def test_contains_4():
assert Order(sin(1 / x**2)).contains(Order(cos(1 / x**2))) is True
assert Order(cos(1 / x**2)).contains(Order(sin(1 / x**2))) is True
def test_Order():
assert precedence(Order(x)) == PRECEDENCE["Atom"]
def test_contains_3():
assert Order(x * y**2).contains(Order(x**2 * y)) is None
assert Order(x**2 * y).contains(Order(x * y**2)) is None
def test_free_symbols():
assert Order(1).free_symbols == set()
assert Order(x).free_symbols == {x}
assert Order(1, x).free_symbols == {x}
assert Order(x * y).free_symbols == {x, y}
assert Order(x, x, y).free_symbols == {x, y}
def test_simple_1():
o = Rational(0)
assert Order(2 * x) == Order(x)
assert Order(x) * 3 == Order(x)
assert -28 * Order(x) == Order(x)
assert Order(Order(x)) == Order(x)
assert Order(Order(x), y) == Order(Order(x), x, y)
assert Order(-23) == Order(1)
assert Order(exp(x)) == Order(1, x)
assert Order(exp(1 / x)).expr == exp(1 / x)
assert Order(x * exp(1 / x)).expr == x * exp(1 / x)
assert Order(x**(o / 3)).expr == x**(o / 3)
assert Order(x**(o * Rational(5, 3))).expr == x**(o * Rational(5, 3))
assert Order(x**2 + x + y, x) == O(1, x)
assert Order(x**2 + x + y, y) == O(1, y)
raises(ValueError, lambda: Order(exp(x), x, x))
raises(TypeError, lambda: Order(x, 2 - x))
def test_multivar_0a():
assert Order(exp(1 / x) * exp(1 / y)).expr == exp(1 / x) * exp(1 / y)
def test_multivar_mul_1():
assert Order(x + y) * x == Order(x**2 + y * x, x, y)
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
from sympy.functions import RisingFactorial
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite([factorial, RisingFactorial], gamma)
if e.is_Mul:
if abs(z0) is S.Infinity:
e = factor_terms(e)
e = e.rewrite(fibonacci, GoldenRatio)
ok = lambda w: (z in w.free_symbols and any(
a.is_polynomial(z) or any(
z in m.free_symbols and m.is_polynomial(z)
for m in Mul.make_args(a)) for a in Add.make_args(w)))
if all(ok(w) for w in e.as_numer_denom()):
u = Dummy(positive=True)
if z0 is S.NegativeInfinity:
inve = e.subs(z, -1 / u)
else:
inve = e.subs(z, 1 / u)
try:
r = limit(inve.as_leading_term(u), u, S.Zero, "+")
if isinstance(r, Limit):
return self
else:
return r
except ValueError:
pass
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError(
"The limit does not exist since "
"left hand limit = %s and right hand limit = %s" %
(l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
def truncate(self, n=6):
ffps = self.ffps
pt_xk = ffps.xk.coeff(n)
x, x0 = ffps.x, ffps.x0
return self.polynomial(n) + Order(pt_xk, (x, x0))
def doit(self, **hints):
"""Evaluates limit"""
from sympy.series.limitseq import limit_seq
from sympy.functions import RisingFactorial
e, z, z0, dir = self.args
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_positive:
e = e.rewrite([factorial, RisingFactorial], gamma)
if e.is_Mul:
if abs(z0) is S.Infinity:
e = factor_terms(e)
ok = lambda w: (z in w.free_symbols and any(
a.is_polynomial(z) or any(
z in m.free_symbols and m.is_polynomial(z)
for m in Mul.make_args(a)) for a in Add.make_args(w)))
if all(ok(w) for w in e.as_numer_denom()):
u = Dummy(positive=(z0 is S.Infinity))
inve = e.subs(z, 1 / u)
r = limit(inve.as_leading_term(u), u, S.Zero,
"+" if z0 is S.Infinity else "-")
if isinstance(r, Limit):
return self
else:
return r
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
try:
r = gruntz(e, z, z0, dir)
if r is S.NaN:
raise PoleError()
except (PoleError, ValueError):
r = heuristics(e, z, z0, dir)
if r is None:
return self
except NotImplementedError:
# Trying finding limits of sequences
if hints.get('sequence', True) and z0 is S.Infinity:
trials = hints.get('trials', 5)
r = limit_seq(e, z, trials)
if r is None:
raise NotImplementedError()
else:
raise NotImplementedError()
return r
def test_series():
e = Symbol("e")
x = Symbol("x")
for c in (Limit, Limit(e, x, 1), Order, Order(e)):
check(c)
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
if z0 is S.NaN:
return S.NaN
if e.has(*_illegal):
