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ai-content-maker/.venv/Lib/site-packages/sympy/functions/elementary/trigonometric.py

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2024-05-03 04:18:51 +03:00
from typing import Tuple as tTuple, Union as tUnion
from sympy.core.add import Add
from sympy.core.cache import cacheit
from sympy.core.expr import Expr
from sympy.core.function import Function, ArgumentIndexError, PoleError, expand_mul
from sympy.core.logic import fuzzy_not, fuzzy_or, FuzzyBool, fuzzy_and
from sympy.core.mod import Mod
from sympy.core.numbers import Rational, pi, Integer, Float, equal_valued
from sympy.core.relational import Ne, Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, Dummy
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.combinatorial.numbers import bernoulli, euler
from sympy.functions.elementary.complexes import arg as arg_f, im, re
from sympy.functions.elementary.exponential import log, exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary._trigonometric_special import (
cos_table, ipartfrac, fermat_coords)
from sympy.logic.boolalg import And
from sympy.ntheory import factorint
from sympy.polys.specialpolys import symmetric_poly
from sympy.utilities.iterables import numbered_symbols
###############################################################################
########################## UTILITIES ##########################################
###############################################################################
def _imaginary_unit_as_coefficient(arg):
""" Helper to extract symbolic coefficient for imaginary unit """
if isinstance(arg, Float):
return None
else:
return arg.as_coefficient(S.ImaginaryUnit)
###############################################################################
########################## TRIGONOMETRIC FUNCTIONS ############################
###############################################################################
class TrigonometricFunction(Function):
"""Base class for trigonometric functions. """
unbranched = True
_singularities = (S.ComplexInfinity,)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational and fuzzy_not(s.args[0].is_zero):
return False
else:
return s.is_rational
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
return False
pi_coeff = _pi_coeff(self.args[0])
if pi_coeff is not None and pi_coeff.is_rational:
return True
else:
return s.is_algebraic
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _as_real_imag(self, deep=True, **hints):
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return (self.args[0].expand(deep, **hints), S.Zero)
else:
return (self.args[0], S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (re, im)
def _period(self, general_period, symbol=None):
f = expand_mul(self.args[0])
if symbol is None:
symbol = tuple(f.free_symbols)[0]
if not f.has(symbol):
return S.Zero
if f == symbol:
return general_period
if symbol in f.free_symbols:
if f.is_Mul:
g, h = f.as_independent(symbol)
if h == symbol:
return general_period/abs(g)
if f.is_Add:
a, h = f.as_independent(symbol)
g, h = h.as_independent(symbol, as_Add=False)
if h == symbol:
return general_period/abs(g)
raise NotImplementedError("Use the periodicity function instead.")
@cacheit
def _table2():
# If nested sqrt's are worse than un-evaluation
# you can require q to be in (1, 2, 3, 4, 6, 12)
# q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
# expressions with 2 or fewer sqrt nestings.
return {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
def _peeloff_pi(arg):
r"""
Split ARG into two parts, a "rest" and a multiple of $\pi$.
This assumes ARG to be an Add.
The multiple of $\pi$ returned in the second position is always a Rational.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _peeloff_pi
>>> from sympy import pi
>>> from sympy.abc import x, y
>>> _peeloff_pi(x + pi/2)
(x, 1/2)
>>> _peeloff_pi(x + 2*pi/3 + pi*y)
(x + pi*y + pi/6, 1/2)
"""
pi_coeff = S.Zero
rest_terms = []
for a in Add.make_args(arg):
K = a.coeff(pi)
if K and K.is_rational:
pi_coeff += K
else:
rest_terms.append(a)
if pi_coeff is S.Zero:
return arg, S.Zero
m1 = (pi_coeff % S.Half)
m2 = pi_coeff - m1
if m2.is_integer or ((2*m2).is_integer and m2.is_even is False):
return Add(*(rest_terms + [m1*pi])), m2
return arg, S.Zero
def _pi_coeff(arg: Expr, cycles: int = 1) -> tUnion[Expr, None]:
r"""
When arg is a Number times $\pi$ (e.g. $3\pi/2$) then return the Number
normalized to be in the range $[0, 2]$, else `None`.
When an even multiple of $\pi$ is encountered, if it is multiplying
something with known parity then the multiple is returned as 0 otherwise
as 2.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _pi_coeff
>>> from sympy import pi, Dummy
>>> from sympy.abc import x
>>> _pi_coeff(3*x*pi)
3*x
>>> _pi_coeff(11*pi/7)
11/7
>>> _pi_coeff(-11*pi/7)
3/7
>>> _pi_coeff(4*pi)
0
>>> _pi_coeff(5*pi)
1
>>> _pi_coeff(5.0*pi)
1
>>> _pi_coeff(5.5*pi)
3/2
>>> _pi_coeff(2 + pi)
>>> _pi_coeff(2*Dummy(integer=True)*pi)
2
>>> _pi_coeff(2*Dummy(even=True)*pi)
0
"""
if arg is pi:
return S.One
elif not arg:
return S.Zero
elif arg.is_Mul:
cx = arg.coeff(pi)
if cx:
c, x = cx.as_coeff_Mul() # pi is not included as coeff
if c.is_Float:
# recast exact binary fractions to Rationals
f = abs(c) % 1
if f != 0:
p = -int(round(log(f, 2).evalf()))
m = 2**p
cm = c*m
i = int(cm)
if equal_valued(i, cm):
c = Rational(i, m)
cx = c*x
else:
c = Rational(int(c))
cx = c*x
if x.is_integer:
c2 = c % 2
if c2 == 1:
return x
elif not c2:
if x.is_even is not None: # known parity
return S.Zero
return Integer(2)
else:
return c2*x
return cx
elif arg.is_zero:
return S.Zero
return None
class sin(TrigonometricFunction):
r"""
The sine function.
Returns the sine of x (measured in radians).
Explanation
===========
This function will evaluate automatically in the
case $x/\pi$ is some rational number [4]_. For example,
if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$.
