def test_to_hyper():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper()
q = 3 * hyper([], [], 2*x)
assert p == q
p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand()
q = 2*x**3 + 6*x**2 + 6*x + 2
assert p == q
p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper()
q = -x**2*hyper((2, 2, 1), (2, 3), -x)/2 + x
assert p == q
p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper()
q = 2*x*hyper((1/2,), (3/2,), -x**2)/sqrt(pi)
assert p == q
p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper())
q = erfc(x)
assert p.rewrite(erfc) == q
p = hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2,
x, 0, [0, S(1)/2]).to_hyper())
q = besselj(1, x)
assert p == q
p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper())
q = besselj(0, x)
assert p == q
def test_mellin_transform2():
MT = mellin_transform
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x)/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)/(x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
def test_lookup_table():
from random import uniform, randrange
from sympy import Add
from sympy.integrals.meijerint import z as z_dummy
table = {}
_create_lookup_table(table)
for _, l in sorted(table.items()):
for formula, terms, cond, hint in sorted(l, key=default_sort_key):
subs = {}
for a in list(formula.free_symbols) + [z_dummy]:
if hasattr(a, "properties") and a.properties:
# these Wilds match positive integers
subs[a] = randrange(1, 10)
else:
subs[a] = uniform(1.5, 2.0)
if not isinstance(terms, list):
terms = terms(subs)
# First test that hyperexpand can do this.
expanded = [hyperexpand(g) for (_, g) in terms]
assert all(x.is_Piecewise or not x.has(meijerg) for x in expanded)
# Now test that the meijer g-function is indeed as advertised.
expanded = Add(*[f * x for (f, x) in terms])
a, b = formula.n(subs=subs), expanded.n(subs=subs)
r = min(abs(a), abs(b))
if r < 1:
assert abs(a - b).n() <= 1e-10
else:
assert (abs(a - b) / r).n() <= 1e-10
def _eval_expand_func(self, **hints):
from sympy import gamma, hyperexpand
if len(self.ap) == 2 and len(self.bq) == 1 and self.argument == 1:
a, b = self.ap
c = self.bq[0]
return gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b)
return hyperexpand(self)
def test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand()
q = 1/(x**2 - 2*x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2/x).integrate((x, 0, x))
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1+x**2)).to_expr()
q = C_2*log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1+x**2)).diff().to_expr()
q = C_0*x/(x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1+x**2)
p = expr_to_holonomic((2*x**2 + 1)**(S(2)/3)).to_expr()
assert p == (2*x**2 + 1)**(S(2)/3)
p = expr_to_holonomic(sqrt(-x**2+2*x)).to_expr()
assert p == sqrt(x)*sqrt(-x + 2)
p = expr_to_holonomic((-2*x**3+7*x)**(S(2)/3)).to_expr()
q = x**(S(2)/3)*(-2*x**2 + 7)**(S(2)/3)
assert p == q
p = from_hyper(hyper((-2, -3), (S(1)/2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1)/2, ), x))
D_0 = Symbol('D_0')
C_0 = Symbol('C_0')
assert (p.to_expr().subs({C_0:1, D_0:0}) - s).simplify() == 0
p.y0 = {0: [1], S(1)/2: [0]}
assert p.to_expr() == s
assert expr_to_holonomic(x**5).to_expr() == x**5
assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \
2*x**3-3*x**2
a = symbols("a")
p = (expr_to_holonomic(1.4*x)*expr_to_holonomic(a*x, x)).to_expr()
q = 1.4*a*x**2
assert p == q
p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(a*x, x)).to_expr()
q = x*(a + 1.4)
assert p == q
p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(x)).to_expr()
assert p == 2.4*x
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 test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand()
q = 1 / (x**2 - 2 * x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2 / x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2 / x).integrate((x, 0, x))
