from __future__ import print_function, division
from sympy.core import S, C, sympify, cacheit
from sympy.core.function import Function, ArgumentIndexError, _coeff_isneg
from sympy.functions.elementary.miscellaneous import sqrt
###############################################################################
########################### HYPERBOLIC FUNCTIONS ##############################
###############################################################################
[docs]class HyperbolicFunction(Function):
"""
Base class for hyperbolic functions.
See Also
========
sinh, cosh, tanh, coth
"""
unbranched = True
[docs]class sinh(HyperbolicFunction):
r"""
The hyperbolic sine function, `\frac{e^x - e^{-x}}{2}`.
* sinh(x) -> Returns the hyperbolic sine of x
See Also
========
cosh, tanh, asinh
"""
[docs] def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return cosh(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return asinh
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * C.sin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.func == asinh:
return arg.args[0]
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1)
if arg.func == atanh:
x = arg.args[0]
return x/sqrt(1 - x**2)
if arg.func == acoth:
x = arg.args[0]
return 1/(sqrt(x - 1) * sqrt(x + 1))
@staticmethod
@cacheit
[docs] def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion.
"""
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 x**(n) / C.factorial(n)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
[docs] def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a complex coordinate.
"""
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, 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 (sinh(re)*C.cos(im), cosh(re)*C.sin(im))
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 _eval_expand_trig(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
x, y = arg.as_two_terms()
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One and coeff.is_Integer and terms is not S.One:
x = terms
y = (coeff - 1)*x
if x is not None:
return (sinh(x)*cosh(y) + sinh(y)*cosh(x)).expand(trig=True)
return sinh(arg)
def _eval_rewrite_as_tractable(self, arg):
return (C.exp(arg) - C.exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg):
return (C.exp(arg) - C.exp(-arg)) / 2
def _eval_rewrite_as_cosh(self, arg):
return -S.ImaginaryUnit*cosh(arg + S.Pi*S.ImaginaryUnit/2)
def _eval_rewrite_as_tanh(self, arg):
tanh_half = tanh(S.Half*arg)
return 2*tanh_half/(1 - tanh_half**2)
def _eval_rewrite_as_coth(self, arg):
coth_half = coth(S.Half*arg)
return 2*coth_half/(coth_half**2 - 1)
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.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_bounded(self):
arg = self.args[0]
if arg.is_imaginary:
return True
def _sage_(self):
import sage.all as sage
return sage.sinh(self.args[0]._sage_())
[docs]class cosh(HyperbolicFunction):
r"""
The hyperbolic cosine function, `\frac{e^x + e^{-x}}{2}`.
* cosh(x) -> Returns the hyperbolic cosine of x
See Also
========
sinh, tanh, acosh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return sinh(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.One
elif arg.is_negative:
return cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cos(i_coeff)
else:
if _coeff_isneg(arg):
return cls(-arg)
if arg.func == asinh:
return sqrt(1 + arg.args[0]**2)
if arg.func == acosh:
return arg.args[0]
if arg.func == atanh:
return 1/sqrt(1 - arg.args[0]**2)
if arg.func == acoth:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
@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 x**(n)/C.factorial(n)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, 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 (cosh(re)*C.cos(im), sinh(re)*C.sin(im))
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 _eval_expand_trig(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
x, y = arg.as_two_terms()
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One and coeff.is_Integer and terms is not S.One:
x = terms
y = (coeff - 1)*x
if x is not None:
return (cosh(x)*cosh(y) + sinh(x)*sinh(y)).expand(trig=True)
return cosh(arg)
def _eval_rewrite_as_tractable(self, arg):
return (C.exp(arg) + C.exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg):
return (C.exp(arg) + C.exp(-arg)) / 2
def _eval_rewrite_as_sinh(self, arg):
return -S.ImaginaryUnit*sinh(arg + S.Pi*S.ImaginaryUnit/2)
def _eval_rewrite_as_tanh(self, arg):
tanh_half = tanh(S.Half*arg)**2
return (1 + tanh_half)/(1 - tanh_half)
def _eval_rewrite_as_coth(self, arg):
coth_half = coth(S.Half*arg)**2
return (coth_half + 1)/(coth_half - 1)
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.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_bounded(self):
arg = self.args[0]
if arg.is_imaginary:
return True
def _sage_(self):
import sage.all as sage
return sage.cosh(self.args[0]._sage_())
[docs]class tanh(HyperbolicFunction):
r"""
The hyperbolic tangent function, `\frac{\sinh(x)}{\cosh(x)}`.
