeff
# XXX: turn this into eval()
inverse = invert(elt.gen + coeff, elt).as_expr()
base = inverse.subs(elt.gen, alg).expand()
if ex.exp == -1:
return bottom_up_scan(base)
else:
ex = base**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
def simpler_inverse(ex):
"""
Returns True if it is more likely that the minimal polynomial
algorithm works better with the inverse
"""
if ex.is_Pow:
if (1 / ex.exp).is_integer and ex.exp < 0:
if ex.base.is_Add:
return True
if ex.is_Mul:
hit = True
a = []
for p in ex.args:
if p.is_Add:
return False
if p.is_Pow:
if p.base.is_Add and p.exp > 0:
return False
if hit:
return True
return False
inverted = False
ex = expand_multinomial(ex)
if ex.is_AlgebraicNumber:
if not polys:
return ex.minpoly.as_expr(x)
else:
return ex.minpoly.replace(x)
elif ex.is_Rational:
result = ex.q * x - ex.p
else:
inverted = simpler_inverse(ex)
if inverted:
ex = ex**-1
res = None
if ex.is_Pow and (1 / ex.exp).is_Integer:
n = 1 / ex.exp
res = _minimal_polynomial_sq(ex.base, n, x)
elif _is_sum_surds(ex):
res = _minimal_polynomial_sq(ex, S.One, x)
if res is not None:
result = res
if res is None:
bus = bottom_up_scan(ex)
F = [x - bus] + mapping.values()
G = groebner(F, symbols.values() + [x], order='lex')
_, factors = factor_list(G[-1])
# by construction G[-1] has root `ex`
result = _choose_factor(factors, x, ex)
if inverted:
result = _invertx(result, x)
if result.coeff(x**degree(result, x)) < 0:
result = expand_mul(-result)
if polys:
return cls(result, x, field=True)
else:
return result
def solve_linear(lhs, rhs=0, x=[], exclude=[]):
""" Return a tuple containing derived from f = lhs - rhs that is either:
(numerator, denominator) of f; if this comes back as (0, 1) it means
that f was actually zero even though it may have had symbols:
e.g. y*cos(x)**2 + y*sin(x)**2 - y = y*(0) = 0 If the numerator
is not zero then the function is guaranteed not to be zero.
or
(symbol, solution) where symbol appears linearly in the numerator of f,
is in x (if given) and is not in exclude (if given).
No simplification is done to f other than and mul=True expansion, so
the solution will correspond strictly to a unique solution.
Examples:
>>> from sympy.solvers.solvers import solve_linear
>>> from sympy.abc import x, y, z
These are linear in x and 1/x:
>>> solve_linear(x + y**2)
(x, -y**2)
>>> solve_linear(1/x - y**2)
(x, y**(-2))
When not linear in x or y then the numerator and denominator are returned.
>>> solve_linear(x**2/y**2 - 3)
(x**2 - 3*y**2, y**2)
If x is allowed to cancel, then this appears linear, but this sort of
cancellation is not done so the solultion will always satisfy the original
expression without causing a division by zero error.
>>> solve_linear(x**2*(1/x - z**2/x))
(x**2*(x - x*z**2), x**2)
You can give a list of what you prefer for x candidates:
>>> solve_linear(x + y + z, x=[y])
(y, -x - z)
You can also indicate what variables you don't want to consider:
>>> solve_linear(x + y + z, exclude=[x, z])
(y, -x - z)
If only x was excluded then a solution for y or z might be obtained.
"""
from sympy import expand_mul, Equality
if isinstance(lhs, Equality):
rhs += lhs.rhs
lhs = lhs.lhs
n, d = (lhs - rhs).as_numer_denom()
ex = expand_mul(n)
if not ex:
return ex, d
exclude = set(exclude)
syms = ex.free_symbols
if not x:
x = syms
else:
x = syms.intersection(x)
x = x.difference(exclude)
for xi in x:
dn = n.diff(xi)
# if not dn then this is a pseudo-function of xi
if dn and not dn.has(xi):
return xi, -(n.subs(xi, 0))/dn
return n, d
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy import expand_mul, Order, Piecewise, piecewise_fold, log
old = self
if old.has(Piecewise):
old = piecewise_fold(old)
# This expansion is the last part of expand_log. expand_log also calls
# expand_mul with factor=True, which would be more expensive
if any(isinstance(a, log) for a in self.args):
logflags = dict(deep=True,
log=True,
mul=False,
power_exp=False,
power_base=False,
multinomial=False,
basic=False,
force=False,
factor=False)
old = old.expand(**logflags)
expr = expand_mul(old)
if not expr.is_Add:
return expr.as_leading_term(x, logx=logx, cdir=cdir)
infinite = [t for t in expr.args if t.is_infinite]
leading_terms = [
t.as_leading_term(x, logx=logx, cdir=cdir) for t in expr.args
]
min, new_expr = Order(0), 0
try:
for term in leading_terms:
order = Order(term, x)
if not min or order not in min:
min = order
new_expr = term
elif min in order:
new_expr += term
except TypeError:
return expr
is_zero = new_expr.is_zero
if is_zero is None:
new_expr = new_expr.trigsimp().cancel()
is_zero = new_expr.is_zero
if is_zero is True:
# simple leading term analysis gave us cancelled terms but we have to send
# back a term, so compute the leading term (via series)
n0 = min.getn()
