def test_match_real_imag():
x, y = symbols('x,y', real=True)
i = Symbol('i', imaginary=True)
assert match_real_imag(S.One) == (1, 0)
assert match_real_imag(I) == (0, 1)
assert match_real_imag(3 - 5 * I) == (3, -5)
assert match_real_imag(-sqrt(3) + S.Half * I) == (-sqrt(3), S.Half)
assert match_real_imag(x + y * I) == (x, y)
assert match_real_imag(x * I + y * I) == (0, x + y)
assert match_real_imag((x + y) * I) == (0, x + y)
assert match_real_imag(Rational(-2, 3) * i * I) == (None, None)
assert match_real_imag(1 - 2 * i) == (None, None)
assert match_real_imag(sqrt(2) * (3 - 5 * I)) == (None, None)
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
n = f.variables[0]
if expr == abs(n):
return S.Naturals0
# f(x) + c and f(-x) + c cover the same integers
# so choose the form that has the fewest negatives
c = f(0)
fx = f(n) - c
f_x = f(-n) - c
neg_count = lambda e: sum(_.could_extract_minus_sign()
for _ in Add.make_args(e))
if neg_count(f_x) < neg_count(fx):
expr = f_x + c
a = Wild('a', exclude=[n])
b = Wild('b', exclude=[n])
match = expr.match(a * n + b)
if match and match[a] and (not match[a].atoms(Float)
and not match[b].atoms(Float)):
# canonical shift
a, b = match[a], match[b]
if a in [1, -1]:
# drop integer addends in b
nonint = []
for bi in Add.make_args(b):
if not bi.is_integer:
nonint.append(bi)
b = Add(*nonint)
if b.is_number and a.is_real:
# avoid Mod for complex numbers, #11391
br, bi = match_real_imag(b)
if br and br.is_comparable and a.is_comparable:
br %= a
b = br + S.ImaginaryUnit * bi
elif b.is_number and a.is_imaginary:
br, bi = match_real_imag(b)
ai = a / S.ImaginaryUnit
if bi and bi.is_comparable and ai.is_comparable:
bi %= ai
b = br + S.ImaginaryUnit * bi
expr = a * n + b
if expr != f.expr:
return ImageSet(Lambda(n, expr), S.Integers)
def eval(cls, arg):
from sympy import atan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit * AccumBounds(-S.Pi / 2, S.Pi / 2)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_zero:
return S.Zero
if isinstance(arg, tanh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
return z
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor(2 * i / pi)
even = f.is_even
m = z - I * f * pi / 2
if even is True:
return m
elif even is False:
return m - I * pi / 2
def eval(cls, arg):
from sympy import asin
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return log(sqrt(2) + 1)
elif arg is S.NegativeOne:
return log(sqrt(2) - 1)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.is_zero:
return S.Zero
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * asin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if isinstance(arg, sinh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
return z
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor((i + pi / 2) / pi)
m = z - I * pi * f
even = f.is_even
if even is True:
return m
elif even is False:
return -m
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.Pi * S.ImaginaryUnit / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi * S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit: log(S.ImaginaryUnit * (1 + sqrt(2))),
-S.ImaginaryUnit: log(-S.ImaginaryUnit * (1 + sqrt(2))),
S.Half: S.Pi / 3,
-S.Half: 2 * S.Pi / 3,
sqrt(2) / 2: S.Pi / 4,
-sqrt(2) / 2: 3 * S.Pi / 4,
1 / sqrt(2): S.Pi / 4,
-1 / sqrt(2): 3 * S.Pi / 4,
sqrt(3) / 2: S.Pi / 6,
-sqrt(3) / 2: 5 * S.Pi / 6,
(sqrt(3) - 1) / sqrt(2**3): 5 * S.Pi / 12,
-(sqrt(3) - 1) / sqrt(2**3): 7 * S.Pi / 12,
sqrt(2 + sqrt(2)) / 2: S.Pi / 8,
-sqrt(2 + sqrt(2)) / 2: 7 * S.Pi / 8,
sqrt(2 - sqrt(2)) / 2: 3 * S.Pi / 8,
-sqrt(2 - sqrt(2)) / 2: 5 * S.Pi / 8,
(1 + sqrt(3)) / (2 * sqrt(2)): S.Pi / 12,
-(1 + sqrt(3)) / (2 * sqrt(2)): 11 * S.Pi / 12,
(sqrt(5) + 1) / 4: S.Pi / 5,
-(sqrt(5) + 1) / 4: 4 * S.Pi / 5
}
if arg in cst_table:
if arg.is_extended_real:
return cst_table[arg] * S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg == S.ImaginaryUnit * S.Infinity:
return S.Infinity + S.ImaginaryUnit * S.Pi / 2
if arg == -S.ImaginaryUnit * S.Infinity:
return S.Infinity - S.ImaginaryUnit * S.Pi / 2
if isinstance(arg, cosh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
from sympy.functions.elementary.complexes import Abs
return Abs(z)
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor(i / pi)
m = z - I * pi * f
even = f.is_even
if even is True:
if r.is_nonnegative:
return m
elif r.is_negative:
return -m
elif even is False:
m -= I * pi
if r.is_nonpositive:
return -m
elif r.is_positive:
return m