Examples
========
>>> from sympy import sin, pi
>>> from sympy.abc import x
>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(1).diff(x)
0
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4
See Also
========
csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] https://dlmf.nist.gov/4.14
.. [3] https://functions.wolfram.com/ElementaryFunctions/Sin
.. [4] https://mathworld.wolfram.com/TrigonometryAngles.html
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return cos(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
from sympy.sets.sets import FiniteSet
min, max = arg.min, arg.max
d = floor(min/(2*pi))
if min is not S.NegativeInfinity:
min = min - d*2*pi
if max is not S.Infinity:
max = max - d*2*pi
if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \
is not S.EmptySet and \
AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2),
pi*Rational(7, 2))) is not S.EmptySet:
return AccumBounds(-1, 1)
elif AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \
is not S.EmptySet:
return AccumBounds(Min(sin(min), sin(max)), 1)
elif AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2), pi*Rational(8, 2))) \
is not S.EmptySet:
return AccumBounds(-1, Max(sin(min), sin(max)))
else:
return AccumBounds(Min(sin(min), sin(max)),
Max(sin(min), sin(max)))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import sinh
return S.ImaginaryUnit*sinh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if (2*pi_coeff).is_integer:
# is_even-case handled above as then pi_coeff.is_integer,
# so check if known to be not even
if pi_coeff.is_even is False:
return S.NegativeOne**(pi_coeff - S.Half)
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
# https://github.com/sympy/sympy/issues/6048
# transform a sine to a cosine, to avoid redundant code
if pi_coeff.is_Rational:
x = pi_coeff % 2
if x > 1:
return -cls((x % 1)*pi)
if 2*x > 1:
return cls((1 - x)*pi)
narg = ((pi_coeff + Rational(3, 2)) % 2)*pi
result = cos(narg)
if not isinstance(result, cos):
return result
if pi_coeff*pi != arg:
return cls(pi_coeff*pi)
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
m = m*pi
return sin(m)*cos(x) + cos(m)*sin(x)
if arg.is_zero:
return S.Zero
if isinstance(arg, asin):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return x/sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return y/sqrt(x**2 + y**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2)
if isinstance(arg, acot):
x = arg.args[0]
return 1/(sqrt(1 + 1/x**2)*x)
if isinstance(arg, acsc):
x = arg.args[0]
return 1/x
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1/x**2)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p*x**2/(n*(n - 1))
else:
return S.NegativeOne**(n//2)*x**n/factorial(n)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
if logx is not None:
arg = arg.subs(log(x), logx)
if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity):
raise PoleError("Cannot expand %s around 0" % (self))
return Function._eval_nseries(self, x, n=n, logx=logx, cdir=cdir)
def _eval_rewrite_as_exp(self, arg, **kwargs):
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
I = S.ImaginaryUnit
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) - exp(-arg*I))/(2*I)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*x**-I/2 - I*x**I /2
def _eval_rewrite_as_cos(self, arg, **kwargs):
return cos(arg - pi/2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)
return 2*tan_half/(1 + tan_half**2)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)
return Piecewise((0, And(Eq(im(arg), 0), Eq(Mod(arg, pi), 0))),
(2*cot_half/(1 + cot_half**2), True))
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self.rewrite(cos).rewrite(pow)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self.rewrite(cos).rewrite(sqrt)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1/csc(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1/sec(arg - pi/2, evaluate=False)
def _eval_rewrite_as_sinc(self, arg, **kwargs):
return arg*sinc(arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy.functions.elementary.hyperbolic import cosh, sinh
re, im = self._as_real_imag(deep=deep, **hints)
return (sin(re)*cosh(im), cos(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy.functions.special.polynomials import chebyshevt, chebyshevu
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
# TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return sx*cy + sy*cx
elif arg.is_Mul:
n, x = arg.as_coeff_Mul(rational=True)
if n.is_Integer: # n will be positive because of .eval
# canonicalization
# See https://mathworld.wolfram.com/Multiple-AngleFormulas.html
if n.is_odd:
return S.NegativeOne**((n - 1)/2)*chebyshevt(n, sin(x))
else:
return expand_mul(S.NegativeOne**(n/2 - 1)*cos(x)*
chebyshevu(n - 1, sin(x)), deep=False)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return sin(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = x0/pi
if n.is_integer:
lt = (arg - n*pi).as_leading_term(x)
return (S.NegativeOne**n)*lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in [S.Infinity, S.NegativeInfinity]:
return AccumBounds(-1, 1)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_extended_real:
return True
def _eval_is_zero(self):
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero:
return pi_mult.is_integer
def _eval_is_complex(self):
if self.args[0].is_extended_real \
or self.args[0].is_complex:
return True
class cos(TrigonometricFunction):
"""
The cosine function.
Returns the cosine of x (measured in radians).
Explanation
===========
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cos, pi
>>> from sympy.abc import x
>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(1).diff(x)
0
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4
See Also
========
sin, csc, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] https://dlmf.nist.gov/4.14
.. [3] https://functions.wolfram.com/ElementaryFunctions/Cos
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return -sin(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.functions.special.polynomials import chebyshevt
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.One
elif arg in (S.Infinity, S.NegativeInfinity):
# In this case it is better to return AccumBounds(-1, 1)
# rather than returning S.NaN, since AccumBounds(-1, 1)
# preserves the information that sin(oo) is between
# -1 and 1, where S.NaN does not do that.
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return sin(arg + pi/2)
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.is_extended_real and arg.is_finite is False:
return AccumBounds(-1, 1)
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import cosh
return cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
if (2*pi_coeff).is_integer:
# is_even-case handled above as then pi_coeff.is_integer,
# so check if known to be not even
if pi_coeff.is_even is False:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
# cosine formula #####################
# https://github.com/sympy/sympy/issues/6048
# explicit calculations are performed for
# cos(k pi/n) for n = 8,10,12,15,20,24,30,40,60,120
# Some other exact values like cos(k pi/240) can be
# calculated using a partial-fraction decomposition
# by calling cos( X ).rewrite(sqrt)
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*pi
return -cls(narg)