* tanh(x) -> Returns the hyperbolic tangent of x
See Also
========
sinh, cosh, atanh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return S.One - tanh(self.args[0])**2
else:
raise ArgumentIndexError(self, argindex)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atanh
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return -S.ImaginaryUnit * C.tan(-i_coeff)
return S.ImaginaryUnit * C.tan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.func == asinh:
x = arg.args[0]
return x/sqrt(1 + x**2)
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1) / x
if arg.func == atanh:
return arg.args[0]
if arg.func == acoth:
return 1/arg.args[0]
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a = 2**(n + 1)
B = C.bernoulli(n + 1)
F = C.factorial(n + 1)
return a*(a - 1) * B/F * x**n
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.cos(im)**2
return (sinh(re)*cosh(re)/denom, C.sin(im)*C.cos(im)/denom)
def _eval_rewrite_as_tractable(self, arg):
neg_exp, pos_exp = C.exp(-arg), C.exp(arg)
return (pos_exp - neg_exp)/(pos_exp + neg_exp)
def _eval_rewrite_as_exp(self, arg):
neg_exp, pos_exp = C.exp(-arg), C.exp(arg)
return (pos_exp - neg_exp)/(pos_exp + neg_exp)
def _eval_rewrite_as_sinh(self, arg):
return S.ImaginaryUnit*sinh(arg)/sinh(S.Pi*S.ImaginaryUnit/2 - arg)
def _eval_rewrite_as_cosh(self, arg):
return S.ImaginaryUnit*cosh(S.Pi*S.ImaginaryUnit/2 - arg)/cosh(arg)
def _eval_rewrite_as_coth(self, arg):
return 1/coth(arg)
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.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_bounded(self):
arg = self.args[0]
if arg.is_real:
return True
def _sage_(self):
import sage.all as sage
return sage.tanh(self.args[0]._sage_())
[docs]class coth(HyperbolicFunction):
r"""
The hyperbolic cotangent function, `\frac{\cosh(x)}{\sinh(x)}`.
* coth(x) -> Returns the hyperbolic cotangent of x
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sinh(self.args[0])**2
else:
raise ArgumentIndexError(self, argindex)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acoth
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return S.ImaginaryUnit * C.cot(-i_coeff)
return -S.ImaginaryUnit * C.cot(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.func == asinh:
x = arg.args[0]
return sqrt(1 + x**2)/x
if arg.func == acosh:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
if arg.func == atanh:
return 1/arg.args[0]
if arg.func == acoth:
return arg.args[0]
@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 = C.bernoulli(n + 1)
F = C.factorial(n + 1)
return 2**(n + 1) * B/F * x**n
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.sin(im)**2
return (sinh(re)*cosh(re)/denom, -C.sin(im)*C.cos(im)/denom)
def _eval_rewrite_as_tractable(self, arg):
neg_exp, pos_exp = C.exp(-arg), C.exp(arg)
return (pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_exp(self, arg):
neg_exp, pos_exp = C.exp(-arg), C.exp(arg)
return (pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_sinh(self, arg):
return -S.ImaginaryUnit*sinh(S.Pi*S.ImaginaryUnit/2 - arg)/sinh(arg)
def _eval_rewrite_as_cosh(self, arg):
return -S.ImaginaryUnit*cosh(arg)/cosh(S.Pi*S.ImaginaryUnit/2 - arg)
def _eval_rewrite_as_tanh(self, arg):
return 1/tanh(arg)
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.Order(1, x).contains(arg):
return S.One
else:
return self.func(arg)
def _sage_(self):
import sage.all as sage
return sage.coth(self.args[0]._sage_())
###############################################################################
############################# HYPERBOLIC INVERSES #############################
###############################################################################
[docs]class asinh(Function):
"""
The inverse hyperbolic sine function.
