def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1 / z
pytest.raises(ArgumentIndexError, lambda: _eis(x).fdiff(2))
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z) * _eis(z)
assert li(z).rewrite('tractable') == z * _eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z) * Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(
z,
n=2) == EulerGamma + log(z) + z * (-log(z) - EulerGamma + 1) + z**2 * (
log(z) / 2 - Rational(3, 4) + EulerGamma / 2) + O(z**2)
l = Limit(Ei(y / x) / exp(y / x), x, 0)
assert l.doit() == l # cover _eis._eval_aseries
def test_order_noncommutative():
A = Symbol('A', commutative=False)
assert O(A + A * x, x) == O(1, x)
assert (A + A * x) * O(x) == O(x)
assert (A * x) * O(x) == O(x**2, x)
assert expand((1 + O(x)) * A * A * x) == A * A * x + O(x**2, x)
assert expand((A * A + O(x)) * x) == A * A * x + O(x**2, x)
assert expand((A + O(x)) * A * x) == A * A * x + O(x**2, x)
def test_expand_frac():
assert expand((x + y)*y/x/(x + 1), frac=True) == \
(x*y + y**2)/(x**2 + x)
assert expand((x + y)*y/x/(x + 1), numer=True) == \
(x*y + y**2)/(x*(x + 1))
assert expand((x + y)*y/x/(x + 1), denom=True) == \
y*(x + y)/(x**2 + x)
eq = (x + 1)**2 / y
assert expand_numer(eq, multinomial=False) == eq
def test_expand_frac():
assert expand((x + y)*y/x/(x + 1), frac=True) == \
(x*y + y**2)/(x**2 + x)
assert expand((x + y)*y/x/(x + 1), numer=True) == \
(x*y + y**2)/(x*(x + 1))
assert expand((x + y)*y/x/(x + 1), denom=True) == \
y*(x + y)/(x**2 + x)
eq = (x + 1)**2/y
assert expand_numer(eq, multinomial=False) == eq
def test_piecewise_fold_expand():
p1 = Piecewise((1, Interval(0, 1, False, True).contains(x)), (0, True))
p2 = piecewise_fold(expand((1 - x)*p1))
assert p2 == Piecewise((1 - x, Interval(0, 1, False, True).contains(x)),
(Piecewise((-x, Interval(0, 1, False, True).contains(x)), (0, True)), True))
p2 = expand(piecewise_fold((1 - x)*p1))
assert p2 == Piecewise(
(1 - x, Interval(0, 1, False, True).contains(x)), (0, True))
def test_piecewise_fold_expand():
p1 = Piecewise((1, Interval(0, 1, False, True).contains(x)), (0, True))
p2 = piecewise_fold(expand((1 - x)*p1))
assert p2 == Piecewise((1 - x, Interval(0, 1, False, True).contains(x)),
(Piecewise((-x, Interval(0, 1, False, True).contains(x)), (0, True)), True))
p2 = expand(piecewise_fold((1 - x)*p1))
assert p2 == Piecewise(
(1 - x, Interval(0, 1, False, True).contains(x)), (0, True))
def decide_termination(program: Program):
"""
The main function, gathering all the information, deciding on and calling a proof-rule
"""
reset_mora()
branch_store.set_program(program)
bound_store.set_program(program)
lgc = get_loop_guard_change(program)
me_pos = create_martingale_expression(program)
me_neg = expand(me_pos * (-1))
log(f"Martingale expression: {me_pos.as_expr()}", LOG_ESSENTIAL)
rules = [
InitialStateRule(lgc, me_pos, program),
RankingSMRule(lgc, me_pos, program),
SupermartingaleRule(lgc, me_pos, program),
RepulsingSMRule(lgc, me_neg, program)
]
result = Result()
for rule in rules:
if rule.is_applicable():
result = rule.run(result)
if result.all_known():
break
return result
def test_sympyissues_5919_6830():
# issue sympy/sympy#5919
n = -1 + 1 / x
z = n / x / (-n)**2 - 1 / n / x
assert expand(
z) == 1 / (x**2 - 2 * x + 1) - 1 / (x - 2 + 1 / x) - 1 / (-x + 1)
# issue sympy/sympy#6830
p = (1 + x)**2
assert expand_multinomial(
(1 + x * p)**2) == (x**2 * (x**4 + 4 * x**3 + 6 * x**2 + 4 * x + 1) +
2 * x * (x**2 + 2 * x + 1) + 1)
assert expand_multinomial(
(1 + (y + x) * p)**2) == (2 * ((x + y) * (x**2 + 2 * x + 1)) +
(x**2 + 2 * x * y + y**2) *
(x**4 + 4 * x**3 + 6 * x**2 + 4 * x + 1) + 1)
A = Symbol('A', commutative=False)
p = (1 + A)**2
assert expand_multinomial(
(1 + x * p)**2) == (x**2 * (1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4) +
2 * x * (1 + 2 * A + A**2) + 1)
assert expand_multinomial(
(1 + (y + x) * p)**2) == ((x + y) * (1 + 2 * A + A**2) * 2 +
(x**2 + 2 * x * y + y**2) *
(1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4) + 1)
assert expand_multinomial((1 + (y + x) * p)**3) == (
(x + y) * (1 + 2 * A + A**2) * 3 + (x**2 + 2 * x * y + y**2) *
(1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4) * 3 +
(x**3 + 3 * x**2 * y + 3 * x * y**2 + y**3) *
(1 + 6 * A + 15 * A**2 + 20 * A**3 + 15 * A**4 + 6 * A**5 + A**6) + 1)
# unevaluate powers
eq = (Pow((x + 1) * ((A + 1)**2), 2, evaluate=False))
# - in this case the base is not an Add so no further
# expansion is done
assert expand_multinomial(eq) == \
(x**2 + 2*x + 1)*(1 + 4*A + 6*A**2 + 4*A**3 + A**4)
# - but here, the expanded base *is* an Add so it gets expanded
eq = (Pow(((A + 1)**2), 2, evaluate=False))
assert expand_multinomial(eq) == 1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4
# coverage
def ok(a, b, n):
e = (a + I * b)**n
return verify_numerically(e, expand_multinomial(e))
for a in [2, Rational(1, 2)]:
for b in [3, Rational(1, 3)]:
for n in range(2, 6):
assert ok(a, b, n)
assert expand_multinomial((x + 1 + O(z))**2) == \
1 + 2*x + x**2 + O(z)
assert expand_multinomial((x + 1 + O(z))**3) == \
1 + 3*x + 3*x**2 + x**3 + O(z)
e = (sin(x) + y)**3
assert (expand_multinomial(e.subs({y: O(x**4)})) ==
expand_multinomial(e).subs({y: O(x**4)}) == sin(x)**3 + O(x**6))
assert expand_multinomial(3**(x + y + 3)) == 27 * 3**(x + y)
def can_do_meijer(a1, a2, b1, b2, numeric=True):
"""
This helper function tries to hyperexpand() the meijer g-function
corresponding to the parameters a1, a2, b1, b2.