# If nested sqrt's are worse than un-evaluation
# you can require q to be in (1, 2, 3, 4, 6, 12)
# q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
# expressions with 2 or fewer sqrt nestings.
table2 = _table2()
if q in table2:
a, b = table2[q]
a, b = p*pi/a, p*pi/b
nvala, nvalb = cls(a), cls(b)
if None in (nvala, nvalb):
return None
return nvala*nvalb + cls(pi/2 - a)*cls(pi/2 - b)
if q > 12:
return None
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1) / 4,
}
if q in cst_table_some:
cts = cst_table_some[pi_coeff.q]
return chebyshevt(pi_coeff.p, cts).expand()
if 0 == q % 2:
narg = (pi_coeff*2)*pi
nval = cls(narg)
if None == nval:
return None
x = (2*pi_coeff + 1)/2
sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x)))
return sign_cos*sqrt( (1 + nval)/2 )
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
m = m*pi
return cos(m)*cos(x) - sin(m)*sin(x)
if arg.is_zero:
return S.One
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1/sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return x/sqrt(x**2 + y**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x ** 2)
if isinstance(arg, acot):
x = arg.args[0]
return 1/sqrt(1 + 1/x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1/x**2)
if isinstance(arg, asec):
x = arg.args[0]
return 1/x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p*x**2/(n*(n - 1))
else:
return S.NegativeOne**(n//2)*x**n/factorial(n)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
if logx is not None:
arg = arg.subs(log(x), logx)
if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity):
raise PoleError("Cannot expand %s around 0" % (self))
return Function._eval_nseries(self, x, n=n, logx=logx, cdir=cdir)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) + exp(-arg*I))/2
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return x**I/2 + x**-I/2
def _eval_rewrite_as_sin(self, arg, **kwargs):
return sin(arg + pi/2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)**2
return (1 - tan_half)/(1 + tan_half)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/sin(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)**2
return Piecewise((1, And(Eq(im(arg), 0), Eq(Mod(arg, 2*pi), 0))),
((cot_half - 1)/(cot_half + 1), True))
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._eval_rewrite_as_sqrt(arg)
def _eval_rewrite_as_sqrt(self, arg: Expr, **kwargs):
from sympy.functions.special.polynomials import chebyshevt
pi_coeff = _pi_coeff(arg)
if pi_coeff is None:
return None
if isinstance(pi_coeff, Integer):
return None
if not isinstance(pi_coeff, Rational):
return None
cst_table_some = cos_table()
if pi_coeff.q in cst_table_some:
rv = chebyshevt(pi_coeff.p, cst_table_some[pi_coeff.q]())
if pi_coeff.q < 257:
rv = rv.expand()
return rv
if not pi_coeff.q % 2: # recursively remove factors of 2
pico2 = pi_coeff * 2
nval = cos(pico2 * pi).rewrite(sqrt)
x = (pico2 + 1) / 2
sign_cos = -1 if int(x) % 2 else 1
return sign_cos * sqrt((1 + nval) / 2)
FC = fermat_coords(pi_coeff.q)
if FC:
denoms = FC
else:
denoms = [b**e for b, e in factorint(pi_coeff.q).items()]
apart = ipartfrac(*denoms)
decomp = (pi_coeff.p * Rational(n, d) for n, d in zip(apart, denoms))
X = [(x[1], x[0]*pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum(x[0] for x in X))._eval_expand_trig().subs(X)
if not FC or len(FC) == 1:
return pcls
return pcls.rewrite(sqrt)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1/sec(arg)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1/sec(arg).rewrite(csc)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy.functions.elementary.hyperbolic import cosh, sinh
re, im = self._as_real_imag(deep=deep, **hints)
return (cos(re)*cosh(im), -sin(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy.functions.special.polynomials import chebyshevt
arg = self.args[0]
x = None
if arg.is_Add: # TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return cx*cy - sx*sy
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer:
return chebyshevt(coeff, cos(terms))
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return cos(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = (x0 + pi/2)/pi
if n.is_integer:
lt = (arg - n*pi + pi/2).as_leading_term(x)
return (S.NegativeOne**n)*lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in [S.Infinity, S.NegativeInfinity]:
return AccumBounds(-1, 1)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_extended_real:
return True
def _eval_is_complex(self):
if self.args[0].is_extended_real \
or self.args[0].is_complex:
return True
def _eval_is_zero(self):
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero and pi_mult:
return (pi_mult - S.Half).is_integer
class tan(TrigonometricFunction):
"""
The tangent function.
Returns the tangent of x (measured in radians).
Explanation
===========
See :class:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import tan, pi
>>> from sympy.abc import x
>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(1).diff(x)
0
>>> tan(pi/8).expand()
-1 + sqrt(2)
See Also
========
sin, csc, cos, sec, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] https://dlmf.nist.gov/4.14
.. [3] https://functions.wolfram.com/ElementaryFunctions/Tan
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.One + self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atan
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
min, max = arg.min, arg.max
d = floor(min/pi)
if min is not S.NegativeInfinity:
min = min - d*pi
if max is not S.Infinity:
max = max - d*pi
from sympy.sets.sets import FiniteSet
if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(3, 2))):
return AccumBounds(S.NegativeInfinity, S.Infinity)
else:
return AccumBounds(tan(min), tan(max))
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import tanh
return S.ImaginaryUnit*tanh(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % q
# ensure simplified results are returned for n*pi/5, n*pi/10
table10 = {
1: sqrt(1 - 2*sqrt(5)/5),
2: sqrt(5 - 2*sqrt(5)),
3: sqrt(1 + 2*sqrt(5)/5),
4: sqrt(5 + 2*sqrt(5))
}
if q in (5, 10):
n = 10*p/q
if n > 5:
n = 10 - n
return -table10[n]
else:
return table10[n]
if not pi_coeff.q % 2:
narg = pi_coeff*pi*2
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return 1/sresult - cresult/sresult
table2 = _table2()
if q in table2:
a, b = table2[q]
nvala, nvalb = cls(p*pi/a), cls(p*pi/b)
if None in (nvala, nvalb):
return None
return (nvala - nvalb)/(1 + nvala*nvalb)
narg = ((pi_coeff + S.Half) % 1 - S.Half)*pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if cresult == 0:
return S.ComplexInfinity
return (sresult/cresult)
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
tanm = tan(m*pi)
if tanm is S.ComplexInfinity:
return -cot(x)
else: # tanm == 0
return tan(x)
if arg.is_zero:
return S.Zero
if isinstance(arg, atan):
return arg.args[0]
if isinstance(arg, atan2):
y, x = arg.args
return y/x
if isinstance(arg, asin):
x = arg.args[0]
return x/sqrt(1 - x**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2)/x
if isinstance(arg, acot):
x = arg.args[0]
return 1/x
if isinstance(arg, acsc):
x = arg.args[0]
return 1/(sqrt(1 - 1/x**2)*x)
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1/x**2)*x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a, b = ((n - 1)//2), 2**(n + 1)
B = bernoulli(n + 1)
F = factorial(n + 1)
return S.NegativeOne**a*b*(b - 1)*B/F*x**n
def _eval_nseries(self, x, n, logx, cdir=0):
i = self.args[0].limit(x, 0)*2/pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return Function._eval_nseries(self, x, n=n, logx=logx)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*(x**-I - x**I)/(x**-I + x**I)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
from sympy.functions.elementary.hyperbolic import cosh, sinh
denom = cos(2*re) + cosh(2*im)
return (sin(2*re)/denom, sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
n = len(arg.args)
TX = []
for x in arg.args:
tx = tan(x, evaluate=False)._eval_expand_trig()
TX.append(tx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n + 1):
p[1 - i % 2] += symmetric_poly(i, Y)*(-1)**((i % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, TX)))
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((1 + I*z)**coeff).expand()
return (im(P)/re(P)).subs([(z, tan(terms))])
return tan(arg)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
def _eval_rewrite_as_sin(self, x, **kwargs):
return 2*sin(x)**2/sin(2*x)
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x - pi/2, evaluate=False)/cos(x)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
return 1/cot(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
sin_in_sec_form = sin(arg).rewrite(sec)
cos_in_sec_form = cos(arg).rewrite(sec)
return sin_in_sec_form/cos_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
sin_in_csc_form = sin(arg).rewrite(csc)
cos_in_csc_form = cos(arg).rewrite(csc)
return sin_in_csc_form/cos_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = 2*x0/pi
if n.is_integer:
lt = (arg - n*pi/2).as_leading_term(x)
return lt if n.is_even else -1/lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
# FIXME: currently tan(pi/2) return zoo
return self.args[0].is_extended_real
def _eval_is_real(self):
arg = self.args[0]
if arg.is_real and (arg/pi - S.Half).is_integer is False:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real and (arg/pi - S.Half).is_integer is False:
return True
if arg.is_imaginary:
return True
def _eval_is_zero(self):
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero:
return pi_mult.is_integer
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_real and (arg/pi - S.Half).is_integer is False:
return True
class cot(TrigonometricFunction):
"""
The cotangent function.
Returns the cotangent of x (measured in radians).