* asinh(x) -> Returns the inverse hyperbolic sine of x
See Also
========
acosh, atanh, sinh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(self.args[0]**2 + 1)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return C.log(sqrt(2) + 1)
elif arg is S.NegativeOne:
return C.log(sqrt(2) - 1)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.ComplexInfinity
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * C.asin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-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 = C.RisingFactorial(S.Half, k)
F = C.factorial(k)
return (-1)**k * R / F * x**n / n
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sinh
def _sage_(self):
import sage.all as sage
return sage.asinh(self.args[0]._sage_())
[docs]class acosh(Function):
"""
The inverse hyperbolic cosine function.
* acosh(x) -> Returns the inverse hyperbolic cosine of x
See Also
========
asinh, atanh, cosh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(self.args[0]**2 - 1)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit: C.log(S.ImaginaryUnit*(1 + sqrt(2))),
-S.ImaginaryUnit: C.log(-S.ImaginaryUnit*(1 + sqrt(2))),
S.Half: S.Pi/3,
-S.Half: 2*S.Pi/3,
sqrt(2)/2: S.Pi/4,
-sqrt(2)/2: 3*S.Pi/4,
1/sqrt(2): S.Pi/4,
-1/sqrt(2): 3*S.Pi/4,
sqrt(3)/2: S.Pi/6,
-sqrt(3)/2: 5*S.Pi/6,
(sqrt(3) - 1)/sqrt(2**3): 5*S.Pi/12,
-(sqrt(3) - 1)/sqrt(2**3): 7*S.Pi/12,
sqrt(2 + sqrt(2))/2: S.Pi/8,
-sqrt(2 + sqrt(2))/2: 7*S.Pi/8,
sqrt(2 - sqrt(2))/2: 3*S.Pi/8,
-sqrt(2 - sqrt(2))/2: 5*S.Pi/8,
(1 + sqrt(3))/(2*sqrt(2)): S.Pi/12,
-(1 + sqrt(3))/(2*sqrt(2)): 11*S.Pi/12,
(sqrt(5) + 1)/4: S.Pi/5,
-(sqrt(5) + 1)/4: 4*S.Pi/5
}
if arg in cst_table:
if arg.is_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
return S.Infinity
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return S.ImaginaryUnit * C.acos(i_coeff)
return S.ImaginaryUnit * C.acos(-i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi*S.ImaginaryUnit / 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 = C.RisingFactorial(S.Half, k)
F = C.factorial(k)
return -R / F * S.ImaginaryUnit * x**n / n
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cosh
def _sage_(self):
import sage.all as sage
return sage.acosh(self.args[0]._sage_())
[docs]class atanh(Function):
"""
The inverse hyperbolic tangent function.
* atanh(x) -> Returns the inverse hyperbolic tangent of x
See Also
========
asinh, acosh, tanh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * C.atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * C.atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * C.atan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return x**n / n
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tanh
def _sage_(self):
import sage.all as sage
return sage.atanh(self.args[0]._sage_())
[docs]class acoth(Function):
"""
The inverse hyperbolic cotangent function.
* acoth(x) -> Returns the inverse hyperbolic cotangent of x
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return 0
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * C.acot(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi*S.ImaginaryUnit / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return x**n / n
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
[docs] def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return coth
def _sage_(self):
import sage.all as sage
return sage.acoth(self.args[0]._sage_())