return self
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
cdir = 0
if str(dir) == "+":
cdir = 1
elif str(dir) == "-":
cdir = -1
def set_signs(expr):
if not expr.args:
return expr
newargs = tuple(set_signs(arg) for arg in expr.args)
if newargs != expr.args:
expr = expr.func(*newargs)
abs_flag = isinstance(expr, Abs)
sign_flag = isinstance(expr, sign)
if abs_flag or sign_flag:
sig = limit(expr.args[0], z, z0, dir)
if sig.is_zero:
sig = limit(1/expr.args[0], z, z0, dir)
if sig.is_extended_real:
if (sig < 0) == True:
return -expr.args[0] if abs_flag else S.NegativeOne
elif (sig > 0) == True:
return expr.args[0] if abs_flag else S.One
return expr
if e.has(Float):
# Convert floats like 0.5 to exact SymPy numbers like S.Half, to
# prevent rounding errors which can lead to unexpected execution
# of conditional blocks that work on comparisons
# Also see comments in https://github.com/sympy/sympy/issues/19453
from sympy.simplify.simplify import nsimplify
e = nsimplify(e)
e = set_signs(e)
if e.is_meromorphic(z, z0):
if abs(z0) is S.Infinity:
newe = e.subs(z, 1/z)
# cdir changes sign as oo- should become 0+
cdir = -cdir
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir=cdir)
except ValueError:
pass
else:
if ex > 0:
return S.Zero
elif ex == 0:
return coeff
if cdir == 1 or not(int(ex) & 1):
return S.Infinity*sign(coeff)
elif cdir == -1:
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
if abs(z0) is S.Infinity:
if e.is_Mul:
e = factor_terms(e)
newe = e.subs(z, 1/z)
# cdir changes sign as oo- should become 0+
cdir = -cdir
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir=cdir)
except (ValueError, NotImplementedError, PoleError):
# The NotImplementedError catching is for custom functions
from sympy.simplify.powsimp import powsimp
e = powsimp(e)
if e.is_Pow:
r = self.pow_heuristics(e)
if r is not None:
return r
else:
if coeff.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity):
return self
if not coeff.has(z):
if ex.is_positive:
return S.Zero
elif ex == 0:
return coeff
elif ex.is_negative:
if ex.is_integer:
if cdir == 1 or ex.is_even:
return S.Infinity*sign(coeff)
elif cdir == -1:
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
else:
if cdir == 1:
return S.Infinity*sign(coeff)
elif cdir == -1:
return S.Infinity*sign(coeff)*S.NegativeOne**ex
else:
return S.ComplexInfinity
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite(factorial, gamma)
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError("The limit does not exist since "
"left hand limit = %s and right hand limit = %s"
% (l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
def test_basic1():
assert limit(x, x, oo) is oo
assert limit(x, x, -oo) is -oo
assert limit(-x, x, oo) is -oo
assert limit(x**2, x, -oo) is oo
assert limit(-x**2, x, oo) is -oo
assert limit(x * log(x), x, 0, dir="+") == 0
assert limit(1 / x, x, oo) == 0
assert limit(exp(x), x, oo) is oo
assert limit(-exp(x), x, oo) is -oo
assert limit(exp(x) / x, x, oo) is oo
assert limit(1 / x - exp(-x), x, oo) == 0
assert limit(x + 1 / x, x, oo) is oo
assert limit(x - x**2, x, oo) is -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == Limit((x + 1)**oo, x, 0)
assert limit((1 + x)**oo, x, 0, dir='-') == Limit((x + 1)**oo,
x,
0,
dir='-')
assert limit((1 + x + y)**oo, x, 0, dir='-') == Limit((x + y + 1)**oo,
x,
0,
dir='-')
assert limit(y / x / log(x), x, 0) == -oo * sign(y)
assert limit(cos(x + y) / x, x, 0) == sign(cos(y)) * oo
assert limit(gamma(1 / x + 3), x, oo) == 2
assert limit(S.NaN, x, -oo) is S.NaN
assert limit(Order(2) * x, x, S.NaN) is S.NaN
assert limit(1 / (x - 1), x, 1, dir="+") is oo
assert limit(1 / (x - 1), x, 1, dir="-") is -oo
assert limit(1 / (5 - x)**3, x, 5, dir="+") is -oo
assert limit(1 / (5 - x)**3, x, 5, dir="-") is oo
assert limit(1 / sin(x), x, pi, dir="+") is -oo
assert limit(1 / sin(x), x, pi, dir="-") is oo
assert limit(1 / cos(x), x, pi / 2, dir="+") is -oo
assert limit(1 / cos(x), x, pi / 2, dir="-") is oo