Explanation
===========
See :class:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cot, pi
>>> from sympy.abc import x
>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)
>>> cot(1).diff(x)
0
>>> cot(pi/12)
sqrt(3) + 2
See Also
========
sin, csc, cos, sec, tan
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] https://dlmf.nist.gov/4.14
.. [3] https://functions.wolfram.com/ElementaryFunctions/Cot
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.NegativeOne - self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acot
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
if arg.is_zero:
return S.ComplexInfinity
elif arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return -tan(arg + pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import coth
return -S.ImaginaryUnit*coth(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.ComplexInfinity
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
if pi_coeff.q in (5, 10):
return tan(pi/2 - arg)
if pi_coeff.q > 2 and not pi_coeff.q % 2:
narg = pi_coeff*pi*2
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
return 1/sresult + cresult/sresult
q = pi_coeff.q
p = pi_coeff.p % q
table2 = _table2()
if q in table2:
a, b = table2[q]
nvala, nvalb = cls(p*pi/a), cls(p*pi/b)
if None in (nvala, nvalb):
return None
return (1 + nvala*nvalb)/(nvalb - nvala)
narg = (((pi_coeff + S.Half) % 1) - S.Half)*pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return cresult/sresult
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
cotm = cot(m*pi)
if cotm is S.ComplexInfinity:
return cot(x)
else: # cotm == 0
return -tan(x)
if arg.is_zero:
return S.ComplexInfinity
if isinstance(arg, acot):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1/x
if isinstance(arg, atan2):
y, x = arg.args
return x/y
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x**2)/x
if isinstance(arg, acos):
x = arg.args[0]
return x/sqrt(1 - x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1/x**2)*x
if isinstance(arg, asec):
x = arg.args[0]
return 1/(sqrt(1 - 1/x**2)*x)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return S.NegativeOne**((n + 1)//2)*2**(n + 1)*B/F*x**n
def _eval_nseries(self, x, n, logx, cdir=0):
i = self.args[0].limit(x, 0)/pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return self.rewrite(tan)._eval_nseries(x, n=n, logx=logx)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
from sympy.functions.elementary.hyperbolic import cosh, sinh
denom = cos(2*re) - cosh(2*im)
return (-sin(2*re)/denom, sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_rewrite_as_exp(self, arg, **kwargs):
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
I = S.ImaginaryUnit
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return -I*(x**-I + x**I)/(x**-I - x**I)
def _eval_rewrite_as_sin(self, x, **kwargs):
return sin(2*x)/(2*(sin(x)**2))
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x)/cos(x - pi/2, evaluate=False)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/sin(arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return 1/tan(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
cos_in_sec_form = cos(arg).rewrite(sec)
sin_in_sec_form = sin(arg).rewrite(sec)
return cos_in_sec_form/sin_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
cos_in_csc_form = cos(arg).rewrite(csc)
sin_in_csc_form = sin(arg).rewrite(csc)
return cos_in_csc_form/sin_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = 2*x0/pi
if n.is_integer:
lt = (arg - n*pi/2).as_leading_term(x)
return 1/lt if n.is_even else -lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
return self.args[0].is_extended_real
def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
n = len(arg.args)
CX = []
for x in arg.args:
cx = cot(x, evaluate=False)._eval_expand_trig()
CX.append(cx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n, -1, -1):
p[(n - i) % 2] += symmetric_poly(i, Y)*(-1)**(((n - i) % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, CX)))
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((z + I)**coeff).expand()
return (re(P)/im(P)).subs([(z, cot(terms))])
return cot(arg) # XXX sec and csc return 1/cos and 1/sin
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
if arg.is_imaginary:
return True
def _eval_is_real(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
def _eval_is_zero(self):
rest, pimult = _peeloff_pi(self.args[0])
if pimult and rest.is_zero:
return (pimult - S.Half).is_integer
def _eval_subs(self, old, new):
arg = self.args[0]
argnew = arg.subs(old, new)
if arg != argnew and (argnew/pi).is_integer:
return S.ComplexInfinity
return cot(argnew)
class ReciprocalTrigonometricFunction(TrigonometricFunction):
"""Base class for reciprocal functions of trigonometric functions. """
_reciprocal_of = None # mandatory, to be defined in subclass
_singularities = (S.ComplexInfinity,)
# _is_even and _is_odd are used for correct evaluation of csc(-x), sec(-x)
# TODO refactor into TrigonometricFunction common parts of
# trigonometric functions eval() like even/odd, func(x+2*k*pi), etc.
# optional, to be defined in subclasses:
_is_even: FuzzyBool = None
_is_odd: FuzzyBool = None
@classmethod
def eval(cls, arg):
if arg.could_extract_minus_sign():
if cls._is_even:
return cls(-arg)
if cls._is_odd:
return -cls(-arg)
pi_coeff = _pi_coeff(arg)
if (pi_coeff is not None
and not (2*pi_coeff).is_integer
and pi_coeff.is_Rational):
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*pi
if cls._is_odd:
return cls(narg)
elif cls._is_even:
return -cls(narg)
if hasattr(arg, 'inverse') and arg.inverse() == cls:
return arg.args[0]
t = cls._reciprocal_of.eval(arg)
if t is None:
return t
elif any(isinstance(i, cos) for i in (t, -t)):
return (1/t).rewrite(sec)
elif any(isinstance(i, sin) for i in (t, -t)):
return (1/t).rewrite(csc)
else:
return 1/t
def _call_reciprocal(self, method_name, *args, **kwargs):
# Calls method_name on _reciprocal_of
o = self._reciprocal_of(self.args[0])
return getattr(o, method_name)(*args, **kwargs)
def _calculate_reciprocal(self, method_name, *args, **kwargs):
# If calling method_name on _reciprocal_of returns a value != None
# then return the reciprocal of that value
t = self._call_reciprocal(method_name, *args, **kwargs)
return 1/t if t is not None else t
def _rewrite_reciprocal(self, method_name, arg):
# Special handling for rewrite functions. If reciprocal rewrite returns
# unmodified expression, then return None
t = self._call_reciprocal(method_name, arg)
if t is not None and t != self._reciprocal_of(arg):
return 1/t
def _period(self, symbol):
f = expand_mul(self.args[0])
return self._reciprocal_of(f).period(symbol)
def fdiff(self, argindex=1):
return -self._calculate_reciprocal("fdiff", argindex)/self**2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sqrt", arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
return (1/self._reciprocal_of(self.args[0])).as_real_imag(deep,
**hints)
def _eval_expand_trig(self, **hints):
return self._calculate_reciprocal("_eval_expand_trig", **hints)
def _eval_is_extended_real(self):
return self._reciprocal_of(self.args[0])._eval_is_extended_real()
def _eval_as_leading_term(self, x, logx=None, cdir=0):
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)
def _eval_is_finite(self):
return (1/self._reciprocal_of(self.args[0])).is_finite
def _eval_nseries(self, x, n, logx, cdir=0):
return (1/self._reciprocal_of(self.args[0]))._eval_nseries(x, n, logx)
class sec(ReciprocalTrigonometricFunction):
"""
The secant function.
