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from __future__ import print_function, division

from sympy.core.add import Add

from sympy.core.basic import sympify, cacheit

from sympy.core.function import Function, ArgumentIndexError

from sympy.core.numbers import igcdex, Rational, pi

from sympy.core.singleton import S

from sympy.core.symbol import Symbol, Wild

from sympy.core.logic import fuzzy_not

from sympy.functions.combinatorial.factorials import factorial, RisingFactorial

from sympy.functions.elementary.miscellaneous import sqrt, Min, Max

from sympy.functions.elementary.exponential import log, exp

from sympy.functions.elementary.integers import floor

from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh,

coth, HyperbolicFunction, sinh, tanh)

from sympy.sets.sets import FiniteSet

from sympy.utilities.iterables import numbered_symbols

from sympy.core.compatibility import range

###############################################################################

########################## TRIGONOMETRIC FUNCTIONS ############################

###############################################################################

class TrigonometricFunction(Function):

"""Base class for trigonometric functions. """

unbranched = True

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_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 = 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:

p, q = Wild('p'), Wild('q')

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.")

def _peeloff_pi(arg):

"""

Split ARG into two parts, a "rest" and a multiple of pi/2.

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 as peel

>>> from sympy import pi

>>> from sympy.abc import x, y

>>> peel(x + pi/2)

(x, pi/2)

>>> peel(x + 2*pi/3 + pi*y)

(x + pi*y + pi/6, pi/2)

"""

for a in Add.make_args(arg):

if a is S.Pi:

K = S.One

break

elif a.is_Mul:

K, p = a.as_two_terms()

if p is S.Pi and K.is_Rational:

break

else:

return arg, S.Zero

m1 = (K % S.Half) * S.Pi

m2 = K*S.Pi - m1

return arg - m2, m2

def _pi_coeff(arg, cycles=1):

"""

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 as coeff

>>> from sympy import pi, Dummy

>>> from sympy.abc import x, y

>>> coeff(3*x*pi)

3*x

>>> coeff(11*pi/7)

11/7

>>> coeff(-11*pi/7)

3/7

>>> coeff(4*pi)

>>> coeff(5*pi)

>>> coeff(5.0*pi)

>>> coeff(5.5*pi)

3/2

>>> coeff(2 + pi)

>>> coeff(2*Dummy(integer=True)*pi)

>>> coeff(2*Dummy(even=True)*pi)

"""

arg = sympify(arg)

if arg is S.Pi:

return S.One

elif not arg:

return S.Zero

elif arg.is_Mul:

cx = arg.coeff(S.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 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 S(2)

else:

return c2*x

return cx

class sin(TrigonometricFunction):

"""

The sine function.

Returns the sine of x (measured in radians).

Notes

=====

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)

>>> sin(pi)

>>> sin(pi/2)

>>> 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] http://en.wikipedia.org/wiki/Trigonometric_functions

.. [2] http://dlmf.nist.gov/4.14

.. [3] http://functions.wolfram.com/ElementaryFunctions/Sin

.. [4] http://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 import AccumBounds

if arg.is_Number:

if arg is S.NaN:

return S.NaN

elif arg is S.Zero:

return S.Zero

elif arg is S.Infinity or arg is S.NegativeInfinity:

return AccumBounds(-1, 1)

if isinstance(arg, AccumBounds):

min, max = arg.min, arg.max

d = floor(min/(2*S.Pi))

if min is not S.NegativeInfinity:

min = min - d*2*S.Pi

if max is not S.Infinity:

max = max - d*2*S.Pi

if AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 5*S.Pi/2)) \

is not S.EmptySet and \

AccumBounds(min, max).intersection(FiniteSet(3*S.Pi/2,

7*S.Pi/2)) is not S.EmptySet:

return AccumBounds(-1, 1)

elif AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 5*S.Pi/2)) \

is not S.EmptySet:

return AccumBounds(Min(sin(min), sin(max)), 1)

elif AccumBounds(min, max).intersection(FiniteSet(3*S.Pi/2, 8*S.Pi/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)))

if arg.could_extract_minus_sign():

return -cls(-arg)

i_coeff = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

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:

if pi_coeff.is_even:

return S.Zero

elif pi_coeff.is_even is False:

return S.NegativeOne**(pi_coeff - S.Half)

if not pi_coeff.is_Rational:

narg = pi_coeff*S.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)*S.Pi)

if 2*x > 1:

return cls((1 - x)*S.Pi)

narg = ((pi_coeff + Rational(3, 2)) % 2)*S.Pi

result = cos(narg)

if not isinstance(result, cos):

return result

if pi_coeff*S.Pi != arg:

return cls(pi_coeff*S.Pi)

return None

if arg.is_Add:

x, m = _peeloff_pi(arg)

if m:

return sin(m)*cos(x) + cos(m)*sin(x)

if arg.func is asin:

return arg.args[0]

if arg.func is atan:

x = arg.args[0]

return x / sqrt(1 + x**2)

if arg.func is atan2:

y, x = arg.args

return y / sqrt(x**2 + y**2)

if arg.func is acos:

x = arg.args[0]

return sqrt(1 - x**2)

if arg.func is acot:

x = arg.args[0]

return 1 / (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)

if len(previous_terms) > 2:

p = previous_terms[-2]

return -p * x**2 / (n*(n - 1))

else:

return (-1)**(n//2) * x**(n)/factorial(n)

def _eval_rewrite_as_exp(self, arg):

I = S.ImaginaryUnit

if isinstance(arg, TrigonometricFunction) or isinstance(arg, 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):

if arg.func is log:

I = S.ImaginaryUnit

x = arg.args[0]

return I*x**-I / 2 - I*x**I /2

def _eval_rewrite_as_cos(self, arg):

return cos(arg - S.Pi / 2, evaluate=False)

def _eval_rewrite_as_tan(self, arg):

tan_half = tan(S.Half*arg)

return 2*tan_half/(1 + tan_half**2)

def _eval_rewrite_as_sincos(self, arg):

return sin(arg)*cos(arg)/cos(arg)

def _eval_rewrite_as_cot(self, arg):

cot_half = cot(S.Half*arg)

return 2*cot_half/(1 + cot_half**2)

def _eval_rewrite_as_pow(self, arg):

return self.rewrite(cos).rewrite(pow)

def _eval_rewrite_as_sqrt(self, arg):

return self.rewrite(cos).rewrite(sqrt)

def _eval_rewrite_as_csc(self, arg):

return 1/csc(arg)

def _eval_rewrite_as_sec(self, arg):

return 1 / sec(arg - S.Pi / 2, evaluate=False)

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)

return (sin(re)*cosh(im), cos(re)*sinh(im))

def _eval_expand_trig(self, **hints):

from sympy import expand_mul

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

else:

n, x = arg.as_coeff_Mul(rational=True)

if n.is_Integer: # n will be positive because of .eval

# canonicalization

# See http://mathworld.wolfram.com/Multiple-AngleFormulas.html

if n.is_odd:

return (-1)**((n - 1)/2)*chebyshevt(n, sin(x))

else:

return expand_mul((-1)**(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):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return arg

else:

return self.func(arg)

def _eval_is_real(self):

return self.args[0].is_real

def _eval_is_finite(self):

arg = self.args[0]

if arg.is_real:

return True

class cos(TrigonometricFunction):

"""

The cosine function.

Returns the cosine of x (measured in radians).

Notes

=====

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)

>>> cos(pi)

-1

>>> cos(pi/2)

>>> 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] http://en.wikipedia.org/wiki/Trigonometric_functions

.. [2] http://dlmf.nist.gov/4.14

.. [3] http://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.util import AccumBounds

if arg.is_Number:

if arg is S.NaN:

return S.NaN

elif arg is S.Zero:

return S.One

elif arg is S.Infinity or arg is 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 isinstance(arg, AccumBounds):

return sin(arg + S.Pi/2)

if arg.could_extract_minus_sign():

return cls(-arg)

i_coeff = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

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:

if pi_coeff.is_even:

return (S.NegativeOne)**(pi_coeff/2)

elif pi_coeff.is_even is False:

return S.Zero

if not pi_coeff.is_Rational:

narg = pi_coeff*S.Pi

if narg != arg:

return cls(narg)

return None

# cosine formula #####################

# https://github.com/sympy/sympy/issues/6048

# explicit calculations are preformed 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)

cst_table_some = {

3: S.Half,

5: (sqrt(5) + 1)/4,

}

if pi_coeff.is_Rational:

q = pi_coeff.q

p = pi_coeff.p % (2*q)

if p > q:

narg = (pi_coeff - 1)*S.Pi

return -cls(narg)

if 2*p > q:

narg = (1 - pi_coeff)*S.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 = {

12: (3, 4),

20: (4, 5),

30: (5, 6),

15: (6, 10),

24: (6, 8),

40: (8, 10),

60: (20, 30),

120: (40, 60)

}

if q in table2:

a, b = p*S.Pi/table2[q][0], p*S.Pi/table2[q][1]

nvala, nvalb = cls(a), cls(b)

if None == nvala or None == nvalb:

return None

return nvala*nvalb + cls(S.Pi/2 - a)*cls(S.Pi/2 - b)

if q > 12:

return None

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)*S.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:

return cos(m)*cos(x) - sin(m)*sin(x)

if arg.func is acos:

return arg.args[0]

if arg.func is atan:

x = arg.args[0]

return 1 / sqrt(1 + x**2)

if arg.func is atan2:

y, x = arg.args

return x / sqrt(x**2 + y**2)

if arg.func is asin:

x = arg.args[0]

return sqrt(1 - x ** 2)

if arg.func is acot:

x = arg.args[0]

return 1 / sqrt(1 + 1 / x**2)

@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 (-1)**(n//2)*x**(n)/factorial(n)

def _eval_rewrite_as_exp(self, arg):

I = S.ImaginaryUnit

if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):

arg = arg.func(arg.args[0]).rewrite(exp)

return (exp(arg*I) + exp(-arg*I)) / 2

def _eval_rewrite_as_Pow(self, arg):

if arg.func is log:

I = S.ImaginaryUnit

x = arg.args[0]

return x**I/2 + x**-I/2

def _eval_rewrite_as_sin(self, arg):

return sin(arg + S.Pi / 2, evaluate=False)

def _eval_rewrite_as_tan(self, arg):

tan_half = tan(S.Half*arg)**2

return (1 - tan_half)/(1 + tan_half)

def _eval_rewrite_as_sincos(self, arg):

return sin(arg)*cos(arg)/sin(arg)

def _eval_rewrite_as_cot(self, arg):

cot_half = cot(S.Half*arg)**2

return (cot_half - 1)/(cot_half + 1)

def _eval_rewrite_as_pow(self, arg):

return self._eval_rewrite_as_sqrt(arg)

def _eval_rewrite_as_sqrt(self, arg):

from sympy.functions.special.polynomials import chebyshevt

def migcdex(x):

# recursive calcuation of gcd and linear combination

# for a sequence of integers.

# Given (x1, x2, x3)

# Returns (y1, y1, y3, g)

# such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0

# Note, that this is only one such linear combination.

if len(x) == 1:

return (1, x[0])

if len(x) == 2:

return igcdex(x[0], x[-1])

g = migcdex(x[1:])

u, v, h = igcdex(x[0], g[-1])

return tuple([u] + [v*i for i in g[0:-1] ] + [h])

def ipartfrac(r, factors=None):

from sympy.ntheory import factorint

if isinstance(r, int):

return r

if not isinstance(r, Rational):

raise TypeError("r is not rational")

n = r.q

if 2 > r.q*r.q:

return r.q

if None == factors:

a = [n//x**y for x, y in factorint(r.q).items()]

else:

a = [n//x for x in factors]

if len(a) == 1:

return [ r ]

h = migcdex(a)

ans = [ r.p*Rational(i*j, r.q) for i, j in zip(h[:-1], a) ]

assert r == sum(ans)

return ans

pi_coeff = _pi_coeff(arg)

if pi_coeff is None:

return None

if pi_coeff.is_integer:

# it was unevaluated

return self.func(pi_coeff*S.Pi)

if not pi_coeff.is_Rational:

return None

def _cospi257():

""" Express cos(pi/257) explicitly as a function of radicals

Based upon the equations in

http://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals

See also http://www.susqu.edu/brakke/constructions/257-gon.m.txt

"""

def f1(a, b):

return (a + sqrt(a**2 + b))/2, (a - sqrt(a**2 + b))/2

def f2(a, b):

return (a - sqrt(a**2 + b))/2

t1, t2 = f1(-1, 256)

z1, z3 = f1(t1, 64)

z2, z4 = f1(t2, 64)

y1, y5 = f1(z1, 4*(5 + t1 + 2*z1))

y6, y2 = f1(z2, 4*(5 + t2 + 2*z2))

y3, y7 = f1(z3, 4*(5 + t1 + 2*z3))

y8, y4 = f1(z4, 4*(5 + t2 + 2*z4))

x1, x9 = f1(y1, -4*(t1 + y1 + y3 + 2*y6))

x2, x10 = f1(y2, -4*(t2 + y2 + y4 + 2*y7))

x3, x11 = f1(y3, -4*(t1 + y3 + y5 + 2*y8))

x4, x12 = f1(y4, -4*(t2 + y4 + y6 + 2*y1))

x5, x13 = f1(y5, -4*(t1 + y5 + y7 + 2*y2))

x6, x14 = f1(y6, -4*(t2 + y6 + y8 + 2*y3))

x15, x7 = f1(y7, -4*(t1 + y7 + y1 + 2*y4))

x8, x16 = f1(y8, -4*(t2 + y8 + y2 + 2*y5))

v1 = f2(x1, -4*(x1 + x2 + x3 + x6))

v2 = f2(x2, -4*(x2 + x3 + x4 + x7))

v3 = f2(x8, -4*(x8 + x9 + x10 + x13))

v4 = f2(x9, -4*(x9 + x10 + x11 + x14))

v5 = f2(x10, -4*(x10 + x11 + x12 + x15))

v6 = f2(x16, -4*(x16 + x1 + x2 + x5))

u1 = -f2(-v1, -4*(v2 + v3))

u2 = -f2(-v4, -4*(v5 + v6))

w1 = -2*f2(-u1, -4*u2)

return sqrt(sqrt(2)*sqrt(w1 + 4)/8 + S.Half)

cst_table_some = {

3: S.Half,

5: (sqrt(5) + 1)/4,

17: sqrt((15 + sqrt(17))/32 + sqrt(2)*(sqrt(17 - sqrt(17)) +

sqrt(sqrt(2)*(-8*sqrt(17 + sqrt(17)) - (1 - sqrt(17))

*sqrt(17 - sqrt(17))) + 6*sqrt(17) + 34))/32),

257: _cospi257()

# 65537 is the only other known Fermat prime and the very

# large expression is intentionally omitted from SymPy; see

# http://www.susqu.edu/brakke/constructions/65537-gon.m.txt

}

def _fermatCoords(n):

# if n can be factored in terms of Fermat primes with

# multiplicity of each being 1, return those primes, else

# False

from sympy import chebyshevt

primes = []

for p_i in cst_table_some:

n, r = divmod(n, p_i)

if not r:

primes.append(p_i)

if n == 1:

return tuple(primes)

return False

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*S.Pi).rewrite(sqrt)

x = (pico2 + 1)/2

sign_cos = -1 if int(x) % 2 else 1

return sign_cos*sqrt( (1 + nval)/2 )

FC = _fermatCoords(pi_coeff.q)

if FC:

decomp = ipartfrac(pi_coeff, FC)

X = [(x[1], x[0]*S.Pi) for x in zip(decomp, numbered_symbols('z'))]

pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)

return pcls.rewrite(sqrt)

else:

decomp = ipartfrac(pi_coeff)

X = [(x[1], x[0]*S.Pi) for x in zip(decomp, numbered_symbols('z'))]

pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)

return pcls

def _eval_rewrite_as_sec(self, arg):

return 1/sec(arg)

def _eval_rewrite_as_csc(self, arg):

return 1 / sec(arg)._eval_rewrite_as_csc(arg)

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)

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

else:

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):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return S.One

else:

return self.func(arg)

def _eval_is_real(self):

return self.args[0].is_real

def _eval_is_finite(self):

arg = self.args[0]

if arg.is_real:

return True

class tan(TrigonometricFunction):

"""

The tangent function.

Returns the tangent of x (measured in radians).