It returns False if this expansion still contains g-functions.
If numeric is True, it also tests the so-obtained formula numerically
(at random values) and returns False if the test fails.
Else it returns True.
"""
r = hyperexpand(meijerg(a1, a2, b1, b2, z))
if r.has(meijerg):
return False
# NOTE hyperexpand() returns a truly branched function, whereas numerical
# evaluation only works on the main branch. Since we are evaluating on
# the main branch, this should not be a problem, but expressions like
# exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get
# rid of them. The expand heuristically does this...
r = unpolarify(expand(r, force=True, power_base=True, power_exp=False,
mul=False, log=False, multinomial=False, basic=False))
if not numeric:
return True
repl = {}
for n, a in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[a] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
def compute_solution_for_recurrence(recurr_coeff: Expr,
inhom_part_solution: Expr,
initial_value: Expr):
"""
Computes the (unique) solution to the recurrence relation:
f(0) = initial_value; f(n+1) = recurr_coeff * f(n) + inhom_part_solution
"""
log(
f"Start compute solution for recurrence, { recurr_coeff }, { inhom_part_solution }, { initial_value }",
LOG_VERBOSE)
n = symbols('n', integer=True, positive=True)
if recurr_coeff.is_zero:
return expand(inhom_part_solution.xreplace({n: n - 1}))
hom_solution = (recurr_coeff**n) * initial_value
k = symbols('_k', integer=True, positive=True)
summand = simplify(
(recurr_coeff**k) * inhom_part_solution.xreplace({n: (n - 1) - k}))
particular_solution = summation(summand, (k, 0, (n - 1)))
particular_solution = without_piecewise(particular_solution)
solution = simplify(hom_solution + particular_solution)
log(
f"End compute solution for recurrence, { recurr_coeff }, { inhom_part_solution }, { initial_value }",
LOG_VERBOSE)
return solution
def compute_recurrence(program: Program, monomial: Poly):
"""
Iteratively splits a monomial on variables which are dependent with respect to the given monomial
"""
log(f"Start compute recurrence, { monomial.as_expr() }", LOG_VERBOSE)
result = monomial.as_expr()
split_variables = set()
for variable, update in reversed(program.updates.items()):
if variable not in result.free_symbols:
continue
if update.is_random_var:
result = replace_rv_in_polynomial(
program, result.as_poly(program.variables),
variable).as_expr()
continue
split_variables.add(variable)
branches = split_expression_on_variable(program, result, variable)
log(f"Start combining {len(branches)} branches", LOG_VERBOSE)
result = sum([prob * branch for branch, prob in branches])
log(f"Mid combining branches", LOG_VERBOSE)
result = expand(result)
log(f"End combining branches", LOG_VERBOSE)
log(f"End compute recurrence, { monomial.as_expr() }", LOG_VERBOSE)
return result.as_poly(program.variables)
def can_do_meijer(a1, a2, b1, b2, numeric=True):
"""
This helper function tries to hyperexpand() the meijer g-function
corresponding to the parameters a1, a2, b1, b2.
It returns False if this expansion still contains g-functions.
If numeric is True, it also tests the so-obtained formula numerically
(at random values) and returns False if the test fails.
Else it returns True.
"""
from diofant import unpolarify, expand
r = hyperexpand(meijerg(a1, a2, b1, b2, z))
if r.has(meijerg):
return False
# NOTE hyperexpand() returns a truly branched function, whereas numerical
# evaluation only works on the main branch. Since we are evaluating on
# the main branch, this should not be a problem, but expressions like
# exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get
# rid of them. The expand heuristically does this...
r = unpolarify(expand(r, force=True, power_base=True, power_exp=False,
mul=False, log=False, multinomial=False, basic=False))
if not numeric:
return True
repl = {}
for n, a in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[a] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
def test_Piecewise():
e1 = x * (x + y) - y * (x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y) * y / x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
def test__erfs():
assert _erfs(z).diff(z) == -2/sqrt(S.Pi) + 2*z*_erfs(z)
assert _erfs(1/z).series(z) == \
z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== erf(z).diff(z)
assert _erfs(z).rewrite("intractable") == (-erf(z) + 1)*exp(z**2)
def get_loop_guard_change(program: Program):
"""
Returns E[LG_{n+1} - LG_{n}]
"""
n = symbols("n", integer=True, positive=True)
lg = sympify(LOOP_GUARD_VAR).as_poly(program.variables)
expected_lg = get_expected(program, lg)
expected_lg_plus = expected_lg.xreplace({n: n + 1})
return expand(expected_lg_plus - expected_lg)
def test__erfs():
assert _erfs(z).diff(z) == -2 / sqrt(pi) + 2 * z * _erfs(z)
pytest.raises(ArgumentIndexError, lambda: _erfs(x).fdiff(2))
assert _erfs(1/z).series(z) == \
z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== erf(z).diff(z)
assert _erfs(z).rewrite('intractable') == (-erf(z) + 1) * exp(z**2)
def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1/z
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z)*_eis(z)
assert li(z).rewrite('tractable') == z*_eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z)*Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(z, n=2) == EulerGamma + log(z) + z*(-log(z) -
EulerGamma + 1) + z**2*(log(z)/2 - Rational(3, 4) + EulerGamma/2) + O(z**2)
def test__erfs():
assert _erfs(z).diff(z) == -2/sqrt(pi) + 2*z*_erfs(z)
pytest.raises(ArgumentIndexError, lambda: _erfs(x).fdiff(2))
assert _erfs(1/z).series(z) == \
z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== erf(z).diff(z)
assert _erfs(z).rewrite("intractable") == (-erf(z) + 1)*exp(z**2)
def create_martingale_expression(program: Program):
"""
Creates the martingale expression E(M_{i+1} - M_i | F_i). Also deterministic variables get substituted
with their representation in n.