res = Order(1)
incr = S.One
while res.is_Order:
res = old._eval_nseries(
x, n=n0 + incr, logx=None,
cdir=cdir).cancel().powsimp().trigsimp()
incr *= 2
return res.as_leading_term(x, logx=logx, cdir=cdir)
elif new_expr is S.NaN:
return old.func._from_args(infinite)
else:
return new_expr
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y, z
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp
mu1, mu2 = symbols('mu1 mu2', real=True, finite=True, bounded=True)
sigma1, sigma2 = symbols('sigma1 sigma2', real=True, finite=True,
bounded=True, positive=True)
rate = Symbol('lambda', real=True, positive=True, bounded=True)
def normal(x, mu, sigma):
return 1/sqrt(2*pi*sigma**2)*exp(-(x-mu)**2/2/sigma**2)
def exponential(x, rate):
return rate*exp(-rate*x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1
assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**2 + sigma1**2
assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**3 + 3*mu1*sigma1**2
assert integrate(normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1
assert integrate(y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu2
assert integrate(x*y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1*mu2
assert integrate((x+y+1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == 1 + mu1 + mu2
assert integrate((x+y-1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == -1 + mu1 + mu2
i = integrate(x**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1**2 + sigma1**2
assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
sigma2**2 + mu2**2
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == 1/rate
assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) \
== 2/rate**2
def E(expr):
res1 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo), meijerg=True)
res2 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo), meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x*y) == mu1/rate
assert E(x*y**2) == mu1**2/rate + sigma1**2/rate
assert simplify(E((x+y+1)**2) - E(x+y+1)**2) == (rate**2*sigma1**2 + 1)/rate**2
assert simplify(E((x+y-1)**2) - E(x+y-1)**2) == (rate**2*sigma1**2 + 1)/rate**2
assert simplify(E((x+y)**2) - E(x+y)**2) == (rate**2*sigma1**2 + 1)/rate**2
# Beta' distribution
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha-1)*(1+x)**(-alpha - beta)*gamma(alpha+beta) \
/gamma(alpha)/gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x*betadist, (x, 0, oo), meijerg=True, conds='separate')
assert (combsimp(i[0]), i[1]) == (alpha/(beta - 1), 1 < beta)
j = integrate(x**2*betadist, (x, 0, oo), meijerg=True, conds='separate')
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
/(beta - 2)/(beta - 1)**2
# Chi distribution
k = Symbol('k', integer=True, positive=True)
chi = 2**(1-k/2)*x**(k-1)*exp(-x**2/2)/gamma(k/2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
assert simplify(integrate(x**2*chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2**(-k/2)/gamma(k/2)*x**(k/2-1)*exp(-x/2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
k*(k + 2)
assert combsimp(integrate(((x-k)/sqrt(2*k))**3*chisquared, (x, 0, oo),
meijerg=True)) == 2*sqrt(2)/sqrt(k)
# Dagum distribution
a, b, p = symbols('a b p', positive=True)
# XXX (x/b)**a does not work
dagum = a*p/x*(x/b)**(a*p)/(1 + x**a/b**a)**(p+1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
assert simplify(integrate(x*dagum, (x, 0, oo), meijerg=True, conds='none')) \
== b*gamma(1 - 1/a)*gamma(p + 1/a)/gamma(p)
assert simplify(integrate(x**2*dagum, (x, 0, oo), meijerg=True, conds='none')) \
== b**2*gamma(1 - 2/a)*gamma(p + 2/a)/gamma(p)
# F-distribution
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1+d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x*f, (x, 0, oo), meijerg=True, conds='none')) == \
d2/(d2 - 2)
assert simplify(integrate(x**2*f, (x, 0, oo), meijerg=True, conds='none')) == \
d2**2*(d1 + 2)/d1/(d2 - 4)/(d2 - 2)
# TODO gamma, inverse gaussian, Levi, log-logistic, rayleigh, weibull
# rice distribution
from sympy import besseli
nu, sigma = symbols('nu sigma', positive=True)
rice = x/sigma**2*exp(-(x**2+ nu**2)/2/sigma**2)*besseli(0, x*nu/sigma**2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol('mu', real=True)
b = Symbol('b', positive=True)
laplace = exp(-abs(x-mu)/b)/2/b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x*laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == 2*b**2 + mu**2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol('k', positive=True)
assert combsimp(expand_mul(integrate(log(x) * x**(k-1) * exp(-x) / gamma(k),
(x, 0, oo)))) == polygamma(0, k)
def minimal_polynomial(ex, x=None, compose=True, polys=False, domain=None):
"""
Computes the minimal polynomial of an algebraic element.