Notes

=====

See :func:`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)

>>> tan(pi/8).expand()

-1 + sqrt(2)

See Also

========

sin, csc, cos, sec, cot

asin, acsc, acos, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Trigonometric_functions

.. [2] http://dlmf.nist.gov/4.14

.. [3] http://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.util import AccumBounds

if arg.is_Number:

if arg is S.NaN:

return S.NaN

elif arg is S.Zero:

return S.Zero

elif arg is S.Infinity or arg is S.NegativeInfinity:

return AccumBounds(S.NegativeInfinity, S.Infinity)

if isinstance(arg, AccumBounds):

min, max = arg.min, arg.max

d = floor(min/S.Pi)

if min is not S.NegativeInfinity:

min = min - d*S.Pi

if max is not S.Infinity:

max = max - d*S.Pi

if AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 3*S.Pi/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 = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

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*S.Pi

if narg != arg:

return cls(narg)

return None

if pi_coeff.is_Rational:

if not pi_coeff.q % 2:

narg = pi_coeff*S.Pi*2

cresult, sresult = cos(narg), cos(narg - S.Pi/2)

if not isinstance(cresult, cos) \

and not isinstance(sresult, cos):

if sresult == 0:

return S.ComplexInfinity

return (1 - cresult)/sresult

table2 = {

12: (3, 4),

20: (4, 5),

30: (5, 6),

15: (6, 10),

24: (6, 8),

40: (8, 10),

60: (20, 30),

120: (40, 60)

}

q = pi_coeff.q

p = pi_coeff.p % q

if q in table2:

nvala, nvalb = cls(p*S.Pi/table2[q][0]), cls(p*S.Pi/table2[q][1])

if None == nvala or None == nvalb:

return None

return (nvala - nvalb)/(1 + nvala*nvalb)

narg = ((pi_coeff + S.Half) % 1 - S.Half)*S.Pi

# see cos() to specify which expressions should be

# expanded automatically in terms of radicals

cresult, sresult = cos(narg), cos(narg - S.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)

tanx = tan(x)

if tanm is S.ComplexInfinity:

return -cot(x)

return (tanm + tanx)/(1 - tanm*tanx)

if arg.func is atan:

return arg.args[0]

if arg.func is atan2:

y, x = arg.args

return y/x

if arg.func is asin:

x = arg.args[0]

return x / sqrt(1 - x**2)

if arg.func is acos:

x = arg.args[0]

return sqrt(1 - x**2) / x

if arg.func is acot:

x = arg.args[0]

return 1 / x

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

from sympy import bernoulli

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 (-1)**a * b*(b - 1) * B/F * x**n

def _eval_nseries(self, x, n, logx):

i = self.args[0].limit(x, 0)*2/S.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):

if arg.func is 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:

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):

from sympy import im, re

arg = self.args[0]

x = None

if arg.is_Add:

from sympy import symmetric_poly

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)))

else:

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):

I = S.ImaginaryUnit

if isinstance(arg, TrigonometricFunction) or isinstance(arg, 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):

return 2*sin(x)**2/sin(2*x)

def _eval_rewrite_as_cos(self, x):

return cos(x - S.Pi / 2, evaluate=False) / cos(x)

def _eval_rewrite_as_sincos(self, arg):

return sin(arg)/cos(arg)

def _eval_rewrite_as_cot(self, arg):

return 1/cot(arg)

def _eval_rewrite_as_sec(self, arg):

sin_in_sec_form = sin(arg)._eval_rewrite_as_sec(arg)

cos_in_sec_form = cos(arg)._eval_rewrite_as_sec(arg)

return sin_in_sec_form / cos_in_sec_form

def _eval_rewrite_as_csc(self, arg):

sin_in_csc_form = sin(arg)._eval_rewrite_as_csc(arg)

cos_in_csc_form = cos(arg)._eval_rewrite_as_csc(arg)

return sin_in_csc_form / cos_in_csc_form

def _eval_rewrite_as_pow(self, arg):

y = self.rewrite(cos).rewrite(pow)

if y.has(cos):

return None

return y

def _eval_rewrite_as_sqrt(self, arg):

y = self.rewrite(cos).rewrite(sqrt)

if y.has(cos):

return None

return y

def _eval_as_leading_term(self, x):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return arg

else:

return self.func(arg)

def _eval_is_real(self):

return self.args[0].is_real

def _eval_is_finite(self):

arg = self.args[0]

if arg.is_imaginary:

return True

class cot(TrigonometricFunction):

"""

The cotangent function.

Returns the cotangent of x (measured in radians).

Notes

=====

See :func:`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)

>>> cot(pi/12)

sqrt(3) + 2

See Also

========

sin, csc, cos, sec, tan

asin, acsc, acos, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Trigonometric_functions

.. [2] http://dlmf.nist.gov/4.14

.. [3] http://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.util import AccumBounds

if arg.is_Number:

if arg is S.NaN:

return S.NaN

if arg is S.Zero:

return S.ComplexInfinity

if isinstance(arg, AccumBounds):

return -tan(arg + S.Pi/2)

if arg.could_extract_minus_sign():

return -cls(-arg)

i_coeff = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

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*S.Pi

if narg != arg:

return cls(narg)

return None

if pi_coeff.is_Rational:

if pi_coeff.q > 2 and not pi_coeff.q % 2:

narg = pi_coeff*S.Pi*2

cresult, sresult = cos(narg), cos(narg - S.Pi/2)

if not isinstance(cresult, cos) \

and not isinstance(sresult, cos):

return (1 + cresult)/sresult

table2 = {

12: (3, 4),

20: (4, 5),

30: (5, 6),

15: (6, 10),

24: (6, 8),

40: (8, 10),

60: (20, 30),

120: (40, 60)

}

q = pi_coeff.q

p = pi_coeff.p % q

if q in table2:

nvala, nvalb = cls(p*S.Pi/table2[q][0]), cls(p*S.Pi/table2[q][1])

if None == nvala or None == nvalb:

return None

return (1 + nvala*nvalb)/(nvalb - nvala)

narg = (((pi_coeff + S.Half) % 1) - S.Half)*S.Pi

# see cos() to specify which expressions should be

# expanded automatically in terms of radicals

cresult, sresult = cos(narg), cos(narg - S.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)

if cotm == 0:

return -tan(x)

cotx = cot(x)

if cotm is S.ComplexInfinity:

return cotx

if cotm.is_Rational:

return (cotm*cotx - 1) / (cotm + cotx)

return None

if arg.func is acot:

return arg.args[0]

if arg.func is atan:

x = arg.args[0]

return 1 / x

if arg.func is atan2:

y, x = arg.args

return x/y

if arg.func is asin:

x = arg.args[0]

return sqrt(1 - x**2) / x

if arg.func is acos:

x = arg.args[0]

return x / sqrt(1 - x**2)