Returns the secant of x (measured in radians).
Explanation
===========
See :class:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import sec
>>> from sympy.abc import x
>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)
>>> sec(1).diff(x)
0
See Also
========
sin, csc, cos, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] https://dlmf.nist.gov/4.14
.. [3] https://functions.wolfram.com/ElementaryFunctions/Sec
"""
_reciprocal_of = cos
_is_even = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half_sq = cot(arg/2)**2
return (cot_half_sq + 1)/(cot_half_sq - 1)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return (1/cos(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/(cos(arg)*sin(arg))
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1/cos(arg).rewrite(sin))
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1/cos(arg).rewrite(tan))
def _eval_rewrite_as_csc(self, arg, **kwargs):
return csc(pi/2 - arg, evaluate=False)
def fdiff(self, argindex=1):
if argindex == 1:
return tan(self.args[0])*sec(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_complex and (arg/pi - S.Half).is_integer is False:
return True
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
# Reference Formula:
# https://functions.wolfram.com/ElementaryFunctions/Sec/06/01/02/01/
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
k = n//2
return S.NegativeOne**k*euler(2*k)/factorial(2*k)*x**(2*k)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = (x0 + pi/2)/pi
if n.is_integer:
lt = (arg - n*pi + pi/2).as_leading_term(x)
return (S.NegativeOne**n)/lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
class csc(ReciprocalTrigonometricFunction):
"""
The cosecant function.
Returns the cosecant of x (measured in radians).
Explanation
===========
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import csc
>>> from sympy.abc import x
>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)
>>> csc(1).diff(x)
0
See Also
========
sin, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] https://dlmf.nist.gov/4.14
.. [3] https://functions.wolfram.com/ElementaryFunctions/Csc
"""
_reciprocal_of = sin
_is_odd = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1/sin(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/(sin(arg)*cos(arg))
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(arg/2)
return (1 + cot_half**2)/(2*cot_half)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return 1/sin(arg).rewrite(cos)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return sec(pi/2 - arg, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1/sin(arg).rewrite(tan))
def fdiff(self, argindex=1):
if argindex == 1:
return -cot(self.args[0])*csc(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = n//2 + 1
return (S.NegativeOne**(k - 1)*2*(2**(2*k - 1) - 1)*
bernoulli(2*k)*x**(2*k - 1)/factorial(2*k))
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = x0/pi
if n.is_integer:
lt = (arg - n*pi).as_leading_term(x)
return (S.NegativeOne**n)/lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
class sinc(Function):
r"""
Represents an unnormalized sinc function:
.. math::
\operatorname{sinc}(x) =
\begin{cases}
\frac{\sin x}{x} & \qquad x \neq 0 \\
1 & \qquad x = 0
\end{cases}
Examples
========
>>> from sympy import sinc, oo, jn
>>> from sympy.abc import x
>>> sinc(x)
sinc(x)
* Automated Evaluation
>>> sinc(0)
1
>>> sinc(oo)
0
* Differentiation
>>> sinc(x).diff()
cos(x)/x - sin(x)/x**2
* Series Expansion
>>> sinc(x).series()
1 - x**2/6 + x**4/120 + O(x**6)
* As zero'th order spherical Bessel Function
>>> sinc(x).rewrite(jn)
jn(0, x)
See also
========
sin
References
==========
.. [1] https://en.wikipedia.org/wiki/Sinc_function
"""
_singularities = (S.ComplexInfinity,)
def fdiff(self, argindex=1):
x = self.args[0]
if argindex == 1:
# We would like to return the Piecewise here, but Piecewise.diff
# currently can't handle removable singularities, meaning things
# like sinc(x).diff(x, 2) give the wrong answer at x = 0. See
# https://github.com/sympy/sympy/issues/11402.
#
# return Piecewise(((x*cos(x) - sin(x))/x**2, Ne(x, S.Zero)), (S.Zero, S.true))
return cos(x)/x - sin(x)/x**2
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.One
if arg.is_Number:
if arg in [S.Infinity, S.NegativeInfinity]:
return S.Zero
elif arg is S.NaN:
return S.NaN
if arg is S.ComplexInfinity:
return S.NaN
if arg.could_extract_minus_sign():
return cls(-arg)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
if fuzzy_not(arg.is_zero):
return S.Zero
elif (2*pi_coeff).is_integer:
return S.NegativeOne**(pi_coeff - S.Half)/arg
def _eval_nseries(self, x, n, logx, cdir=0):
x = self.args[0]
return (sin(x)/x)._eval_nseries(x, n, logx)
def _eval_rewrite_as_jn(self, arg, **kwargs):
from sympy.functions.special.bessel import jn
return jn(0, arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return Piecewise((sin(arg)/arg, Ne(arg, S.Zero)), (S.One, S.true))
def _eval_is_zero(self):
if self.args[0].is_infinite:
return True
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero:
return fuzzy_and([pi_mult.is_integer, pi_mult.is_nonzero])
if rest.is_Number and pi_mult.is_integer:
return False
def _eval_is_real(self):
if self.args[0].is_extended_real or self.args[0].is_imaginary:
return True
_eval_is_finite = _eval_is_real
###############################################################################
########################### TRIGONOMETRIC INVERSES ############################
###############################################################################
class InverseTrigonometricFunction(Function):
"""Base class for inverse trigonometric functions."""
_singularities = (S.One, S.NegativeOne, S.Zero, S.ComplexInfinity) # type: tTuple[Expr, ...]
@staticmethod
@cacheit
def _asin_table():
# Only keys with could_extract_minus_sign() == False
# are actually needed.
return {
sqrt(3)/2: pi/3,
sqrt(2)/2: pi/4,
1/sqrt(2): pi/4,
sqrt((5 - sqrt(5))/8): pi/5,
sqrt(2)*sqrt(5 - sqrt(5))/4: pi/5,
sqrt((5 + sqrt(5))/8): pi*Rational(2, 5),
sqrt(2)*sqrt(5 + sqrt(5))/4: pi*Rational(2, 5),
S.Half: pi/6,
sqrt(2 - sqrt(2))/2: pi/8,
sqrt(S.Half - sqrt(2)/4): pi/8,
sqrt(2 + sqrt(2))/2: pi*Rational(3, 8),
sqrt(S.Half + sqrt(2)/4): pi*Rational(3, 8),
(sqrt(5) - 1)/4: pi/10,
(1 - sqrt(5))/4: -pi/10,
(sqrt(5) + 1)/4: pi*Rational(3, 10),
sqrt(6)/4 - sqrt(2)/4: pi/12,
-sqrt(6)/4 + sqrt(2)/4: -pi/12,
(sqrt(3) - 1)/sqrt(8): pi/12,
(1 - sqrt(3))/sqrt(8): -pi/12,
sqrt(6)/4 + sqrt(2)/4: pi*Rational(5, 12),
(1 + sqrt(3))/sqrt(8): pi*Rational(5, 12)
}
@staticmethod
@cacheit
def _atan_table():