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1 + x**2)).to_expr()
q = C_2 * log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1 + x**2)).diff().to_expr()
q = C_0 * x / (x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1 + x**2)
p = expr_to_holonomic((2 * x**2 + 1)**(S(2) / 3)).to_expr()
assert p == (2 * x**2 + 1)**(S(2) / 3)
p = expr_to_holonomic(sqrt(-x**2 + 2 * x)).to_expr()
assert p == sqrt(x) * sqrt(-x + 2)
p = expr_to_holonomic((-2 * x**3 + 7 * x)**(S(2) / 3)).to_expr()
q = x**(S(2) / 3) * (-2 * x**2 + 7)**(S(2) / 3)
assert p == q
p = from_hyper(hyper((-2, -3), (S(1) / 2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1) / 2, ), x))
D_0 = Symbol('D_0')
C_0 = Symbol('C_0')
assert (p.to_expr().subs({C_0: 1, D_0: 0}) - s).simplify() == 0
p.y0 = {0: [1], S(1) / 2: [0]}
assert p.to_expr() == s
assert expr_to_holonomic(x**5).to_expr() == x**5
assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \
2*x**3-3*x**2
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 test_laplace_transform():
LT = laplace_transform
a, b, c, = symbols('a b c', positive=True)
t = symbols('t')
# 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(t**a, t, s) == (s**(-a - 1) * gamma(a + 1), 0, True)
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(1 - exp(-a * t), t, s) == (a / (s * (a + s)), 0, True)
assert LT((exp(2*t)-1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
assert LT(exp(t), t, s)[0:2] == (1 / (s - 1), 1)
assert LT(exp(2 * t), t, s)[0:2] == (1 / (s - 2), 2)
assert LT(exp(a * t), t, s)[0:2] == (1 / (s - a), a)
lt = LT(log(t / a), t, s)
assert lt[1:] == (0, True)
# TODO hyperexpand is not clever enough to recognise this on its own
assert hyperexpand(lt[0],
allow_hyper=True) == (-log(a * s) - EulerGamma) / s
assert LT(erf(t), t, s) == ((-erf(s / 2) + 1) * exp(s**2 / 4) / s, 0, True)
assert LT(sin(a * t), t, s) == (a / (a**2 + s**2), 0, True)
assert LT(cos(a * t), t, s) == (s / (a**2 + s**2), 0, True)
# TODO would be nice to have these come out better
assert LT(exp(-a * t) * sin(b * t), t,
s) == (1 / b / (1 + (a + s)**2 / b**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
(sqrt((a + s)**2)/b**2/(1 + (a + s)**2/b**2), -a, True)
# TODO sinh, cosh have delicate cancellation
# TODO conditions are a mess
assert LT(besselj(0, t), t, s, noconds=True) == 1 / sqrt(1 + s**2)
assert LT(besselj(1, t), t, s, noconds=True) == 1 - 1 / sqrt(1 + 1 / s**2)
def test_laplace_transform():
LT = laplace_transform
a, b, c, = symbols('a b c', positive=True)
t = symbols('t')
# 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(t**a, t, s) == (s**(-a - 1)*gamma(a + 1), 0, True)
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(1 - exp(-a*t), t, s) == (a/(s*(a + s)), 0, True)
assert LT((exp(2*t)-1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
assert LT(exp(t), t, s)[0:2] == (1/(s-1), 1)
assert LT(exp(2*t), t, s)[0:2] == (1/(s-2), 2)
assert LT(exp(a*t), t, s)[0:2] == (1/(s-a), a)
lt = LT(log(t/a), t, s)
assert lt[1:] == (0, True)
# TODO hyperexpand is not clever enough to recognise this on its own
assert hyperexpand(lt[0], allow_hyper=True) == (-log(a*s) - EulerGamma)/s
assert LT(erf(t), t, s) == ((-erf(s/2) + 1)*exp(s**2/4)/s, 0, True)
assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
# TODO would be nice to have these come out better
assert LT(exp(-a*t)*sin(b*t), t, s) == (1/b/(1 + (a + s)**2/b**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
(sqrt((a + s)**2)/b**2/(1 + (a + s)**2/b**2), -a, True)
# TODO sinh, cosh have delicate cancellation
# TODO conditions are a mess
assert LT(besselj(0, t), t, s, noconds=True) == 1/sqrt(1 + s**2)
assert LT(besselj(1, t), t, s, noconds=True) == 1 - 1/sqrt(1 + 1/s**2)
def _eval_simplify(self, ratio, measure, rational, inverse):