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

from sympy import bernoulli

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 (-1)**((n + 1)//2) * 2**(n + 1) * B/F * x**n

def _eval_nseries(self, x, n, logx):

i = self.args[0].limit(x, 0)/S.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:

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):

I = S.ImaginaryUnit

if isinstance(arg, TrigonometricFunction) or isinstance(arg, 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):

if arg.func is 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):

return 2*sin(2*x)/sin(x)**2

def _eval_rewrite_as_cos(self, x):

return cos(x) / cos(x - S.Pi / 2, evaluate=False)

def _eval_rewrite_as_sincos(self, arg):

return cos(arg)/sin(arg)

def _eval_rewrite_as_tan(self, arg):

return 1/tan(arg)

def _eval_rewrite_as_sec(self, arg):

cos_in_sec_form = cos(arg)._eval_rewrite_as_sec(arg)

sin_in_sec_form = sin(arg)._eval_rewrite_as_sec(arg)

return cos_in_sec_form / sin_in_sec_form

def _eval_rewrite_as_csc(self, arg):

cos_in_csc_form = cos(arg)._eval_rewrite_as_csc(arg)

sin_in_csc_form = sin(arg)._eval_rewrite_as_csc(arg)

return cos_in_csc_form / sin_in_csc_form

def _eval_rewrite_as_pow(self, arg):

y = self.rewrite(cos).rewrite(pow)

if y.has(cos):

return None

return y

def _eval_rewrite_as_sqrt(self, arg):

y = self.rewrite(cos).rewrite(sqrt)

if y.has(cos):

return None

return y

def _eval_as_leading_term(self, x):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return 1/arg

else:

return self.func(arg)

def _eval_is_real(self):

return self.args[0].is_real

def _eval_expand_trig(self, **hints):

from sympy import im, re

arg = self.args[0]

x = None

if arg.is_Add:

from sympy import symmetric_poly

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)))

else:

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)

def _eval_is_finite(self):

arg = self.args[0]

if arg.is_imaginary:

return True

def _eval_subs(self, old, new):

if self == old:

return new

arg = self.args[0]

argnew = arg.subs(old, new)

if arg != argnew and (argnew/S.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

# _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.

_is_even = None # optional, to be defined in subclass

_is_odd = None # optional, to be defined in subclass

@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)*S.Pi

return -cls(narg)

if 2*p > q:

narg = (1 - pi_coeff)*S.Pi

if cls._is_odd:

return cls(narg)

elif cls._is_even:

return -cls(narg)

t = cls._reciprocal_of.eval(arg)

if hasattr(arg, 'inverse') and arg.inverse() == cls:

return arg.args[0]

return 1/t if t != None else 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 != 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 != None and t != self._reciprocal_of(arg):

return 1/t

def _period(self, symbol):

f = 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):

return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)

def _eval_rewrite_as_Pow(self, arg):

return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg)

def _eval_rewrite_as_sin(self, arg):

return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg)

def _eval_rewrite_as_cos(self, arg):

return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg)

def _eval_rewrite_as_tan(self, arg):

return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg)

def _eval_rewrite_as_pow(self, arg):

return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg)

def _eval_rewrite_as_sqrt(self, arg):

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_real(self):

return self._reciprocal_of(self.args[0])._eval_is_real()

def _eval_as_leading_term(self, x):

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):

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).

Notes

=====

See :func:`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)

See Also

========

sin, csc, cos, tan, cot

asin, acsc, acos, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Trigonometric_functions

.. [2] http://dlmf.nist.gov/4.14

.. [3] http://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):

cot_half_sq = cot(arg/2)**2

return (cot_half_sq + 1)/(cot_half_sq - 1)

def _eval_rewrite_as_cos(self, arg):

return (1/cos(arg))

def _eval_rewrite_as_sincos(self, arg):

return sin(arg)/(cos(arg)*sin(arg))

def _eval_rewrite_as_sin(self, arg):

return (1 / cos(arg)._eval_rewrite_as_sin(arg))

def _eval_rewrite_as_tan(self, arg):

return (1 / cos(arg)._eval_rewrite_as_tan(arg))

def _eval_rewrite_as_csc(self, arg):

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)

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

# Reference Formula:

# http://functions.wolfram.com/ElementaryFunctions/Sec/06/01/02/01/

from sympy.functions.combinatorial.numbers import euler

if n < 0 or n % 2 == 1:

return S.Zero

else:

x = sympify(x)

k = n//2

return (-1)**k*euler(2*k)/factorial(2*k)*x**(2*k)

class csc(ReciprocalTrigonometricFunction):

"""

The cosecant function.

Returns the cosecant of x (measured in radians).

Notes

=====

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)

See Also

========

sin, cos, sec, tan, cot

asin, acsc, acos, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Trigonometric_functions

.. [2] http://dlmf.nist.gov/4.14

.. [3] http://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):

return (1/sin(arg))

def _eval_rewrite_as_sincos(self, arg):

return cos(arg)/(sin(arg)*cos(arg))

def _eval_rewrite_as_cot(self, arg):

cot_half = cot(arg/2)

return (1 + cot_half**2)/(2*cot_half)

def _eval_rewrite_as_cos(self, arg):

return (1 / sin(arg)._eval_rewrite_as_cos(arg))

def _eval_rewrite_as_sec(self, arg):

return sec(pi / 2 - arg, evaluate=False)

def _eval_rewrite_as_tan(self, arg):

return (1 / sin(arg)._eval_rewrite_as_tan(arg))

def fdiff(self, argindex=1):

if argindex == 1:

return -cot(self.args[0])*csc(self.args[0])

else:

raise ArgumentIndexError(self, argindex)