# Only keys with could_extract_minus_sign() == False
# are actually needed.
return {
sqrt(3)/3: pi/6,
1/sqrt(3): pi/6,
sqrt(3): pi/3,
sqrt(2) - 1: pi/8,
1 - sqrt(2): -pi/8,
1 + sqrt(2): pi*Rational(3, 8),
sqrt(5 - 2*sqrt(5)): pi/5,
sqrt(5 + 2*sqrt(5)): pi*Rational(2, 5),
sqrt(1 - 2*sqrt(5)/5): pi/10,
sqrt(1 + 2*sqrt(5)/5): pi*Rational(3, 10),
2 - sqrt(3): pi/12,
-2 + sqrt(3): -pi/12,
2 + sqrt(3): pi*Rational(5, 12)
}
@staticmethod
@cacheit
def _acsc_table():
# Keys for which could_extract_minus_sign()
# will obviously return True are omitted.
return {
2*sqrt(3)/3: pi/3,
sqrt(2): pi/4,
sqrt(2 + 2*sqrt(5)/5): pi/5,
1/sqrt(Rational(5, 8) - sqrt(5)/8): pi/5,
sqrt(2 - 2*sqrt(5)/5): pi*Rational(2, 5),
1/sqrt(Rational(5, 8) + sqrt(5)/8): pi*Rational(2, 5),
2: pi/6,
sqrt(4 + 2*sqrt(2)): pi/8,
2/sqrt(2 - sqrt(2)): pi/8,
sqrt(4 - 2*sqrt(2)): pi*Rational(3, 8),
2/sqrt(2 + sqrt(2)): pi*Rational(3, 8),
1 + sqrt(5): pi/10,
sqrt(5) - 1: pi*Rational(3, 10),
-(sqrt(5) - 1): pi*Rational(-3, 10),
sqrt(6) + sqrt(2): pi/12,
sqrt(6) - sqrt(2): pi*Rational(5, 12),
-(sqrt(6) - sqrt(2)): pi*Rational(-5, 12)
}
class asin(InverseTrigonometricFunction):
r"""
The inverse sine function.
Returns the arcsine of x in radians.
Explanation
===========
``asin(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
result is a rational multiple of $\pi$ (see the ``eval`` class method).
A purely imaginary argument will lead to an asinh expression.
Examples
========
>>> from sympy import asin, oo
>>> asin(1)
pi/2
>>> asin(-1)
-pi/2
>>> asin(-oo)
oo*I
>>> asin(oo)
-oo*I
See Also
========
sin, csc, cos, sec, tan, cot
acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://dlmf.nist.gov/4.23
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSin
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self._eval_is_extended_real() and self.args[0].is_positive
def _eval_is_negative(self):
return self._eval_is_extended_real() and self.args[0].is_negative
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.NegativeInfinity*S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.Infinity*S.ImaginaryUnit
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return pi/2
elif arg is S.NegativeOne:
return -pi/2
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
asin_table = cls._asin_table()
if arg in asin_table:
return asin_table[arg]
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import asinh
return S.ImaginaryUnit*asinh(i_coeff)
if arg.is_zero:
return S.Zero
if isinstance(arg, sin):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, cos): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acos(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p*(n - 2)**2/(n*(n - 1))*x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return R/F*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # asin
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0.is_zero:
return arg.as_leading_term(x)
# Handling branch points
if x0 in (-S.One, S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_negative:
return -pi - self.func(x0)
elif im(ndir).is_positive:
if x0.is_positive:
return pi - self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # asin
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = asin(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else pi/2 + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = asin(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else -pi/2 + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - arg0**2).is_negative:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_negative:
return -pi - res
elif im(ndir).is_positive:
if arg0.is_positive:
return pi - res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_acos(self, x, **kwargs):
return pi/2 - acos(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return 2*atan(x/(1 + sqrt(1 - x**2)))
def _eval_rewrite_as_log(self, x, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_acot(self, arg, **kwargs):
return 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return pi/2 - asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return acsc(1/arg)
def _eval_is_extended_real(self):
x = self.args[0]
return x.is_extended_real and (1 - abs(x)).is_nonnegative
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sin
class acos(InverseTrigonometricFunction):
r"""
The inverse cosine function.
Explanation
===========
Returns the arc cosine of x (measured in radians).
``acos(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when
the result is a rational multiple of $\pi$ (see the eval class method).
``acos(zoo)`` evaluates to ``zoo``
(see note in :class:`sympy.functions.elementary.trigonometric.asec`)
A purely imaginary argument will be rewritten to asinh.
Examples
========
>>> from sympy import acos, oo
>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://dlmf.nist.gov/4.23
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCos
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity*S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.NegativeInfinity*S.ImaginaryUnit
elif arg.is_zero:
return pi/2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return pi
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.is_number:
asin_table = cls._asin_table()
if arg in asin_table:
return pi/2 - asin_table[arg]
elif -arg in asin_table:
return pi/2 + asin_table[-arg]
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return pi/2 - asin(arg)
if isinstance(arg, cos):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, sin): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asin(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return pi/2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p*(n - 2)**2/(n*(n - 1))*x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R/F*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acos
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 == 1:
return sqrt(2)*sqrt((S.One - arg).as_leading_term(x))
if x0 in (-S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_negative:
return 2*pi - self.func(x0)
elif im(ndir).is_positive:
if x0.is_positive:
return -self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_is_extended_real(self):
x = self.args[0]
return x.is_extended_real and (1 - abs(x)).is_nonnegative
def _eval_is_nonnegative(self):
return self._eval_is_extended_real()
def _eval_nseries(self, x, n, logx, cdir=0): # acos
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = acos(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = acos(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else pi + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - arg0**2).is_negative:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_negative:
return 2*pi - res
elif im(ndir).is_positive:
if arg0.is_positive:
return -res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return pi/2 + S.ImaginaryUnit*\
log(S.ImaginaryUnit*x + sqrt(1 - x**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asin(self, x, **kwargs):
return pi/2 - asin(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cos
def _eval_rewrite_as_acot(self, arg, **kwargs):
return pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return pi/2 - acsc(1/arg)
def _eval_conjugate(self):
z = self.args[0]
r = self.func(self.args[0].conjugate())
if z.is_extended_real is False:
return r
elif z.is_extended_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive:
return r
class atan(InverseTrigonometricFunction):
r"""
The inverse tangent function.
Returns the arc tangent of x (measured in radians).
Explanation
===========
``atan(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
result is a rational multiple of $\pi$ (see the eval class method).