"""
lg = symbols(LOOP_GUARD_VAR).as_poly(program.variables)
expected_guard = get_recurrence(program, lg)
lg = program.updates[symbols(LOOP_GUARD_VAR)].branches[0][0]
expression = expand(expected_guard - lg).as_expr()
expression = substitute_deterministic_variables(expression, program)
return simplify(expression)
def test_sympyissues_5919_6830():
# issue sympy/sympy#5919
n = -1 + 1/x
z = n/x/(-n)**2 - 1/n/x
assert expand(z) == 1/(x**2 - 2*x + 1) - 1/(x - 2 + 1/x) - 1/(-x + 1)
# issue sympy/sympy#6830
p = (1 + x)**2
assert expand_multinomial((1 + x*p)**2) == (
x**2*(x**4 + 4*x**3 + 6*x**2 + 4*x + 1) + 2*x*(x**2 + 2*x + 1) + 1)
assert expand_multinomial((1 + (y + x)*p)**2) == (
2*((x + y)*(x**2 + 2*x + 1)) + (x**2 + 2*x*y + y**2) *
(x**4 + 4*x**3 + 6*x**2 + 4*x + 1) + 1)
A = Symbol('A', commutative=False)
p = (1 + A)**2
assert expand_multinomial((1 + x*p)**2) == (
x**2*(1 + 4*A + 6*A**2 + 4*A**3 + A**4) + 2*x*(1 + 2*A + A**2) + 1)
assert expand_multinomial((1 + (y + x)*p)**2) == (
(x + y)*(1 + 2*A + A**2)*2 + (x**2 + 2*x*y + y**2) *
(1 + 4*A + 6*A**2 + 4*A**3 + A**4) + 1)
assert expand_multinomial((1 + (y + x)*p)**3) == (
(x + y)*(1 + 2*A + A**2)*3 + (x**2 + 2*x*y + y**2)*(1 + 4*A +
6*A**2 + 4*A**3 + A**4)*3 + (x**3 + 3*x**2*y + 3*x*y**2 + y**3)*(1 + 6*A
+ 15*A**2 + 20*A**3 + 15*A**4 + 6*A**5 + A**6) + 1)
# unevaluate powers
eq = (Pow((x + 1)*((A + 1)**2), 2, evaluate=False))
# - in this case the base is not an Add so no further
# expansion is done
assert expand_multinomial(eq) == \
(x**2 + 2*x + 1)*(1 + 4*A + 6*A**2 + 4*A**3 + A**4)
# - but here, the expanded base *is* an Add so it gets expanded
eq = (Pow(((A + 1)**2), 2, evaluate=False))
assert expand_multinomial(eq) == 1 + 4*A + 6*A**2 + 4*A**3 + A**4
# coverage
def ok(a, b, n):
e = (a + I*b)**n
return verify_numerically(e, expand_multinomial(e))
for a in [2, Rational(1, 2)]:
for b in [3, Rational(1, 3)]:
for n in range(2, 6):
assert ok(a, b, n)
assert expand_multinomial((x + 1 + O(z))**2) == \
1 + 2*x + x**2 + O(z)
assert expand_multinomial((x + 1 + O(z))**3) == \
1 + 3*x + 3*x**2 + x**3 + O(z)
e = (sin(x) + y)**3
assert (expand_multinomial(e.subs({y: O(x**4)})) ==
expand_multinomial(e).subs({y: O(x**4)}) == sin(x)**3 + O(x**6))
assert expand_multinomial(3**(x + y + 3)) == 27*3**(x + y)
def test_Piecewise():
e1 = x * (x + y) - y * (x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y) * y / x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
# trigsimp tries not to touch non-trig containing args
assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
Piecewise((e1, e3 < s2), (e3, True))
def test_Piecewise():
e1 = x*(x + y) - y*(x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y)*y/x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
# trigsimp tries not to touch non-trig containing args
assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
Piecewise((e1, e3 < s2), (e3, True))
def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1/z
pytest.raises(ArgumentIndexError, lambda: _eis(x).fdiff(2))
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z)*_eis(z)
assert li(z).rewrite('tractable') == z*_eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z)*Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(z, n=2) == EulerGamma + log(z) + z*(-log(z) -
EulerGamma + 1) + z**2*(log(z)/2 - Rational(3, 4) + EulerGamma/2) + O(z**2)
l = Limit(Ei(y/x)/exp(y/x), x, 0)
assert l.doit() == l # cover _eis._eval_aseries
def get_inhom_part_solution(program: Program, inhom_part: Poly):
"""
For a given inhomogenous part of the assignment of a monomial replace the monomials in the inhom part by their
closed form solutions.
"""
log(f"Start get inhom_part_solution, { inhom_part.as_expr() }",
LOG_VERBOSE)
monomials = get_monoms(inhom_part)
result = inhom_part.coeff_monomial(1)
for monomial in monomials:
solution = get_solution(program, monomial)
monomial = monomial.as_expr()
result += inhom_part.coeff_monomial(monomial) * solution
log(f"End get inhom_part_solution, {inhom_part.as_expr()}", LOG_VERBOSE)
return expand(result)
def test_expand_non_commutative():
A = Symbol('A', commutative=False)
B = Symbol('B', commutative=False)
C = Symbol('C', commutative=False)
a = Symbol('a')
b = Symbol('b')
i = Symbol('i', integer=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
p = Symbol('p', polar=True)
np = Symbol('p', polar=False)
assert (C * (A + B)).expand() == C * A + C * B
assert (C * (A + B)).expand() != A * C + B * C
assert ((A + B)**2).expand() == A**2 + A * B + B * A + B**2
assert ((A +
B)**3).expand() == (A**2 * B + B**2 * A + A * B**2 + B * A**2 +
A**3 + B**3 + A * B * A + B * A * B)
# issue 6219
assert ((a * A * B * A**-1)**2).expand() == a**2 * A * B**2 / A
# Note that (a*A*B*A**-1)**2 is automatically converted to a**2*(A*B*A**-1)**2
assert ((a * A * B *
A**-1)**2).expand(deep=False) == a**2 * (A * B * A**-1)**2
assert ((a * A * B * A**-1)**2).expand() == a**2 * (A * B**2 * A**-1)
assert ((a * A * B *
A**-1)**2).expand(force=True) == a**2 * A * B**2 * A**(-1)
assert ((a * A * B)**2).expand() == a**2 * A * B * A * B
assert ((a * A)**2).expand() == a**2 * A**2
assert ((a * A * B)**i).expand() == a**i * (A * B)**i
assert ((a * A *
(B *
(A * B / A)**2))**i).expand() == a**i * (A * B * A * B**2 / A)**i
# issue 6558
assert (A * B * (A * B)**-1).expand() == A * B * (A * B)**-1
assert ((a * A)**i).expand() == a**i * A**i
assert ((a * A * B * A**-1)**3).expand() == a**3 * A * B**3 / A
assert ((a*A*B*A*B/A)**3).expand() == \
a**3*A*B*(A*B**2)*(A*B**2)*A*B*A**(-1)
assert ((a*A*B*A*B/A)**-3).expand() == \
a**-3*(A*B*(A*B**2)*(A*B**2)*A*B*A**(-1))**-1
assert ((a * b * A * B *
A**-1)**i).expand() == a**i * b**i * (A * B / A)**i