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

from sympy import bernoulli

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 ((-1)**(k - 1)*2*(2**(2*k - 1) - 1)*

bernoulli(2*k)*x**(2*k - 1)/factorial(2*k))

class sinc(TrigonometricFunction):

r"""Represents unnormalized sinc function

Examples

========

>>> from sympy import sinc, oo, jn, Product, Symbol

>>> from sympy.abc import x

>>> sinc(x)

sinc(x)

* Automated Evaluation

>>> sinc(0)

>>> sinc(oo)

* Differentiation

>>> sinc(x).diff()

(x*cos(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)

References

==========

.. [1] http://en.wikipedia.org/wiki/Sinc_function

"""

def fdiff(self, argindex=1):

x = self.args[0]

if argindex == 1:

return (x*cos(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.Infinity]:

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):

x = self.args[0]

return (sin(x)/x)._eval_nseries(x, n, logx)

def _eval_rewrite_as_jn(self, arg):

from sympy.functions.special.bessel import jn

return jn(0, arg)

def _eval_rewrite_as_sin(self, arg):

return sin(arg) / arg

###############################################################################

########################### TRIGONOMETRIC INVERSES ############################

###############################################################################

class InverseTrigonometricFunction(Function):

"""Base class for inverse trigonometric functions."""

pass

class asin(InverseTrigonometricFunction):

"""

The inverse sine function.

Returns the arcsine of x in radians.

Notes

=====

asin(x) will evaluate automatically in the cases oo, -oo, 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 asin, oo, pi

>>> asin(1)

pi/2

>>> asin(-1)

-pi/2

See Also

========

sin, csc, cos, sec, tan, cot

acsc, acos, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://dlmf.nist.gov/4.23

.. [3] http://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):

if self.args[0].is_positive:

return (self.args[0] - 1).is_negative

if self.args[0].is_negative:

return not (self.args[0] + 1).is_positive

@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 S.Zero:

return S.Zero

elif arg is S.One:

return S.Pi / 2

elif arg is S.NegativeOne:

return -S.Pi / 2

if arg.could_extract_minus_sign():

return -cls(-arg)

if arg.is_number:

cst_table = {

sqrt(3)/2: 3,

-sqrt(3)/2: -3,

sqrt(2)/2: 4,

-sqrt(2)/2: -4,

1/sqrt(2): 4,

-1/sqrt(2): -4,

sqrt((5 - sqrt(5))/8): 5,

-sqrt((5 - sqrt(5))/8): -5,

S.Half: 6,

-S.Half: -6,

sqrt(2 - sqrt(2))/2: 8,

-sqrt(2 - sqrt(2))/2: -8,

(sqrt(5) - 1)/4: 10,

(1 - sqrt(5))/4: -10,

(sqrt(3) - 1)/sqrt(2**3): 12,

(1 - sqrt(3))/sqrt(2**3): -12,

(sqrt(5) + 1)/4: S(10)/3,

-(sqrt(5) + 1)/4: -S(10)/3

}

if arg in cst_table:

return S.Pi / cst_table[arg]

i_coeff = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

return S.ImaginaryUnit * asinh(i_coeff)

@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):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return arg

else:

return self.func(arg)

def _eval_rewrite_as_acos(self, x):

return S.Pi/2 - acos(x)

def _eval_rewrite_as_atan(self, x):

return 2*atan(x/(1 + sqrt(1 - x**2)))

def _eval_rewrite_as_log(self, x):

return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2))

def _eval_rewrite_as_acot(self, arg):

return 2*acot((1 + sqrt(1 - arg**2))/arg)

def _eval_rewrite_as_asec(self, arg):

return S.Pi/2 - asec(1/arg)

def _eval_rewrite_as_acsc(self, arg):

return acsc(1/arg)

def _eval_is_real(self):

x = self.args[0]

return x.is_real and (1 - abs(x)).is_nonnegative

def inverse(self, argindex=1):

"""

Returns the inverse of this function.

"""

return sin

class acos(InverseTrigonometricFunction):

"""

The inverse cosine function.

Returns the arc cosine of x (measured in radians).

Notes

=====

``acos(x)`` will evaluate automatically in the cases

``oo``, ``-oo``, ``0``, ``1``, ``-1``.

``acos(zoo)`` evaluates to ``zoo``

(see note in :py:class`sympy.functions.elementary.trigonometric.asec`)

Examples

========

>>> from sympy import acos, oo, pi

>>> acos(1)

>>> acos(0)

pi/2

>>> acos(oo)

oo*I

See Also

========

sin, csc, cos, sec, tan, cot

asin, acsc, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://dlmf.nist.gov/4.23

.. [3] http://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

def _eval_is_positive(self):

x = self.args[0]

return (1 - abs(x)).is_nonnegative

@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 S.Zero:

return S.Pi / 2

elif arg is S.One:

return S.Zero

elif arg is S.NegativeOne:

return S.Pi

if arg is S.ComplexInfinity:

return S.ComplexInfinity

if arg.is_number:

cst_table = {

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,

}

if arg in cst_table:

return cst_table[arg]

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

if n == 0:

return S.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):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return arg

else:

return self.func(arg)

def _eval_is_real(self):

x = self.args[0]

return x.is_real and (1 - abs(x)).is_nonnegative

def _eval_rewrite_as_log(self, x):

return S.Pi/2 + S.ImaginaryUnit * \

log(S.ImaginaryUnit * x + sqrt(1 - x**2))

def _eval_rewrite_as_asin(self, x):

return S.Pi/2 - asin(x)

def _eval_rewrite_as_atan(self, x):

return atan(sqrt(1 - x**2)/x) + (S.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):

return S.Pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg)

def _eval_rewrite_as_asec(self, arg):

return asec(1/arg)

def _eval_rewrite_as_acsc(self, arg):

return S.Pi/2 - acsc(1/arg)

def _eval_conjugate(self):

z = self.args[0]

r = self.func(self.args[0].conjugate())

if z.is_real is False:

return r

elif z.is_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive:

return r

class atan(InverseTrigonometricFunction):

"""

The inverse tangent function.