from sympy.simplify.hyperexpand import hyperexpand
return hyperexpand(self)
def test_J14():
p = hyper([S(1)/2, S(1)/2], [S(3)/2], z**2)
assert hyperexpand(p) == asin(z)/z
def test_mellin_transform():
from sympy 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, -re(beta) + 1), re(-beta) < 0)
assert MT((1 + x)**(-rho), x, s) == \
(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
# TODO also the conditions should be simplified
assert MT(abs(1 - x)**(-rho), x,
s) == (2 * sin(pi * rho / 2) * gamma(1 - rho) *
cos(pi * (rho / 2 - s)) * gamma(s) * gamma(rho - s) / pi,
(0, re(rho)), And(re(rho) - 1 < 0,
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), True)
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(-re(a), 0), Min(1 - re(a), 1)), 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(1)/2), 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)
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x) / (x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2 / (x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x) / (x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
def test_mellin_transform():
from sympy import Max, Min, Ne
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, -re(beta) + 1), re(-beta) < 0)
assert MT((1 + x)**(-rho), x, s) == \
(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
# TODO also the conditions should be simplified
assert MT(abs(1 - x)**(-rho), x, s) == (
cos(pi*(rho/2 - s))*gamma(s)*gamma(rho - s)/(cos(pi*rho/2)*gamma(rho)),
(0, re(rho)), And(re(rho) - 1 < 0, 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), True)
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(-re(a), 0), Min(1 - re(a), 1)), 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(1)/2), 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)
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x)/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)/(x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
def test_to_meijerg():
x = symbols('x')
assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
assert hyperexpand(expr_to_holonomic(
log(x)).to_meijerg()).simplify() == log(x)
assert expr_to_holonomic(4 * x**2 / 3 + 7).to_meijerg() == 4 * x**2 / 3 + 7
assert hyperexpand(
expr_to_holonomic(besselj(2, x),
lenics=3).to_meijerg()) == besselj(2, x)
p = hyper((Rational(-1, 2), -3), (), x)
assert from_hyper(p).to_meijerg() == hyperexpand(p)
p = hyper((S.One, S(3)), (S(2), ), x)
assert (hyperexpand(from_hyper(p).to_meijerg()) -
hyperexpand(p)).expand() == 0
p = from_hyper(hyper((-2, -3), (S.Half, ), x))
s = hyperexpand(hyper((-2, -3), (S.Half, ), x))
C_0 = Symbol('C_0')
C_1 = Symbol('C_1')
D_0 = Symbol('D_0')
assert (hyperexpand(p.to_meijerg()).subs({
C_0: 1,
D_0: 0
}) - s).simplify() == 0
p.y0 = {0: [1], S.Half: [0]}
assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
p = expr_to_holonomic(besselj(S.Half, x), initcond=False)
assert (
p.to_expr() -
(D_0 * sin(x) + C_0 * cos(x) + C_1 * sin(x)) / sqrt(x)).simplify() == 0
p = expr_to_holonomic(
besselj(S.Half, x),
y0={Rational(-1, 2): [sqrt(2) / sqrt(pi),
sqrt(2) / sqrt(pi)]})
assert (p.to_expr() - besselj(S.Half, x) -
besselj(Rational(-1, 2), x)).simplify() == 0
def test_mellin_transform():
from sympy 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, -re(beta) + 1), re(-beta) < 0)
assert MT(
(1 + x)**(-rho), x,
s) == (gamma(s) * gamma(rho - s) / gamma(rho), (0, re(rho)), True)
# TODO also the conditions should be simplified
assert MT(abs(1-x)**(-rho), x, s) == \
(cos(pi*rho/2 - pi*s)*gamma(s)*gamma(rho-s)/(cos(pi*rho/2)*gamma(rho)),\
(0, re(rho)), And(re(rho) - 1 < 0, 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), True)
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(-re(a), 0), Min(1 - re(a), 1)), 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(1)/2), True)