assert ((a * (a * b)**i)**i).expand() == a**i * a**(i**2) * b**(i**2)
e = Pow(Mul(a, 1 / a, A, B, evaluate=False), Integer(2), evaluate=False)
assert e.expand() == A * B * A * B
assert sqrt(a * (A * b)**i).expand() == sqrt(a * b**i * A**i)
assert (sqrt(-a)**a).expand() == sqrt(-a)**a
assert expand((-2 * n)**(i / 3)) == 2**(i / 3) * (-n)**(i / 3)
assert expand(
(-2 * n * m)**(i / a)) == (-2)**(i / a) * (-n)**(i / a) * (-m)**(i / a)
assert expand((-2 * a * p)**b) == 2**b * p**b * (-a)**b
assert expand((-2 * a * np)**b) == 2**b * (-a * np)**b
assert expand(sqrt(A * B)) == sqrt(A * B)
assert expand(sqrt(-2 * a * b)) == sqrt(2) * sqrt(-a * b)
def test_expand_non_commutative():
A = Symbol('A', commutative=False)
B = Symbol('B', commutative=False)
C = Symbol('C', commutative=False)
a = Symbol('a')
b = Symbol('b')
i = Symbol('i', integer=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
p = Symbol('p', polar=True)
np = Symbol('p', polar=False)
assert (C*(A + B)).expand() == C*A + C*B
assert (C*(A + B)).expand() != A*C + B*C
assert ((A + B)**2).expand() == A**2 + A*B + B*A + B**2
assert ((A + B)**3).expand() == (A**2*B + B**2*A + A*B**2 + B*A**2 +
A**3 + B**3 + A*B*A + B*A*B)
# issue sympy/sympy#6219
assert ((a*A*B*A**-1)**2).expand() == a**2*A*B**2/A
# Note that (a*A*B*A**-1)**2 is automatically converted to a**2*(A*B*A**-1)**2
assert ((a*A*B*A**-1)**2).expand(deep=False) == a**2*(A*B*A**-1)**2
assert ((a*A*B*A**-1)**2).expand() == a**2*(A*B**2*A**-1)
assert ((a*A*B*A**-1)**2).expand(force=True) == a**2*A*B**2*A**(-1)
assert ((a*A*B)**2).expand() == a**2*A*B*A*B
assert ((a*A)**2).expand() == a**2*A**2
assert ((a*A*B)**i).expand() == a**i*(A*B)**i
assert ((a*A*(B*(A*B/A)**2))**i).expand() == a**i*(A*B*A*B**2/A)**i
# issue sympy/sympy#6558
assert (A*B*(A*B)**-1).expand() == A*B*(A*B)**-1
assert ((a*A)**i).expand() == a**i*A**i
assert ((a*A*B*A**-1)**3).expand() == a**3*A*B**3/A
assert ((a*A*B*A*B/A)**3).expand() == \
a**3*A*B*(A*B**2)*(A*B**2)*A*B*A**(-1)
assert ((a*A*B*A*B/A)**-3).expand() == \
a**-3*(A*B*(A*B**2)*(A*B**2)*A*B*A**(-1))**-1
assert ((a*b*A*B*A**-1)**i).expand() == a**i*b**i*(A*B/A)**i
assert ((a*(a*b)**i)**i).expand() == a**i*a**(i**2)*b**(i**2)
e = Pow(Mul(a, 1/a, A, B, evaluate=False), Integer(2), evaluate=False)
assert e.expand() == A*B*A*B
assert sqrt(a*(A*b)**i).expand() == sqrt(a*b**i*A**i)
assert (sqrt(-a)**a).expand() == sqrt(-a)**a
assert expand((-2*n)**(i/3)) == 2**(i/3)*(-n)**(i/3)
assert expand((-2*n*m)**(i/a)) == (-2)**(i/a)*(-n)**(i/a)*(-m)**(i/a)
assert expand((-2*a*p)**b) == 2**b*p**b*(-a)**b
assert expand((-2*a*np)**b) == 2**b*(-a*np)**b
assert expand(sqrt(A*B)) == sqrt(A*B)
assert expand(sqrt(-2*a*b)) == sqrt(2)*sqrt(-a*b)
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3*I) == I
assert sign(-3*I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
assert sign(2 + 2*I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2*x) == sign(x)
assert sign(2*x) == sign(x)
assert sign(I*x) == I*sign(x)
assert sign(-2*I*x) == -I*sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2*p*x) == sign(x)
assert sign(n*x) == -sign(x)
assert sign(n*m*x) == sign(x)
x = Symbol('x', imaginary=True)
xn = Symbol('xn', imaginary=True, nonzero=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(-I*x)
assert sign(xn).doit() == xn / Abs(xn)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', extended_real=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
assert sign(sin(x)).nseries(x) == 1
y = Symbol('y')
assert sign(x*y).nseries(x).removeO() == sign(y)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is True
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_extended_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', extended_real=True)).is_nonnegative is None
assert sign(Symbol('x', extended_real=True)).is_nonpositive is None
assert sign(Symbol('x', extended_real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', extended_real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2*Heaviside(x)-1
assert sign(y).rewrite(Heaviside) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I*pi)*pi) is Integer(-1)
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq) == 0
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = cbrt(expand(q**3))
d = p - q
assert sign(d) == 0
assert abs(sign(z)) == Abs(sign(z), evaluate=False)
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
def test_meijerint():
s, t, mu = symbols('s t mu', extended_real=True)
assert integrate(meijerg([], [], [0], [], s*t)
* meijerg([], [], [mu/2], [-mu/2], t**2/4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(integrate(x**s*meijerg([[], []], [[0], []], x),
(x, 0, oo), meijerg=False),
Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1)**3*exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu)/(2*sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi)*sigma*(erf(mu/(2*sigma)) + 1)
assert c
i, _ = meijerint_definite(exp(-mu*x)*exp(sigma*x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1/(mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2*x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(exp(-((x - mu)/sigma)**2/2)/sqrt(2*pi*sigma**2),
x, -oo, oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x)*sin(x), x, 0, oo) == (Rational(1, 2), True)
# Test a bug
def res(n):
return (1/(1 + x**2)).diff(x, n).subs({x: 1})*(-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x)*sin(x + a), (x, 0, oo), meijerg=True)
) == sqrt(2)*sin(a + pi/4)/2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
* meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
* gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, Rational(1, 2))/4).expand()