Returns the arc tangent of x (measured in radians).

Notes

=====

atan(x) will evaluate automatically in the cases

oo, -oo, 0, 1, -1.

Examples

========

>>> from sympy import atan, oo, pi

>>> atan(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] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://dlmf.nist.gov/4.23

.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan

"""

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_positive

def _eval_is_nonnegative(self):

return self.args[0].is_nonnegative

@classmethod

def eval(cls, arg):

if arg.is_Number:

if arg is S.NaN:

return S.NaN

elif arg is S.Infinity:

return S.Pi / 2

elif arg is S.NegativeInfinity:

return -S.Pi / 2

elif arg is S.Zero:

return S.Zero

elif arg is S.One:

return S.Pi / 4

elif arg is S.NegativeOne:

return -S.Pi / 4

if arg.could_extract_minus_sign():

return -cls(-arg)

if arg.is_number:

cst_table = {

sqrt(3)/3: 6,

-sqrt(3)/3: -6,

1/sqrt(3): 6,

-1/sqrt(3): -6,

sqrt(3): 3,

-sqrt(3): -3,

(1 + sqrt(2)): S(8)/3,

-(1 + sqrt(2)): S(8)/3,

(sqrt(2) - 1): 8,

(1 - sqrt(2)): -8,

sqrt((5 + 2*sqrt(5))): S(5)/2,

-sqrt((5 + 2*sqrt(5))): -S(5)/2,

(2 - sqrt(3)): 12,

-(2 - sqrt(3)): -12

}

if arg in cst_table:

return S.Pi / cst_table[arg]

i_coeff = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

return S.ImaginaryUnit * atanh(i_coeff)

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

if n < 0 or n % 2 == 0:

return S.Zero

else:

x = sympify(x)

return (-1)**((n - 1)//2) * x**n / n

def _eval_as_leading_term(self, x):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return arg

else:

return self.func(arg)

def _eval_is_real(self):

return self.args[0].is_real

def _eval_rewrite_as_log(self, x):

return S.ImaginaryUnit/2 * (log(

(S(1) - S.ImaginaryUnit * x)/(S(1) + S.ImaginaryUnit * x)))

def _eval_aseries(self, n, args0, x, logx):

if args0[0] == S.Infinity:

return (S.Pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)

elif args0[0] == S.NegativeInfinity:

return (-S.Pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)

else:

return super(atan, self)._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):

return sqrt(arg**2)/arg*(S.Pi/2 - asin(1/sqrt(1 + arg**2)))

def _eval_rewrite_as_acos(self, arg):

return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2))

def _eval_rewrite_as_acot(self, arg):

return acot(1/arg)

def _eval_rewrite_as_asec(self, arg):

return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2))

def _eval_rewrite_as_acsc(self, arg):

return sqrt(arg**2)/arg*(S.Pi/2 - acsc(sqrt(1 + arg**2)))

class acot(InverseTrigonometricFunction):

"""

The inverse cotangent function.

Returns the arc cotangent of x (measured in radians).

See Also

========

sin, csc, cos, sec, tan, cot

asin, acsc, acos, asec, atan, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://dlmf.nist.gov/4.23

.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCot

"""

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_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 S.Zero:

return S.Pi/ 2

elif arg is S.One:

return S.Pi / 4

elif arg is S.NegativeOne:

return -S.Pi / 4

if arg.could_extract_minus_sign():

return -cls(-arg)

if arg.is_number:

cst_table = {

sqrt(3)/3: 3,

-sqrt(3)/3: -3,

1/sqrt(3): 3,

-1/sqrt(3): -3,

sqrt(3): 6,

-sqrt(3): -6,

(1 + sqrt(2)): 8,

-(1 + sqrt(2)): -8,

(1 - sqrt(2)): -S(8)/3,

(sqrt(2) - 1): S(8)/3,

sqrt(5 + 2*sqrt(5)): 10,

-sqrt(5 + 2*sqrt(5)): -10,

(2 + sqrt(3)): 12,

-(2 + sqrt(3)): -12,

(2 - sqrt(3)): S(12)/5,

-(2 - sqrt(3)): -S(12)/5,

}

if arg in cst_table:

return S.Pi / cst_table[arg]

i_coeff = arg.as_coefficient(S.ImaginaryUnit)

if i_coeff is not None:

return -S.ImaginaryUnit * acoth(i_coeff)

@staticmethod

@cacheit

def taylor_term(n, x, *previous_terms):

if n == 0:

return S.Pi / 2 # FIX THIS

elif n < 0 or n % 2 == 0:

return S.Zero

else:

x = sympify(x)

return (-1)**((n + 1)//2) * x**n / n

def _eval_as_leading_term(self, x):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if x in arg.free_symbols and Order(1, x).contains(arg):

return arg

else:

return self.func(arg)

def _eval_is_real(self):

return self.args[0].is_real

def _eval_aseries(self, n, args0, x, logx):

if args0[0] == S.Infinity:

return (S.Pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)

elif args0[0] == S.NegativeInfinity:

return (3*S.Pi/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):

return S.ImaginaryUnit/2 * \

(log((x - S.ImaginaryUnit)/(x + S.ImaginaryUnit)))

def inverse(self, argindex=1):

"""

Returns the inverse of this function.

"""

return cot

def _eval_rewrite_as_asin(self, arg):

return (arg*sqrt(1/arg**2)*

(S.Pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1))))

def _eval_rewrite_as_acos(self, arg):

return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1))

def _eval_rewrite_as_atan(self, arg):

return atan(1/arg)

def _eval_rewrite_as_asec(self, arg):

return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2))

def _eval_rewrite_as_acsc(self, arg):

return arg*sqrt(1/arg**2)*(S.Pi/2 - acsc(sqrt((1 + arg**2)/arg**2)))

class asec(InverseTrigonometricFunction):

"""

The inverse secant function.