# TODO does not work with bneg, argument wrong. Needs changes to matching.
#assert MT(expr.subs(b, -bpos), x, s) == \
# ((-1)**(a+1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s) \
# *gamma(1 - a - 2*s)/gamma(1 - s),
# (-re(a), -re(a)/2 + S(1)/2), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, bpos), x, s) == \
(-2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
(0, -re(a)/2), True)
#assert MT(expr.subs(b, -bpos), x, s) == \
# (2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*s)*gamma(a + s)/gamma(-s + 1),
# (-re(a), -re(a)/2), True)
# Test exponent 1:
#assert MT(expr.subs({b: -bpos, a:1}), x, s) == \
# (-bpos**(2*s + 1)*gamma(s)*gamma(-s - S(1)/2)/(2*sqrt(pi)),
# (-1, -S(1)/2), 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)
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x) / (x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2 / (x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x) / (x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
# 8.4.19
assert MT(besselj(a, 2*sqrt(x)), x, s) == \
(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, S(3)/4), True)
assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(S(1)/2 - 2*s)*gamma((a+1)/2 + s) \
/ (gamma(1 - s- a/2)*gamma(1 + a - 2*s)),
(-(re(a) + 1)/2, S(1)/4), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
/ (gamma(S(1)/2 - s - a/2)*gamma(a - 2*s + 1)),
(-re(a)/2, S(1)/4), True)
assert MT(besselj(a, sqrt(x))**2, x, s) == \
(gamma(a + s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, S(1)/2), True)
# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(gamma(1-s)*gamma(a + s - S(1)/2)
/ (sqrt(pi)*gamma(S(3)/2 - s)*gamma(a - s + S(1)/2)),
(S(1)/2 - re(a), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
(4**s*gamma(1 - 2*s)*gamma((a+b)/2 + s)
/ (gamma(1 - s + (b-a)/2)*gamma(1 - s + (a-b)/2)
*gamma( 1 - s + (a+b)/2)),
(-(re(a) + re(b))/2, S(1)/2), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), S(1)/2), True)
# Section 8.4.20
assert MT(bessely(a, 2*sqrt(x)), x, s) == \
(-cos(pi*a/2 - pi*s)*gamma(s - a/2)*gamma(s + a/2)/pi,
(Max(-re(a)/2, re(a)/2), S(3)/4), True)
assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*sin(pi*a/2 - pi*s)*gamma(S(1)/2 - 2*s)
* gamma((1-a)/2 + s)*gamma((1+a)/2 + s)
/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
(Max(-(re(a) + 1)/2, (re(a) - 1)/2), S(1)/4), True)
assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*cos(pi*a/2 - pi*s)*gamma(s - a/2)*gamma(s + a/2)*gamma(S(1)/2 - 2*s)
/ (sqrt(pi)*gamma(S(1)/2 - s - a/2)*gamma(S(1)/2 - s + a/2)),
(Max(-re(a)/2, re(a)/2), S(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(1)/2 - s)
/ (pi**S('3/2')*gamma(1 + a - s)),
(Max(-re(a), 0), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
(-4**s*cos(pi*a/2 - pi*b/2 + pi*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(1)/2), True)
assert MT(bessely(a, sqrt(x))**2, x, s) == \
((2*cos(pi*a - pi*s)*gamma(s)*gamma(-a + s)*gamma(-s + 1)*gamma(a - s + 1) \
+ pi*gamma(-s + S(1)/2)*gamma(s + S(1)/2)) \
*gamma(a + s)/(pi**(S(3)/2)*gamma(-s + 1)*gamma(s + S(1)/2)*gamma(a - s + 1)),
(Max(-re(a), 0, re(a)), S(1)/2), True)
# TODO bessely(a, sqrt(x))*bessely(b, sqrt(x)) is a mess
# (no matter what way you look at it ...)
# 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)
# TODO this result needs expansion of F(a, b; c; 1) using gauss-summation
assert MT(exp(-x/2)*besselk(a, x/2), x, s)[1:] == \
((Max(-re(a), re(a)), oo), True)
def _eval_simplify(self, **kwargs):
from sympy.simplify.hyperexpand import hyperexpand
return hyperexpand(self)
def _eval_expand_func(self, **hints):
from sympy import hyperexpand
return hyperexpand(self)
def test_mellin_transform():
from sympy 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, -re(beta) + 1), re(-beta) < 0)
assert MT((1+x)**(-rho), x, s) == (gamma(s)*gamma(rho-s)/gamma(rho),
(0, re(rho)), True)
# TODO also the conditions should be simplified
assert MT(abs(1-x)**(-rho), x, s) == \
(cos(pi*rho/2 - pi*s)*gamma(s)*gamma(rho-s)/(cos(pi*rho/2)*gamma(rho)),\
(0, re(rho)), And(re(rho) - 1 < 0, 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), True)
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(-re(a), 0), Min(1 - re(a), 1)), 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(1)/2), True)