# Test hyperexpand bug.
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + Rational(1, 2),
alpha/2 + 1)), ((0, 0, Rational(1, 2)), (-Rational(1, 2),)), alpha**2/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - Rational(1, 2))*((-1)**s + 1)*gamma(s/2 + Rational(1, 2))/2
def test_Abs():
pytest.raises(TypeError, lambda: Abs(Interval(2, 3))) # issue sympy/sympy#8717
x, y = symbols('x,y')
assert sign(sign(x)) == sign(x)
assert isinstance(sign(x*y), sign)
assert Abs(0) == 0
assert Abs(1) == 1
assert Abs(-1) == 1
assert Abs(I) == 1
assert Abs(-I) == 1
assert Abs(nan) == nan
assert Abs(I * pi) == pi
assert Abs(-I * pi) == pi
assert Abs(I * x) == Abs(x)
assert Abs(-I * x) == Abs(x)
assert Abs(-2*x) == 2*Abs(x)
assert Abs(-2.0*x) == 2.0*Abs(x)
assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
assert Abs(conjugate(x)) == Abs(x)
assert conjugate(Abs(x)) == Abs(x)
a = cos(1)**2 + sin(1)**2 - 1
assert Abs(a*x).series(x).simplify() == 0
a = Symbol('a', positive=True)
assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)
x = Symbol('x', extended_real=True)
n = Symbol('n', integer=True)
assert Abs((-1)**n) == 1
assert x**(2*n) == Abs(x)**(2*n)
assert Abs(x).diff(x) == sign(x)
assert Abs(-x).fdiff() == sign(x)
assert abs(x) == Abs(x) # Python built-in
assert Abs(x)**3 == x**2*Abs(x)
assert Abs(x)**4 == x**4
assert (
Abs(x)**(3*n)).args == (Abs(x), 3*n) # leave symbolic odd unchanged
assert (1/Abs(x)).args == (Abs(x), -1)
assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
assert Abs(x)**-3 == Abs(x)/(x**4)
assert Abs(x**3) == x**2*Abs(x)
assert Abs(x**pi) == Abs(x**pi, evaluate=False)
x = Symbol('x', imaginary=True)
assert Abs(x).diff(x) == -sign(x)
pytest.raises(ArgumentIndexError, lambda: Abs(z).fdiff(2))
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when you can and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert abs(eq) == 0
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = cbrt(expand(q**3))
d = p - q
assert abs(d) == 0
assert Abs(4*exp(pi*I/4)) == 4
assert Abs(3**(2 + I)) == 9
assert Abs((-3)**(1 - I)) == 3*exp(pi)
assert Abs(oo) is oo
assert Abs(-oo) is oo
assert Abs(oo + I) is oo
assert Abs(oo + I*oo) is oo
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert abs(sign(z)) == Abs(sign(z), evaluate=False)
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3*I) == I
assert sign(-3*I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
assert sign(2 + 2*I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2*x) == sign(x)
assert sign(2*x) == sign(x)
assert sign(I*x) == I*sign(x)
assert sign(-2*I*x) == -I*sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2*p*x) == sign(x)
assert sign(n*x) == -sign(x)
assert sign(n*m*x) == sign(x)
x = Symbol('x', imaginary=True)
xn = Symbol('xn', imaginary=True, nonzero=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(-I*x)
assert sign(xn).doit() == xn / Abs(xn)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', extended_real=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
assert sign(sin(x)).nseries(x) == 1
y = Symbol('y')
assert sign(x*y).nseries(x).removeO() == sign(y)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is True
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_extended_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', extended_real=True)).is_nonnegative is None
assert sign(Symbol('x', extended_real=True)).is_nonpositive is None
assert sign(Symbol('x', extended_real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', extended_real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2*Heaviside(x)-1
assert sign(y).rewrite(Heaviside) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I*pi)*pi) is Integer(-1)
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq) == 0
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = cbrt(expand(q**3))
d = p - q
assert sign(d) == 0
assert abs(sign(z)) == Abs(sign(z), evaluate=False)
def test_expand():
assert expand((A*B)**2) == A*B*A*B
assert expand(A*B - B*A) == A*B - B*A
assert expand((A*B/A)**2) == A*B*B/A
assert expand(B*A*(A + B)*B) == B*A**2*B + B*A*B**2
assert expand(B*A*(A + C)*B) == B*A**2*B + B*A*C*B
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I*im(x) + I*im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
def test_expand_log():
t = Symbol('t', positive=True)
# after first expansion, -2*log(2) + log(4); then 0 after second
assert expand(log(t**2) - log(t**2 / 4) - 2 * log(2)) == 0
def mysimp(expr):
return expand(powsimp(logcombine(expr, force=True),
force=True,
deep=True),
force=True).replace(exp_polar, exp)
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I * im(x) + I * im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
def test_minimal_polynomial():
assert minimal_polynomial(-7)(x) == x + 7
assert minimal_polynomial(-1)(x) == x + 1
assert minimal_polynomial(+0)(x) == x
assert minimal_polynomial(+1)(x) == x - 1
assert minimal_polynomial(+7)(x) == x - 7
assert minimal_polynomial(Rational(1, 3),
method='groebner')(x) == 3 * x - 1
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(pi, method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(sin(sqrt(2)), method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(2**pi, method='groebner'))
pytest.raises(ValueError, lambda: minimal_polynomial(1, method='spam'))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(5))(x) == x**2 - 5
assert minimal_polynomial(sqrt(6))(x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2))(x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5))(x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6))(x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3)(x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6)(x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7)(x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3)(x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6)(x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7)(x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)))(x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)))(x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I +
sqrt(2 + I))(x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3))(x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) +
sqrt(6))(x) == x**4 - 22 * x**2 - 48 * x - 23
e = 1 / sqrt(sqrt(1 + sqrt(3)) - 4)
assert minimal_polynomial(e) == minimal_polynomial(e, method='groebner')
assert minimal_polynomial(e)(x) == (222 * x**8 + 240 * x**6 + 94 * x**4 +
16 * x**2 + 1)
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a)(x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a))(x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1)))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(2)) == PurePoly(x**2 - 2)
assert minimal_polynomial(sqrt(2), method='groebner') == PurePoly(x**2 - 2)
a = sqrt(2)
b = sqrt(3)
assert minimal_polynomial(b)(x) == x**2 - 3
assert minimal_polynomial(a) == PurePoly(x**2 - 2)
assert minimal_polynomial(b) == PurePoly(x**2 - 3)
assert minimal_polynomial(sqrt(a)) == PurePoly(x**4 - 2)
assert minimal_polynomial(a + 1) == PurePoly(x**2 - 2 * x - 1)
assert minimal_polynomial(sqrt(a / 2 +