Parameters
==========
ex : Expr
Element or expression whose minimal polynomial is to be calculated.
x : Symbol, optional
Independent variable of the minimal polynomial
compose : boolean, optional (default=True)
Method to use for computing minimal polynomial. If ``compose=True``
(default) then ``_minpoly_compose`` is used, if ``compose=False`` then
groebner bases are used.
polys : boolean, optional (default=False)
If ``True`` returns a ``Poly`` object else an ``Expr`` object.
domain : Domain, optional
Ground domain
Notes
=====
By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
are computed, then the arithmetic operations on them are performed using the resultant
and factorization.
If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
The default algorithm stalls less frequently.
If no ground domain is given, it will be generated automatically from the expression.
Examples
========
>>> from sympy import minimal_polynomial, sqrt, solve, QQ
>>> from sympy.abc import x, y
>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y
"""
from sympy.polys.polytools import degree
from sympy.polys.domains import FractionField
from sympy.core.basic import preorder_traversal
ex = sympify(ex)
if ex.is_number:
# not sure if it's always needed but try it for numbers (issue 8354)
ex = _mexpand(ex, recursive=True)
for expr in preorder_traversal(ex):
if expr.is_AlgebraicNumber:
compose = False
break
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not domain:
if ex.free_symbols:
domain = FractionField(QQ, list(ex.free_symbols))
else:
domain = QQ
if hasattr(domain, 'symbols') and x in domain.symbols:
raise GeneratorsError("the variable %s is an element of the ground "
"domain %s" % (x, domain))
if compose:
result = _minpoly_compose(ex, x, domain)
result = result.primitive()[1]
c = result.coeff(x**degree(result, x))
if c.is_negative:
result = expand_mul(-result)
return cls(result, x, field=True) if polys else result.collect(x)
if not domain.is_QQ:
raise NotImplementedError("groebner method only works for QQ")
result = _minpoly_groebner(ex, x, cls)
return cls(result, x, field=True) if polys else result.collect(x)
def _eval_Abs(self):
from sympy import expand_mul
return sqrt( expand_mul(self * self.conjugate()) )
def test_asech():
x = Symbol('x')
assert unchanged(asech, -x)
# values at fixed points
assert asech(1) == 0
assert asech(-1) == pi * I
assert asech(0) is oo
assert asech(2) == I * pi / 3
assert asech(-2) == 2 * I * pi / 3
assert asech(nan) is nan
# at infinites
assert asech(oo) == I * pi / 2
assert asech(-oo) == I * pi / 2
assert asech(zoo) == I * AccumBounds(-pi / 2, pi / 2)
assert asech(I) == log(1 + sqrt(2)) - I * pi / 2
assert asech(-I) == log(1 + sqrt(2)) + I * pi / 2
assert asech(sqrt(2) - sqrt(6)) == 11 * I * pi / 12
assert asech(sqrt(2 - 2 / sqrt(5))) == I * pi / 10
assert asech(-sqrt(2 - 2 / sqrt(5))) == 9 * I * pi / 10
assert asech(2 / sqrt(2 + sqrt(2))) == I * pi / 8
assert asech(-2 / sqrt(2 + sqrt(2))) == 7 * I * pi / 8
assert asech(sqrt(5) - 1) == I * pi / 5
assert asech(1 - sqrt(5)) == 4 * I * pi / 5
assert asech(-sqrt(2 * (2 + sqrt(2)))) == 5 * I * pi / 8
# properties
# asech(x) == acosh(1/x)
assert asech(sqrt(2)) == acosh(1 / sqrt(2))
assert asech(2 / sqrt(3)) == acosh(sqrt(3) / 2)
assert asech(2 / sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2)) / 2)
assert asech(2) == acosh(S.Half)
# asech(x) == I*acos(1/x)
# (Note: the exact formula is asech(x) == +/- I*acos(1/x))
assert asech(-sqrt(2)) == I * acos(-1 / sqrt(2))
assert asech(-2 / sqrt(3)) == I * acos(-sqrt(3) / 2)
assert asech(-S(2)) == I * acos(Rational(-1, 2))
assert asech(-2 / sqrt(2)) == I * acos(-sqrt(2) / 2)
# sech(asech(x)) / x == 1
assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) /
(sqrt(6) - sqrt(2))) == 1
assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) /
(sqrt(6) + sqrt(2))) == 1
assert (sech(asech(sqrt(2 + 2 / sqrt(5)))) /
(sqrt(2 + 2 / sqrt(5)))).simplify() == 1
assert (sech(asech(-sqrt(2 + 2 / sqrt(5)))) /
(-sqrt(2 + 2 / sqrt(5)))).simplify() == 1
assert (sech(asech(sqrt(2 * (2 + sqrt(2))))) /
(sqrt(2 * (2 + sqrt(2))))).simplify() == 1
assert expand_mul(sech(asech(1 + sqrt(5))) / (1 + sqrt(5))) == 1
assert expand_mul(sech(asech(-1 - sqrt(5))) / (-1 - sqrt(5))) == 1
assert expand_mul(sech(asech(-sqrt(6) - sqrt(2))) /
(-sqrt(6) - sqrt(2))) == 1
# numerical evaluation
assert str(asech(5 * I).n(6)) == '0.19869 - 1.5708*I'
assert str(asech(-5 * I).n(6)) == '0.19869 + 1.5708*I'