# TODO does not work with bneg, argument wrong. Needs changes to matching.
#assert MT(expr.subs(b, -bpos), x, s) == \
# ((-1)**(a+1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s) \
# *gamma(1 - a - 2*s)/gamma(1 - s),
# (-re(a), -re(a)/2 + S(1)/2), True)
expr = (sqrt(x+b**2)+b)**a
assert MT(expr.subs(b, bpos), x, s) == \
(-2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
(0, -re(a)/2), True)
#assert MT(expr.subs(b, -bpos), x, s) == \
# (2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*s)*gamma(a + s)/gamma(-s + 1),
# (-re(a), -re(a)/2), True)
# Test exponent 1:
#assert MT(expr.subs({b: -bpos, a:1}), x, s) == \
# (-bpos**(2*s + 1)*gamma(s)*gamma(-s - S(1)/2)/(2*sqrt(pi)),
# (-1, -S(1)/2), 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)
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x)/(x+1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2/(x+1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)/(x+1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
# 8.4.19
assert MT(besselj(a, 2*sqrt(x)), x, s) == \
(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, S(3)/4), True)
assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(S(1)/2 - 2*s)*gamma((a+1)/2 + s) \
/ (gamma(1 - s- a/2)*gamma(1 + a - 2*s)),
(-(re(a) + 1)/2, S(1)/4), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
/ (gamma(S(1)/2 - s - a/2)*gamma(a - 2*s + 1)),
(-re(a)/2, S(1)/4), True)
assert MT(besselj(a, sqrt(x))**2, x, s) == \
(gamma(a + s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, S(1)/2), True)
# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(gamma(1-s)*gamma(a + s - S(1)/2)
/ (sqrt(pi)*gamma(S(3)/2 - s)*gamma(a - s + S(1)/2)),
(S(1)/2 - re(a), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
(4**s*gamma(1 - 2*s)*gamma((a+b)/2 + s)
/ (gamma(1 - s + (b-a)/2)*gamma(1 - s + (a-b)/2)
*gamma( 1 - s + (a+b)/2)),
(-(re(a) + re(b))/2, S(1)/2), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), S(1)/2), True)
# Section 8.4.20
assert MT(bessely(a, 2*sqrt(x)), x, s) == \
(-cos(pi*a/2 - pi*s)*gamma(s - a/2)*gamma(s + a/2)/pi,
(Max(-re(a)/2, re(a)/2), S(3)/4), True)
assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*sin(pi*a/2 - pi*s)*gamma(S(1)/2 - 2*s)
* gamma((1-a)/2 + s)*gamma((1+a)/2 + s)
/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
(Max(-(re(a) + 1)/2, (re(a) - 1)/2), S(1)/4), True)
assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*cos(pi*a/2 - pi*s)*gamma(s - a/2)*gamma(s + a/2)*gamma(S(1)/2 - 2*s)
/ (sqrt(pi)*gamma(S(1)/2 - s - a/2)*gamma(S(1)/2 - s + a/2)),
(Max(-re(a)/2, re(a)/2), S(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(1)/2 - s)
/ (pi**S('3/2')*gamma(1 + a - s)),
(Max(-re(a), 0), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
(-4**s*cos(pi*a/2 - pi*b/2 + pi*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(1)/2), True)
assert MT(bessely(a, sqrt(x))**2, x, s) == \
((2*cos(pi*a - pi*s)*gamma(s)*gamma(-a + s)*gamma(-s + 1)*gamma(a - s + 1) \
+ pi*gamma(-s + S(1)/2)*gamma(s + S(1)/2)) \
*gamma(a + s)/(pi**(S(3)/2)*gamma(-s + 1)*gamma(s + S(1)/2)*gamma(a - s + 1)),
(Max(-re(a), 0, re(a)), S(1)/2), True)