17))(x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 +
17))(x) == 4 * x**4 - 136 * x**2 + 1153
# issue diofant/diofant#431
K = QQ.algebraic_field(sqrt(2))
theta = K.to_expr(K([17, Rational(1, 2)]))
assert minimal_polynomial(theta)(x) == 2 * x**2 - 68 * x + 577
K = QQ.algebraic_field(RootOf(x**7 + x - 1, 3))
theta = K.to_expr(K([1, 0, 0, 2, 1]))
ans = minimal_polynomial(theta)(x)
assert ans == (x**7 - 7 * x**6 + 19 * x**5 - 27 * x**4 + 63 * x**3 -
115 * x**2 + 82 * x - 147)
assert minimal_polynomial(theta, method='groebner')(x) == ans
K = QQ.algebraic_field(RootOf(x**5 + 5 * x - 1, 2))
theta = K.to_expr(K([1, -1, 1]))
ans = (x**30 - 15 * x**28 - 10 * x**27 + 135 * x**26 + 330 * x**25 -
705 * x**24 - 150 * x**23 + 3165 * x**22 - 6850 * x**21 +
7182 * x**20 + 3900 * x**19 + 4435 * x**18 + 11970 * x**17 -
259725 * x**16 - 18002 * x**15 + 808215 * x**14 - 200310 * x**13 -
647115 * x**12 + 299280 * x**11 - 1999332 * x**10 + 910120 * x**9 +
2273040 * x**8 - 5560320 * x**7 + 5302000 * x**6 - 2405376 * x**5 +
1016640 * x**4 - 804480 * x**3 + 257280 * x**2 - 53760 * x + 1280)
assert minimal_polynomial(sqrt(theta) + root(theta, 3),
method='groebner')(x) == ans
K1 = QQ.algebraic_field(RootOf(x**3 + 4 * x - 15, 1))
K2 = QQ.algebraic_field(RootOf(x**3 - x + 1, 0))
theta = sqrt(1 + 1 / (K1.to_expr(K1([1, 0, 1])) + 1 /
(sqrt(3) + K2.to_expr(K2([-1, 2, 1])))))
ans = (2262264837876687263 * x**36 - 38939909597855051866 * x**34 +
315720420314462950715 * x**32 - 1601958657418182606114 * x**30 +
5699493671077371036494 * x**28 - 15096777696140985506150 * x**26 +
30847690820556462893974 * x**24 - 49706549068200640994022 * x**22 +
64013601241426223813103 * x**20 - 66358713088213594372990 * x**18 +
55482571280904904971976 * x**16 - 37309340229165533529076 * x**14 +
20016999328983554519040 * x**12 - 8446273798231518826782 * x**10 +
2738866994867366499481 * x**8 - 657825125060873756424 * x**6 +
110036313740049140508 * x**4 - 11416087328869938298 * x**2 +
551322649782053543)
assert minimal_polynomial(theta)(x) == ans
a = sqrt(2) / 3 + 7
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)))(x) == f
assert minimal_polynomial(a**Rational(
3, 2))(x) == 729 * x**4 - 506898 * x**2 + 84604519
K = QQ.algebraic_field(RootOf(x**3 + x - 1, 0))
a = K.to_expr(K([0, 1]))
assert minimal_polynomial(1 / a**2)(x) == x**3 - x**2 - 2 * x - 1
# issue sympy/sympy#5994
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000)) +
2 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000))))
assert minimal_polynomial(eq)(x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex)(x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex)(x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = cbrt(expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(-sqrt(5) / 2 - Rational(1, 2) +
(-sqrt(5) / 2 - Rational(1, 2))**2)(x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3)(x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, method='groebner')(x) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, method='groebner')(x) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I,
method='groebner')(x) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1,
method='groebner')(x) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
method='groebner')(x) == x**4 + 18 * x**2 + 49
assert minimal_polynomial(exp_polar(0))(x) == x - 1
def mysimp(expr):
from diofant import expand, logcombine, powsimp
return expand(powsimp(logcombine(expr, force=True),
force=True,
deep=True),
force=True).replace(exp_polar, exp)
def test_Abs():
pytest.raises(TypeError, lambda: Abs(Interval(2, 3))) # issue sympy/sympy#8717
x, y = symbols('x,y')
assert sign(sign(x)) == sign(x)
assert isinstance(sign(x*y), sign)
assert Abs(0) == 0
assert Abs(1) == 1
assert Abs(-1) == 1
assert Abs(I) == 1
assert Abs(-I) == 1
assert Abs(nan) == nan
assert Abs(I * pi) == pi
assert Abs(-I * pi) == pi
assert Abs(I * x) == Abs(x)
assert Abs(-I * x) == Abs(x)
assert Abs(-2*x) == 2*Abs(x)
assert Abs(-2.0*x) == 2.0*Abs(x)
assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
assert Abs(conjugate(x)) == Abs(x)
assert conjugate(Abs(x)) == Abs(x)
a = cos(1)**2 + sin(1)**2 - 1
assert Abs(a*x).series(x).simplify() == 0
a = Symbol('a', positive=True)
assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)
x = Symbol('x', extended_real=True)
n = Symbol('n', integer=True)
assert Abs((-1)**n) == 1
assert x**(2*n) == Abs(x)**(2*n)
assert Abs(x).diff(x) == sign(x)
assert abs(x) == Abs(x) # Python built-in
assert Abs(x)**3 == x**2*Abs(x)
assert Abs(x)**4 == x**4
assert (
Abs(x)**(3*n)).args == (Abs(x), 3*n) # leave symbolic odd unchanged
assert (1/Abs(x)).args == (Abs(x), -1)
assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
assert Abs(x)**-3 == Abs(x)/(x**4)
assert Abs(x**3) == x**2*Abs(x)
assert Abs(x**pi) == Abs(x**pi, evaluate=False)
x = Symbol('x', imaginary=True)
assert Abs(x).diff(x) == -sign(x)
pytest.raises(ArgumentIndexError, lambda: Abs(z).fdiff(2))
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when you can and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert abs(eq) == 0
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = cbrt(expand(q**3))
d = p - q
assert abs(d) == 0
assert Abs(4*exp(pi*I/4)) == 4
assert Abs(3**(2 + I)) == 9
assert Abs((-3)**(1 - I)) == 3*exp(pi)
assert Abs(oo) is oo
assert Abs(-oo) is oo
assert Abs(oo + I) is oo
assert Abs(oo + I*oo) is oo
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert abs(sign(z)) == Abs(sign(z), evaluate=False)
def test_expand_log():
t = Symbol('t', positive=True)
# after first expansion, -2*log(2) + log(4); then 0 after second
assert expand(log(t**2) - log(t**2/4) - 2*log(2)) == 0
def test_inverse_mellin_transform():
from diofant import (sin, simplify, Max, Min, expand, powsimp, exp_polar,
cos, cot)
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1 / x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
(-x/2 + 1/(2*x))*Heaviside(-x + 1)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1) * exp(-x) / x
# test factorisation of polys
r = symbols('r', extended_real=True)
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
).subs(x, r).rewrite(sin).simplify() \
== sin(r)*Heaviside(1 - exp(-r))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s / _a) * factorial(s / _a) / s, s, x,
(0, oo)) == exp(-_b * x**_a)
assert IMT(factorial(_a / _b + s / _b) / (_a + s), s, x,
(-_a, oo)) == x**_a * exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False),
force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', extended_real=True, finite=True)
assert IMT(-1 / (nu + s), s, x, (-oo, None)) == x**nu * Heaviside(x - 1)
assert IMT(1 / (nu + s), s, x, (None, oo)) == x**nu * Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
* gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
* gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/ gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24 / s**5, s, x, (0, oo)) == log(x)**4 * Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi / (s * sin(pi * s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi / (s * sin(pi * s / 2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi / (s * sin(2 * pi * s)), s, x,