assert limit(1 / tan(x**3), x, (2 * pi)**Rational(1, 3), dir="+") is oo
assert limit(1 / tan(x**3), x, (2 * pi)**Rational(1, 3), dir="-") is -oo
assert limit(1 / cot(x)**3, x, (pi * Rational(3, 2)), dir="+") is -oo
assert limit(1 / cot(x)**3, x, (pi * Rational(3, 2)), dir="-") is oo
# test bi-directional limits
assert limit(sin(x) / x, x, 0, dir="+-") == 1
assert limit(x**2, x, 0, dir="+-") == 0
assert limit(1 / x**2, x, 0, dir="+-") is oo
# test failing bi-directional limits
assert limit(1 / x, x, 0, dir="+-") is zoo
# approaching 0
# from dir="+"
assert limit(1 + 1 / x, x, 0) is oo
# from dir='-'
# Add
assert limit(1 + 1 / x, x, 0, dir='-') is -oo
# Pow
assert limit(x**(-2), x, 0, dir='-') is oo
assert limit(x**(-3), x, 0, dir='-') is -oo
assert limit(1 / sqrt(x), x, 0, dir='-') == (-oo) * I
assert limit(x**2, x, 0, dir='-') == 0
assert limit(sqrt(x), x, 0, dir='-') == 0
assert limit(x**-pi, x, 0, dir='-') == oo * sign((-1)**(-pi))
assert limit((1 + cos(x))**oo, x, 0) == Limit((cos(x) + 1)**oo, x, 0)
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
from sympy import sign
from sympy.functions import RisingFactorial
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
if e.is_meromorphic(z, z0):
if abs(z0) is S.Infinity:
newe = e.subs(z, -1/z)
else:
newe = e.subs(z, z + z0)
try:
coeff, exp = newe.leadterm(z)
if coeff is not S.ComplexInfinity:
if exp > 0:
return S.Zero
elif exp == 0:
return coeff
if str(dir) == "+" or not(int(exp) & 1):
return S.Infinity*sign(coeff)
elif str(dir) == "-":
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
except ValueError:
pass
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite([factorial, RisingFactorial], gamma)
if e.is_Mul and abs(z0) is S.Infinity:
e = factor_terms(e)
u = Dummy('u', positive=True)
if z0 is S.NegativeInfinity:
inve = e.subs(z, -1/u)
else:
inve = e.subs(z, 1/u)
try:
f = inve.as_leading_term(u)
if f.is_meromorphic(u, S.Zero):
r = limit(f, u, S.Zero, "+")
if isinstance(r, Limit):
return self
else:
return r
except (ValueError, NotImplementedError, PoleError):
pass
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError("The limit does not exist since "
"left hand limit = %s and right hand limit = %s"
% (l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
def test_contains():
assert Order(1, x) not in Order(1)
assert Order(1) in Order(1, x)
raises(TypeError, lambda: Order(x * y**2) in Order(x**2 * y))
def test_order_oo():
x = Symbol('x', positive=True)
assert Order(x)*oo != Order(1, x)
assert limit(oo/(x**2 - 4), x, oo) is oo
def test_add_1():
assert Order(x + x) == Order(x)
assert Order(3 * x - 2 * x**2) == Order(x)
assert Order(1 + x) == Order(1, x)
assert Order(1 + 1 / x) == Order(1 / x)
assert Order(log(x) + 1 / log(x)) == Order(log(x))
assert Order(exp(1 / x) + x) == Order(exp(1 / x))
assert Order(exp(1 / x) + 1 / x**20) == Order(exp(1 / x))
def test_as_expr_variables():
assert Order(x).as_expr_variables(None) == (x, ((x, 0), ))
assert Order(x).as_expr_variables(((x, 0), )) == (x, ((x, 0), ))
assert Order(y).as_expr_variables(((x, 0), )) == (y, ((x, 0), (y, 0)))
assert Order(y).as_expr_variables(
((x, 0), (y, 0))) == (y, ((x, 0), (y, 0)))
def test_multivar_2():
assert Order(x**2 * y + y**2 * x, x, y).expr == x**2 * y + y**2 * x
def test_contains_0():
assert Order(1, x).contains(Order(1, x))
assert Order(1, x).contains(Order(1))
assert Order(1).contains(Order(1, x)) is False
def test_multivar_3():
assert (Order(x) + Order(y)).args in [(Order(x), Order(y)),
(Order(y), Order(x))]
assert Order(x) + Order(y) + Order(x + y) == Order(x + y)
assert (Order(x**2 * y) + Order(y**2 * x)).args in [
(Order(x * y**2), Order(y * x**2)), (Order(y * x**2), Order(x * y**2))
]
assert (Order(x**2 * y) + Order(y * x)) == Order(x * y)
def test_contains_1():
assert Order(x).contains(Order(x))
assert Order(x).contains(Order(x**2))
assert not Order(x**2).contains(Order(x))
assert not Order(x).contains(Order(1 / x))