# TODO bessely(a, sqrt(x))*bessely(b, sqrt(x)) is a mess
# (no matter what way you look at it ...)
# 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)
# TODO this result needs expansion of F(a, b; c; 1) using gauss-summation
assert MT(exp(-x/2)*besselk(a, x/2), x, s)[1:] == \
((Max(-re(a), re(a)), oo), True)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy import hyperexpand
return hyperexpand(self).as_leading_term(x, logx=logx, cdir=cdir)
def test_J14():
p = hyper([S(1)/2, S(1)/2], [S(3)/2], z**2)
assert hyperexpand(p) == asin(z)/z
def _eval_simplify(self, ratio, measure):
from sympy.simplify.hyperexpand import hyperexpand
return hyperexpand(self)
def test_to_meijerg():
x = symbols('x')
assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
assert hyperexpand(expr_to_holonomic(log(x)).to_meijerg()).simplify() == log(x)
assert expr_to_holonomic(4*x**2/3 + 7).to_meijerg() == 4*x**2/3 + 7
assert hyperexpand(expr_to_holonomic(besselj(2, x), lenics=3).to_meijerg()) == besselj(2, x)
p = hyper((-S(1)/2, -3), (), x)
assert from_hyper(p).to_meijerg() == hyperexpand(p)
p = hyper((S(1), S(3)), (S(2), ), x)
assert (hyperexpand(from_hyper(p).to_meijerg()) - hyperexpand(p)).expand() == 0
p = from_hyper(hyper((-2, -3), (S(1)/2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1)/2, ), x))
C_0 = Symbol('C_0')
C_1 = Symbol('C_1')
D_0 = Symbol('D_0')
assert (hyperexpand(p.to_meijerg()).subs({C_0:1, D_0:0}) - s).simplify() == 0
p.y0 = {0: [1], S(1)/2: [0]}
assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
p = expr_to_holonomic(besselj(S(1)/2, x), initcond=False)
assert (p.to_expr() - (D_0*sin(x) + C_0*cos(x) + C_1*sin(x))/sqrt(x)).simplify() == 0
p = expr_to_holonomic(besselj(S(1)/2, x), y0={S(-1)/2: [sqrt(2)/sqrt(pi), sqrt(2)/sqrt(pi)]})
assert (p.to_expr() - besselj(S(1)/2, x) - besselj(S(-1)/2, x)).simplify() == 0
def _eval_expand_func(self, deep=True, **hints):
from sympy import hyperexpand
return hyperexpand(self)
def test_to_meijerg():
x = symbols('x')
assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
assert hyperexpand(expr_to_holonomic(
log(x)).to_meijerg()).simplify() == log(x)
assert expr_to_holonomic(4 * x**2 / 3 + 7).to_meijerg() == 4 * x**2 / 3 + 7
assert hyperexpand(
expr_to_holonomic(besselj(2, x),
lenics=3).to_meijerg()) == besselj(2, x)
p = hyper((-S(1) / 2, -3), (), x)
assert from_hyper(p).to_meijerg() == hyperexpand(p)
p = hyper((S(1), S(3)), (S(2), ), x)
assert (hyperexpand(from_hyper(p).to_meijerg()) -
hyperexpand(p)).expand() == 0
p = from_hyper(hyper((-2, -3), (S(1) / 2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1) / 2, ), x))
C_0 = Symbol('C_0')
C_1 = Symbol('C_1')
D_0 = Symbol('D_0')
assert (hyperexpand(p.to_meijerg()).subs({
C_0: 1,
D_0: 0
}) - s).simplify() == 0
p.singular_ics = [(0, [1]), (S(1) / 2, [0])]
assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
p = expr_to_holonomic(besselj(S(1) / 2, x), initcond=False)
assert (
p.to_expr() -
(D_0 * sin(x) + C_0 * cos(x) + C_1 * sin(x)) / sqrt(x)).simplify() == 0
p = expr_to_holonomic(
besselj(S(1) / 2, x),
singular_ics=((S(-1) / 2, [sqrt(2) / sqrt(pi),
sqrt(2) / sqrt(pi)]), ))
assert (p.to_expr() - besselj(S(1) / 2, x) -
besselj(S(-1) / 2, x)).simplify() == 0