(-Rational(1, 2), 0)) == log(sqrt(x) + 1)
assert IMT(pi / (s * sin(pi * s)), s, x, (0, 1)) == log(1 + 1 / x)
# TODO
def mysimp(expr):
from diofant import expand, logcombine, powsimp
return expand(powsimp(logcombine(expr, force=True),
force=True,
deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi / (s * tan(pi * s)), s, x, (-1, 0)))) in [
log(1 - x) * Heaviside(1 - x) + log(x - 1) * Heaviside(x - 1),
log(x) * Heaviside(x - 1) + log(1 - 1 / x) * Heaviside(x - 1) +
log(-x + 1) * Heaviside(-x + 1)
]
# test passing cot
assert mysimp(IMT(pi * cot(pi * s) / s, s, x, (0, 1))) in [
log(1 / x - 1) * Heaviside(1 - x) + log(1 - 1 / x) * Heaviside(x - 1),
-log(x) * Heaviside(-x + 1) + log(1 - 1 / x) * Heaviside(x - 1) +
log(-x + 1) * Heaviside(-x + 1),
]
# 8.4.14
assert IMT(-gamma(s + Rational(1, 2))/(sqrt(pi)*s), s, x, (-Rational(1, 2), 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(Rational(1, 2) - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(Rational(1, 2) - 2*s)
/ (gamma(Rational(1, 2) - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, Rational(1, 4)))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), Rational(1, 2)))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, Rational(1, 2)))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
* gamma(a/2 + b/2 - s + 1)),
s, x, (-(re(a) + re(b))/2, Rational(1, 2)))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), Rational(1, 2)))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi / cos(pi * s), s, x,
(0, Rational(1, 2))) == sqrt(x) / (x + 1)
def mysimp(expr):
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
def test_minimal_polynomial():
assert minimal_polynomial(-7, x) == x + 7
assert minimal_polynomial(-1, x) == x + 1
assert minimal_polynomial(0, x) == x
assert minimal_polynomial(1, x) == x - 1
assert minimal_polynomial(7, x) == x - 7
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(5), x) == x**2 - 5
assert minimal_polynomial(sqrt(6), x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2), x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5), x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6), x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3, x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6, x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7, x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3, x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6, x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7, x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I + sqrt(2 + I),
x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6),
x) == x**4 - 22 * x**2 - 48 * x - 23
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a, x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a), x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True,
compose=False) == Poly(x**2 - 2)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(b, x) == x**2 - 3
assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3)
assert minimal_polynomial(sqrt(a / 2 + 17),
x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 + 17),
x) == 4 * x**4 - 136 * x**2 + 1153
a, b = sqrt(2) / 3 + 7, AlgebraicNumber(sqrt(2) / 3 + 7)
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
assert minimal_polynomial(a**Q(3, 2),
x) == 729 * x**4 - 506898 * x**2 + 84604519
# issue 5994
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 *
(Rational(-1, 17280000) + sqrt(15) * I / 28800000)**Rational(1, 3)) +
2 *
(Rational(-1, 17280000) + sqrt(15) * I / 28800000)**Rational(1, 3))))
assert minimal_polynomial(eq, x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex, x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex, x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = (expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))**Rational(1, 3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(
-sqrt(5) / 2 - S.Half + (-sqrt(5) / 2 - S.Half)**2, x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3, x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I, x,
compose=False) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1, x,
compose=False) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
x,
compose=False) == x**4 + 18 * x**2 + 49
def test_meijerint():
s, t, mu = symbols('s t mu', extended_real=True)
assert integrate(
meijerg([], [], [0], [], s * t) *
meijerg([], [], [mu / 2], [-mu / 2], t**2 / 4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(
integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=False), Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1)**3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (erf(mu / (2 * sigma)) + 1)
assert c
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(
exp(-((x - mu) / sigma)**2 / 2) / sqrt(2 * pi * sigma**2), x, -oo,
oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0,
oo) == (Rational(1, 2), True)
# Test a bug
def res(n):
return (1 / (1 + x**2)).diff(x, n).subs({x: 1}) * (-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo),
meijerg=True)) == sqrt(2) * sin(a + pi / 4) / 2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
* meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
* gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, Rational(1, 2))/4).expand()
# Test hyperexpand bug.
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + Rational(1, 2),
alpha/2 + 1)), ((0, 0, Rational(1, 2)), (-Rational(1, 2),)), alpha**2/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - Rational(1, 2))*((-1)**s + 1)*gamma(s/2 + Rational(1, 2))/2
def test_minimal_polynomial():
assert minimal_polynomial(-7)(x) == x + 7
assert minimal_polynomial(-1)(x) == x + 1
assert minimal_polynomial( 0)(x) == x
assert minimal_polynomial( 1)(x) == x - 1
assert minimal_polynomial( 7)(x) == x - 7
assert minimal_polynomial(Rational(1, 3), method='groebner')(x) == 3*x - 1
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(pi, method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(sin(sqrt(2)), method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(2**pi, method='groebner'))
pytest.raises(ValueError, lambda: minimal_polynomial(1, method='spam'))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(5))(x) == x**2 - 5
assert minimal_polynomial(sqrt(6))(x) == x**2 - 6
assert minimal_polynomial(2*sqrt(2))(x) == x**2 - 8
assert minimal_polynomial(3*sqrt(5))(x) == x**2 - 45
assert minimal_polynomial(4*sqrt(6))(x) == x**2 - 96
assert minimal_polynomial(2*sqrt(2) + 3)(x) == x**2 - 6*x + 1
assert minimal_polynomial(3*sqrt(5) + 6)(x) == x**2 - 12*x - 9
assert minimal_polynomial(4*sqrt(6) + 7)(x) == x**2 - 14*x - 47
assert minimal_polynomial(2*sqrt(2) - 3)(x) == x**2 + 6*x + 1
assert minimal_polynomial(3*sqrt(5) - 6)(x) == x**2 + 12*x - 9
assert minimal_polynomial(4*sqrt(6) - 7)(x) == x**2 + 14*x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)))(x) == x**4 - 2*x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)))(x) == x**8 - 10*x**4 + 49
assert minimal_polynomial(2*I + sqrt(2 + I))(x) == x**4 + 4*x**2 + 8*x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3))(x) == x**4 - 10*x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6))(x) == x**4 - 22*x**2 - 48*x - 23