Examples
========
>>> from sympy import atan, oo
>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://dlmf.nist.gov/4.23
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan
"""
args: tTuple[Expr]
_singularities = (S.ImaginaryUnit, -S.ImaginaryUnit)
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_extended_positive
def _eval_is_nonnegative(self):
return self.args[0].is_extended_nonnegative
def _eval_is_zero(self):
return self.args[0].is_zero
def _eval_is_real(self):
return self.args[0].is_extended_real
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return pi/2
elif arg is S.NegativeInfinity:
return -pi/2
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return pi/4
elif arg is S.NegativeOne:
return -pi/4
if arg is S.ComplexInfinity:
from sympy.calculus.accumulationbounds import AccumBounds
return AccumBounds(-pi/2, pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
atan_table = cls._atan_table()
if arg in atan_table:
return atan_table[arg]
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import atanh
return S.ImaginaryUnit*atanh(i_coeff)
if arg.is_zero:
return S.Zero
if isinstance(arg, tan):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
if isinstance(arg, cot): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - acot(arg)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return S.NegativeOne**((n - 1)//2)*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # atan
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0.is_zero:
return arg.as_leading_term(x)
# Handling branch points
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
# Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
if (1 + x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_negative:
if im(x0).is_positive:
return self.func(x0) - pi
elif re(ndir).is_positive:
if im(x0).is_negative:
return self.func(x0) + pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # atan
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
ndir = self.args[0].dir(x, cdir if cdir else 1)
if arg0 is S.ComplexInfinity:
if re(ndir) > 0:
return res - pi
return res
# Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
if (1 + arg0**2).is_negative:
if re(ndir).is_negative:
if im(arg0).is_positive:
return res - pi
elif re(ndir).is_positive:
if im(arg0).is_negative:
return res + pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2*(log(S.One - S.ImaginaryUnit*x)
- log(S.One + S.ImaginaryUnit*x))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_aseries(self, n, args0, x, logx):
if args0[0] is S.Infinity:
return (pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] is S.NegativeInfinity:
return (-pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super()._eval_aseries(n, args0, x, logx)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tan
def _eval_rewrite_as_asin(self, arg, **kwargs):
return sqrt(arg**2)/arg*(pi/2 - asin(1/sqrt(1 + arg**2)))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return acot(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return sqrt(arg**2)/arg*(pi/2 - acsc(sqrt(1 + arg**2)))
class acot(InverseTrigonometricFunction):
r"""
The inverse cotangent function.
Returns the arc cotangent of x (measured in radians).
Explanation
===========
``acot(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$
and for some instances when the result is a rational multiple of $\pi$
(see the eval class method).
A purely imaginary argument will lead to an ``acoth`` expression.
``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous
at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$.
Examples
========
>>> from sympy import acot, sqrt
>>> acot(0)
pi/2
>>> acot(1)
pi/4
>>> acot(sqrt(3) - 2)
-5*pi/12
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, atan2
References
==========
.. [1] https://dlmf.nist.gov/4.23
.. [2] https://functions.wolfram.com/ElementaryFunctions/ArcCot
"""
_singularities = (S.ImaginaryUnit, -S.ImaginaryUnit)
def fdiff(self, argindex=1):
if argindex == 1:
return -1/(1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_nonnegative
def _eval_is_negative(self):
return self.args[0].is_negative
def _eval_is_extended_real(self):
return self.args[0].is_extended_real
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg.is_zero:
return pi/ 2
elif arg is S.One:
return pi/4
elif arg is S.NegativeOne:
return -pi/4
if arg is S.ComplexInfinity:
return S.Zero
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
atan_table = cls._atan_table()
if arg in atan_table:
ang = pi/2 - atan_table[arg]
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi
return ang
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import acoth
return -S.ImaginaryUnit*acoth(i_coeff)
if arg.is_zero:
return pi*S.Half
if isinstance(arg, cot):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi;
return ang
if isinstance(arg, tan): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - atan(arg)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return pi/2 # FIX THIS
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return S.NegativeOne**((n + 1)//2)*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acot
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0 is S.ComplexInfinity:
return (1/arg).as_leading_term(x)
# Handling branch points
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
# Handling points lying on branch cuts [-I, I]
if x0.is_imaginary and (1 + x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_positive:
if im(x0).is_positive:
return self.func(x0) + pi
elif re(ndir).is_negative:
if im(x0).is_negative:
return self.func(x0) - pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acot
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
ndir = self.args[0].dir(x, cdir if cdir else 1)
if arg0.is_zero:
if re(ndir) < 0:
return res - pi
return res
# Handling points lying on branch cuts [-I, I]
if arg0.is_imaginary and (1 + arg0**2).is_positive:
if re(ndir).is_positive:
if im(arg0).is_positive:
return res + pi
elif re(ndir).is_negative:
if im(arg0).is_negative:
return res - pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_aseries(self, n, args0, x, logx):
if args0[0] is S.Infinity:
return (pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] is S.NegativeInfinity:
return (pi*Rational(3, 2) - acot(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super(atan, self)._eval_aseries(n, args0, x, logx)
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2*(log(1 - S.ImaginaryUnit/x)
- log(1 + S.ImaginaryUnit/x))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cot
def _eval_rewrite_as_asin(self, arg, **kwargs):
return (arg*sqrt(1/arg**2)*
(pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1))))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1))
def _eval_rewrite_as_atan(self, arg, **kwargs):
return atan(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*(pi/2 - acsc(sqrt((1 + arg**2)/arg**2)))
class asec(InverseTrigonometricFunction):
r"""
The inverse secant function.
Returns the arc secant of x (measured in radians).
Explanation
===========
``asec(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
result is a rational multiple of $\pi$ (see the eval class method).
``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments,
it can be defined [4]_ as
.. math::
\operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}
At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For
negative branch cut, the limit
.. math::
\lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}
simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which
ultimately evaluates to ``zoo``.
As ``acos(x) = asec(1/x)``, a similar argument can be given for
``acos(x)``.
Examples
========
>>> from sympy import asec, oo
>>> asec(1)
0
>>> asec(-1)
pi
>>> asec(0)
zoo
>>> asec(-oo)
pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://dlmf.nist.gov/4.23
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSec
.. [4] https://reference.wolfram.com/language/ref/ArcSec.html
"""
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.ComplexInfinity
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return pi
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return pi/2
if arg.is_number:
acsc_table = cls._acsc_table()
if arg in acsc_table:
return pi/2 - acsc_table[arg]
elif -arg in acsc_table:
return pi/2 + acsc_table[-arg]
if arg.is_infinite:
return pi/2
if isinstance(arg, sec):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, csc): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acsc(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sec
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.ImaginaryUnit*log(2 / x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return -S.ImaginaryUnit * R / F * x**n / 4
def _eval_as_leading_term(self, x, logx=None, cdir=0): # asec
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 == 1:
return sqrt(2)*sqrt((arg - S.One).as_leading_term(x))
if x0 in (-S.One, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-1, 1)
if x0.is_real and (1 - x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_positive:
return -self.func(x0)
elif im(ndir).is_positive:
if x0.is_negative:
return 2*pi - self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # asec
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = asec(S.One + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = asec(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-1, 1)
if arg0.is_real and (1 - arg0**2).is_positive:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_positive:
return -res
elif im(ndir).is_positive:
if arg0.is_negative:
return 2*pi - res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_is_extended_real(self):
x = self.args[0]
if x.is_extended_real is False:
return False
return fuzzy_or(((x - 1).is_nonnegative, (-x - 1).is_nonnegative))
def _eval_rewrite_as_log(self, arg, **kwargs):
return pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asin(self, arg, **kwargs):
return pi/2 - asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return acos(1/arg)
def _eval_rewrite_as_atan(self, x, **kwargs):
sx2x = sqrt(x**2)/x
return pi/2*(1 - sx2x) + sx2x*atan(sqrt(x**2 - 1))
def _eval_rewrite_as_acot(self, x, **kwargs):
sx2x = sqrt(x**2)/x
return pi/2*(1 - sx2x) + sx2x*acot(1/sqrt(x**2 - 1))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return pi/2 - acsc(arg)
class acsc(InverseTrigonometricFunction):
r"""
The inverse cosecant function.