assert not Order(1 / x).contains(Order(exp(1 / x)))
assert not Order(x).contains(Order(exp(1 / x)))
assert Order(1 / x).contains(Order(x))
assert Order(exp(1 / x)).contains(Order(x))
assert Order(exp(1 / x)).contains(Order(1 / x))
assert Order(exp(1 / x)).contains(Order(exp(1 / x)))
assert Order(exp(2 / x)).contains(Order(exp(1 / x)))
assert not Order(exp(1 / x)).contains(Order(exp(2 / x)))
def test_eval():
assert Order(x).subs(Order(x), 1) == 1
assert Order(x).subs(x, y) == Order(y)
assert Order(x).subs(y, x) == Order(x)
assert Order(x).subs(x, x + y) == Order(x + y, (x, -y))
assert (O(1)**x).is_Pow
def test_contains_2():
assert Order(x).contains(Order(y)) is None
assert Order(x).contains(Order(y * x))
assert Order(y * x).contains(Order(x))
assert Order(y).contains(Order(x * y))
assert Order(x).contains(Order(y**2 * x))
def test_order_conjugate_transpose():
x = Symbol('x', real=True)
y = Symbol('y', imaginary=True)
assert conjugate(Order(x)) == Order(conjugate(x))
assert conjugate(Order(y)) == Order(conjugate(y))
assert conjugate(Order(x**2)) == Order(conjugate(x)**2)
assert conjugate(Order(y**2)) == Order(conjugate(y)**2)
assert transpose(Order(x)) == Order(transpose(x))
assert transpose(Order(y)) == Order(transpose(y))
assert transpose(Order(x**2)) == Order(transpose(x)**2)
assert transpose(Order(y**2)) == Order(transpose(y)**2)
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
from sympy import Abs, exp, log, sign
from sympy.calculus.util import AccumBounds
from sympy.functions import RisingFactorial
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
cdir = 0
if str(dir) == "+":
cdir = 1
elif str(dir) == "-":
cdir = -1
def remove_abs(expr):
if not expr.args:
return expr
newargs = tuple(remove_abs(arg) for arg in expr.args)
if newargs != expr.args:
expr = expr.func(*newargs)
if isinstance(expr, Abs):
sig = limit(expr.args[0], z, z0, dir)
if sig.is_zero:
sig = limit(1/expr.args[0], z, z0, dir)
if sig.is_extended_real:
if (sig < 0) == True:
return -expr.args[0]
elif (sig > 0) == True:
return expr.args[0]
return expr
e = remove_abs(e)
if e.is_meromorphic(z, z0):
if abs(z0) is S.Infinity:
newe = e.subs(z, -1/z)
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir)
except ValueError:
pass
else:
if ex > 0:
return S.Zero
elif ex == 0:
return coeff
if str(dir) == "+" or not(int(ex) & 1):
return S.Infinity*sign(coeff)
elif str(dir) == "-":
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite([factorial, RisingFactorial], gamma)
if e.is_Mul and abs(z0) is S.Infinity:
e = factor_terms(e)
u = Dummy('u', positive=True)
if z0 is S.NegativeInfinity:
inve = e.subs(z, -1/u)
else:
inve = e.subs(z, 1/u)
try:
f = inve.as_leading_term(u).gammasimp()
if f.is_meromorphic(u, S.Zero):
r = limit(f, u, S.Zero, "+")
if isinstance(r, Limit):
return self
else:
return r
except (ValueError, NotImplementedError, PoleError):
pass
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
if e.is_Pow:
if e.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.NaN):
return self
b1, e1 = e.base, e.exp
f1 = e1*log(b1)
if f1.is_meromorphic(z, z0):
res = limit(f1, z, z0)
return exp(res)
ex_lim = limit(e1, z, z0)
base_lim = limit(b1, z, z0)
if base_lim is S.One:
if ex_lim in (S.Infinity, S.NegativeInfinity):
res = limit(e1*(b1 - 1), z, z0)
return exp(res)
elif ex_lim.is_real:
return S.One
if base_lim in (S.Zero, S.Infinity, S.NegativeInfinity) and ex_lim is S.Zero:
res = limit(f1, z, z0)
return exp(res)
if base_lim is S.NegativeInfinity:
if ex_lim is S.NegativeInfinity:
return S.Zero
if ex_lim is S.Infinity:
return S.ComplexInfinity
if not isinstance(base_lim, AccumBounds) and not isinstance(ex_lim, AccumBounds):
res = base_lim**ex_lim
if res is not S.ComplexInfinity and not res.is_Pow:
return res
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError("The limit does not exist since "
"left hand limit = %s and right hand limit = %s"
% (l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r