e = 1/sqrt(sqrt(1 + sqrt(3)) - 4)
assert minimal_polynomial(e) == minimal_polynomial(e, method='groebner')
assert minimal_polynomial(e)(x) == (222*x**8 + 240*x**6 +
94*x**4 + 16*x**2 + 1)
a = 1 - 9*sqrt(2) + 7*sqrt(3)
assert minimal_polynomial(1/a)(x) == 392*x**4 - 1232*x**3 + 612*x**2 + 4*x - 1
assert minimal_polynomial(1/sqrt(a))(x) == 392*x**8 - 1232*x**6 + 612*x**4 + 4*x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1)))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(2)) == PurePoly(x**2 - 2)
assert minimal_polynomial(sqrt(2), method='groebner') == PurePoly(x**2 - 2)
a = sqrt(2)
b = sqrt(3)
assert minimal_polynomial(b)(x) == x**2 - 3
assert minimal_polynomial(a) == PurePoly(x**2 - 2)
assert minimal_polynomial(b) == PurePoly(x**2 - 3)
assert minimal_polynomial(sqrt(a)) == PurePoly(x**4 - 2)
assert minimal_polynomial(a + 1) == PurePoly(x**2 - 2*x - 1)
assert minimal_polynomial(sqrt(a/2 + 17))(x) == 2*x**4 - 68*x**2 + 577
assert minimal_polynomial(sqrt(b/2 + 17))(x) == 4*x**4 - 136*x**2 + 1153
# issue diofant/diofant#431
K = QQ.algebraic_field(sqrt(2))
theta = K.to_expr(K([Rational(1, 2), 17]))
assert minimal_polynomial(theta)(x) == 2*x**2 - 68*x + 577
K = QQ.algebraic_field(RootOf(x**7 + x - 1, 3))
theta = K.to_expr(K([1, 2, 0, 0, 1]))
ans = minimal_polynomial(theta)(x)
assert ans == (x**7 - 7*x**6 + 19*x**5 - 27*x**4 + 63*x**3 -
115*x**2 + 82*x - 147)
assert minimal_polynomial(theta, method='groebner')(x) == ans
K = QQ.algebraic_field(RootOf(x**5 + 5*x - 1, 2))
theta = K.to_expr(K([1, -1, 1]))
ans = (x**30 - 15*x**28 - 10*x**27 + 135*x**26 + 330*x**25 - 705*x**24 -
150*x**23 + 3165*x**22 - 6850*x**21 + 7182*x**20 + 3900*x**19 +
4435*x**18 + 11970*x**17 - 259725*x**16 - 18002*x**15 +
808215*x**14 - 200310*x**13 - 647115*x**12 + 299280*x**11 -
1999332*x**10 + 910120*x**9 + 2273040*x**8 - 5560320*x**7 +
5302000*x**6 - 2405376*x**5 + 1016640*x**4 - 804480*x**3 +
257280*x**2 - 53760*x + 1280)
assert minimal_polynomial(sqrt(theta) + root(theta, 3),
method='groebner')(x) == ans
K1 = QQ.algebraic_field(RootOf(x**3 + 4*x - 15, 1))
K2 = QQ.algebraic_field(RootOf(x**3 - x + 1, 0))
theta = sqrt(1 + 1/(K1.to_expr(K1([1, 0, 1])) +
1/(sqrt(3) + K2.to_expr(K2([1, 2, -1])))))
ans = (2262264837876687263*x**36 - 38939909597855051866*x**34 +
315720420314462950715*x**32 - 1601958657418182606114*x**30 +
5699493671077371036494*x**28 - 15096777696140985506150*x**26 +
30847690820556462893974*x**24 - 49706549068200640994022*x**22 +
64013601241426223813103*x**20 - 66358713088213594372990*x**18 +
55482571280904904971976*x**16 - 37309340229165533529076*x**14 +
20016999328983554519040*x**12 - 8446273798231518826782*x**10 +
2738866994867366499481*x**8 - 657825125060873756424*x**6 +
110036313740049140508*x**4 - 11416087328869938298*x**2 +
551322649782053543)
assert minimal_polynomial(theta)(x) == ans
a = sqrt(2)/3 + 7
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)))(x) == f
assert minimal_polynomial(a**Rational(3, 2))(x) == 729*x**4 - 506898*x**2 + 84604519
K = QQ.algebraic_field(RootOf(x**3 + x - 1, 0))
a = K.to_expr(K([1, 0]))
assert minimal_polynomial(1/a**2)(x) == x**3 - x**2 - 2*x - 1
# issue sympy/sympy#5994
eq = (-1/(800*sqrt(Rational(-1, 240) + 1/(18000*cbrt(Rational(-1, 17280000) +
sqrt(15)*I/28800000)) + 2*cbrt(Rational(-1, 17280000) +
sqrt(15)*I/28800000))))
assert minimal_polynomial(eq)(x) == 8000*x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex)(x)
assert mp == x**4 - 4*x**3 - 4*x**2 + 16*x - 8
ex = 1/(1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex)(x)
assert mp == 8*x**4 - 16*x**3 + 4*x**2 + 4*x - 1
p = cbrt(expand((1 + sqrt(2) - 2*sqrt(3) + sqrt(7))**3))
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 8*x**7 - 56*x**6 + 448*x**5 + 480*x**4 - 5056*x**3 + 1984*x**2 + 7424*x - 3008
p = expand((1 + sqrt(2) - 2*sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 512*x**7 - 118208*x**6 + 31131136*x**5 + 647362560*x**4 - 56026611712*x**3 + 116994310144*x**2 + 404854931456*x - 27216576512
assert minimal_polynomial(-sqrt(5)/2 - Rational(1, 2) + (-sqrt(5)/2 - Rational(1, 2))**2)(x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a*sqrt(2) + a)**3)(x) == x**2 - 198*x + 1
p = 1/(1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(p, method='groebner')(x) == 8*x**4 - 16*x**3 + 4*x**2 + 4*x - 1
p = 2/(1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(p, method='groebner')(x) == x**4 - 4*x**3 + 2*x**2 + 4*x - 2
assert minimal_polynomial(1 + sqrt(2)*I, method='groebner')(x) == x**2 - 2*x + 3
assert minimal_polynomial(1/(1 + sqrt(2)) + 1, method='groebner')(x) == x**2 - 2
assert minimal_polynomial(sqrt(2)*I + I*(1 + sqrt(2)),
method='groebner')(x) == x**4 + 18*x**2 + 49
assert minimal_polynomial(exp_polar(0))(x) == x - 1
def test_expand():
assert expand((A * B)**2) == A * B * A * B
assert expand(A * B - B * A) == A * B - B * A
assert expand((A * B / A)**2) == A * B * B / A
assert expand(B * A * (A + B) * B) == B * A**2 * B + B * A * B**2
assert expand(B * A * (A + C) * B) == B * A**2 * B + B * A * C * B
def test_inverse_mellin_transform():
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
(-x/2 + 1/(2*x))*Heaviside(-x + 1)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x
# test factorisation of polys
r = symbols('r', extended_real=True)
assert (IMT(1/(s**2 + 1), s, exp(-x),
(None, oo)).subs({x: r}).rewrite(sin).simplify() ==
sin(r)*Heaviside(1 - exp(-r)))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a)
assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', real=True)
assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
* gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
* gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/ gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi/(s*sin(2*pi*s)), s, x, (-Rational(1, 2), 0)) == log(sqrt(x) + 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x)
# TODO
def mysimp(expr):
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [
log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1),
log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1)]
# test passing cot
assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [
log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1),
-log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1), ]
# 8.4.14
assert IMT(-gamma(s + Rational(1, 2))/(sqrt(pi)*s), s, x, (-Rational(1, 2), 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(Rational(1, 2) - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(Rational(1, 2) - 2*s)
/ (gamma(Rational(1, 2) - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, Rational(1, 4)))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), Rational(1, 2)))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, Rational(1, 2)))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
* gamma(a/2 + b/2 - s + 1)),
s, x, (-(re(a) + re(b))/2, Rational(1, 2)))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), Rational(1, 2)))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi/cos(pi*s), s, x, (0, Rational(1, 2))) == sqrt(x)/(x + 1)