Returns the arc cosecant of x (measured in radians).
Explanation
===========
``acsc(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the
result is a rational multiple of $\pi$ (see the ``eval`` class method).
Examples
========
>>> from sympy import acsc, oo
>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2
>>> acsc(oo)
0
>>> acsc(-oo) == acsc(oo)
True
>>> acsc(0)
zoo
See Also
========
sin, csc, cos, sec, tan, cot
asin, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://dlmf.nist.gov/4.23
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsc
"""
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.ComplexInfinity
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return pi/2
elif arg is S.NegativeOne:
return -pi/2
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return S.Zero
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_infinite:
return S.Zero
if arg.is_number:
acsc_table = cls._acsc_table()
if arg in acsc_table:
return acsc_table[arg]
if isinstance(arg, csc):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, sec): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asec(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return -1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return csc
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return pi/2 - S.ImaginaryUnit*log(2) + S.ImaginaryUnit*log(x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return S.ImaginaryUnit * R / F * x**n / 4
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acsc
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 in (-S.One, S.One, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
if x0 is S.ComplexInfinity:
return (1/arg).as_leading_term(x)
# Handling points lying on branch cuts (-1, 1)
if x0.is_real and (1 - x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_positive:
return pi - self.func(x0)
elif im(ndir).is_positive:
if x0.is_negative:
return -pi - self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acsc
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = acsc(S.One + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = acsc(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-1, 1)
if arg0.is_real and (1 - arg0**2).is_positive:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_positive:
return pi - res
elif im(ndir).is_positive:
if arg0.is_negative:
return -pi - res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, arg, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asin(self, arg, **kwargs):
return asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return pi/2 - acos(1/arg)
def _eval_rewrite_as_atan(self, x, **kwargs):
return sqrt(x**2)/x*(pi/2 - atan(sqrt(x**2 - 1)))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return sqrt(arg**2)/arg*(pi/2 - acot(1/sqrt(arg**2 - 1)))
def _eval_rewrite_as_asec(self, arg, **kwargs):
return pi/2 - asec(arg)
class atan2(InverseTrigonometricFunction):
r"""
The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking
two arguments `y` and `x`. Signs of both `y` and `x` are considered to
determine the appropriate quadrant of `\operatorname{atan}(y/x)`.
The range is `(-\pi, \pi]`. The complete definition reads as follows:
.. math::
\operatorname{atan2}(y, x) =
\begin{cases}
\arctan\left(\frac y x\right) & \qquad x > 0 \\
\arctan\left(\frac y x\right) + \pi& \qquad y \ge 0, x < 0 \\
\arctan\left(\frac y x\right) - \pi& \qquad y < 0, x < 0 \\
+\frac{\pi}{2} & \qquad y > 0, x = 0 \\
-\frac{\pi}{2} & \qquad y < 0, x = 0 \\
\text{undefined} & \qquad y = 0, x = 0
\end{cases}
Attention: Note the role reversal of both arguments. The `y`-coordinate
is the first argument and the `x`-coordinate the second.
If either `x` or `y` is complex:
.. math::
\operatorname{atan2}(y, x) =
-i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right)
Examples
========
Going counter-clock wise around the origin we find the
following angles:
>>> from sympy import atan2
>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4
which are all correct. Compare this to the results of the ordinary
`\operatorname{atan}` function for the point `(x, y) = (-1, 1)`
>>> from sympy import atan, S
>>> atan(S(1)/-1)
-pi/4
>>> atan2(1, -1)
3*pi/4
where only the `\operatorname{atan2}` function reurns what we expect.
We can differentiate the function with respect to both arguments:
>>> from sympy import diff
>>> from sympy.abc import x, y
>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)
>>> diff(atan2(y, x), y)
x/(x**2 + y**2)
We can express the `\operatorname{atan2}` function in terms of
complex logarithms:
>>> from sympy import log
>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))
and in terms of `\operatorname(atan)`:
>>> from sympy import atan
>>> atan2(y, x).rewrite(atan)
Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True))
but note that this form is undefined on the negative real axis.
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://en.wikipedia.org/wiki/Atan2
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan2
"""
@classmethod
def eval(cls, y, x):
from sympy.functions.special.delta_functions import Heaviside
if x is S.NegativeInfinity:
if y.is_zero:
# Special case y = 0 because we define Heaviside(0) = 1/2
return pi
return 2*pi*(Heaviside(re(y))) - pi
elif x is S.Infinity:
return S.Zero
elif x.is_imaginary and y.is_imaginary and x.is_number and y.is_number:
x = im(x)
y = im(y)
if x.is_extended_real and y.is_extended_real:
if x.is_positive:
return atan(y/x)
elif x.is_negative:
if y.is_negative:
return atan(y/x) - pi
elif y.is_nonnegative:
return atan(y/x) + pi
elif x.is_zero:
if y.is_positive:
return pi/2
elif y.is_negative:
return -pi/2
elif y.is_zero:
return S.NaN
if y.is_zero:
if x.is_extended_nonzero:
return pi*(S.One - Heaviside(x))
if x.is_number:
return Piecewise((pi, re(x) < 0),
(0, Ne(x, 0)),
(S.NaN, True))
if x.is_number and y.is_number:
return -S.ImaginaryUnit*log(
(x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))
def _eval_rewrite_as_log(self, y, x, **kwargs):
return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))
def _eval_rewrite_as_atan(self, y, x, **kwargs):
return Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)),
(pi, re(x) < 0),
(0, Ne(x, 0)),
(S.NaN, True))
def _eval_rewrite_as_arg(self, y, x, **kwargs):
if x.is_extended_real and y.is_extended_real:
return arg_f(x + y*S.ImaginaryUnit)
n = x + S.ImaginaryUnit*y
d = x**2 + y**2
return arg_f(n/sqrt(d)) - S.ImaginaryUnit*log(abs(n)/sqrt(abs(d)))
def _eval_is_extended_real(self):
return self.args[0].is_extended_real and self.args[1].is_extended_real
def _eval_conjugate(self):
return self.func(self.args[0].conjugate(), self.args[1].conjugate())
def fdiff(self, argindex):
y, x = self.args
if argindex == 1:
# Diff wrt y
return x/(x**2 + y**2)
elif argindex == 2:
# Diff wrt x
return -y/(x**2 + y**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
y, x = self.args
if x.is_extended_real and y.is_extended_real:
return super()._eval_evalf(prec)