Returns the arc secant of x (measured in radians).

Notes

=====

``asec(x)`` will evaluate automatically in the cases

``oo``, ``-oo``, ``0``, ``1``, ``-1``.

``asec(x)`` has branch cut in the interval [-1, 1]. For complex arguments,

it can be defined [4]_ as

.. math::

sec^{-1}(z) = -i*(log(\sqrt{1 - z^2} + 1) / 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*(log(-\sqrt{1 - z^2} + 1) / z)

simplifies to :math:`-i*log(z/2 + O(z^3))` which ultimately evaluates to

``zoo``.

As ``asex(x)`` = ``asec(1/x)``, a similar argument can be given for

``acos(x)``.

Examples

========

>>> from sympy import asec, oo, pi

>>> asec(1)

>>> asec(-1)

See Also

========

sin, csc, cos, sec, tan, cot

asin, acsc, acos, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://dlmf.nist.gov/4.23

.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec

.. [4] http://refrence.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 S.Pi

if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:

return S.Pi/2

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

def _eval_as_leading_term(self, x):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if Order(1,x).contains(arg):

return log(arg)

else:

return self.func(arg)

def _eval_is_real(self):

x = self.args[0]

if x.is_real is False:

return False

return (x - 1).is_nonnegative or (-x - 1).is_nonnegative

def _eval_rewrite_as_log(self, arg):

return S.Pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))

def _eval_rewrite_as_asin(self, arg):

return S.Pi/2 - asin(1/arg)

def _eval_rewrite_as_acos(self, arg):

return acos(1/arg)

def _eval_rewrite_as_atan(self, arg):

return sqrt(arg**2)/arg*(-S.Pi/2 + 2*atan(arg + sqrt(arg**2 - 1)))

def _eval_rewrite_as_acot(self, arg):

return sqrt(arg**2)/arg*(-S.Pi/2 + 2*acot(arg - sqrt(arg**2 - 1)))

def _eval_rewrite_as_acsc(self, arg):

return S.Pi/2 - acsc(arg)

class acsc(InverseTrigonometricFunction):

"""

The inverse cosecant function.

Returns the arc cosecant of x (measured in radians).

Notes

=====

acsc(x) will evaluate automatically in the cases

oo, -oo, 0, 1, -1.

Examples

========

>>> from sympy import acsc, oo, pi

>>> acsc(1)

pi/2

>>> acsc(-1)

-pi/2

See Also

========

sin, csc, cos, sec, tan, cot

asin, acos, asec, atan, acot, atan2

References

==========

.. [1] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://dlmf.nist.gov/4.23

.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc

"""

@classmethod

def eval(cls, arg):

if arg.is_Number:

if arg is S.NaN:

return S.NaN

elif arg is S.One:

return S.Pi/2

elif arg is S.NegativeOne:

return -S.Pi/2

if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:

return S.Zero

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

def _eval_as_leading_term(self, x):

from sympy import Order

arg = self.args[0].as_leading_term(x)

if Order(1,x).contains(arg):

return log(arg)

else:

return self.func(arg)

def _eval_rewrite_as_log(self, arg):

return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))

def _eval_rewrite_as_asin(self, arg):

return asin(1/arg)

def _eval_rewrite_as_acos(self, arg):

return S.Pi/2 - acos(1/arg)

def _eval_rewrite_as_atan(self, arg):

return sqrt(arg**2)/arg*(S.Pi/2 - atan(sqrt(arg**2 - 1)))

def _eval_rewrite_as_acot(self, arg):

return sqrt(arg**2)/arg*(S.Pi/2 - acot(1/sqrt(arg**2 - 1)))

def _eval_rewrite_as_asec(self, arg):

return S.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.

Examples

========

Going counter-clock wise around the origin we find the

following angles:

>>> from sympy import atan2

>>> atan2(0, 1)

>>> atan2(1, 1)

pi/4

>>> atan2(1, 0)

pi/2

>>> atan2(1, -1)

3*pi/4

>>> atan2(0, -1)

>>> 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)

2*atan(y/(x + sqrt(x**2 + y**2)))

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] http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

.. [2] http://en.wikipedia.org/wiki/Atan2

.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan2

"""

@classmethod

def eval(cls, y, x):

from sympy import Heaviside, im, re

if x is S.NegativeInfinity:

if y.is_zero:

# Special case y = 0 because we define Heaviside(0) = 1/2

return S.Pi

return 2*S.Pi*(Heaviside(re(y))) - S.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_real and y.is_real:

if x.is_positive:

return atan(y / x)

elif x.is_negative:

if y.is_negative:

return atan(y / x) - S.Pi

elif y.is_nonnegative:

return atan(y / x) + S.Pi

elif x.is_zero:

if y.is_positive:

return S.Pi/2

elif y.is_negative:

return -S.Pi/2

elif y.is_zero:

return S.NaN

if y.is_zero and x.is_real and fuzzy_not(x.is_zero):

return S.Pi * (S.One - Heaviside(x))

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):

return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y) / sqrt(x**2 + y**2))

def _eval_rewrite_as_atan(self, y, x):

return 2*atan(y / (sqrt(x**2 + y**2) + x))

def _eval_rewrite_as_arg(self, y, x):

from sympy import arg

if x.is_real and y.is_real:

return arg(x + y*S.ImaginaryUnit)

I = S.ImaginaryUnit

n = x + I*y

d = x**2 + y**2

return arg(n/sqrt(d)) - I*log(abs(n)/sqrt(abs(d)))

def _eval_is_real(self):

return self.args[0].is_real and self.args[1].is_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_real and y.is_real:

super(atan2, self)._eval_evalf(prec)

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