Source code for sympy.functions.elementary.hyperbolic

from __future__ import print_function, division

from sympy.core import S, sympify, cacheit
from sympy.core.function import Function, ArgumentIndexError, _coeff_isneg

from sympy.functions.elementary.miscellaneous import sqrt

from sympy.functions.elementary.exponential import exp, log
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial


def _rewrite_hyperbolics_as_exp(expr):
    expr = sympify(expr)
    return expr.xreplace(dict([(h, h.rewrite(exp))
        for h in expr.atoms(HyperbolicFunction)]))


###############################################################################
########################### 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): from sympy import sin 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 * 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) / 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. """ from sympy import cos, sin 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)*cos(im), cosh(re)*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 (exp(arg) - exp(-arg)) / 2 def _eval_rewrite_as_exp(self, arg): return (exp(arg) - 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): 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
[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): from sympy import cos 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 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)/factorial(n) def _eval_conjugate(self): return self.func(self.args[0].conjugate()) def as_real_imag(self, deep=True, **hints): from sympy import cos, sin 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)*cos(im), sinh(re)*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 (exp(arg) + exp(-arg)) / 2 def _eval_rewrite_as_exp(self, arg): return (exp(arg) + 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): 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_imaginary: return True
[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): from sympy import tan 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 * tan(-i_coeff) return S.ImaginaryUnit * 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): from sympy import bernoulli if n < 0 or n % 2 == 0: return S.Zero else: x = sympify(x) a = 2**(n + 1) B = bernoulli(n + 1) F = 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): from sympy import cos, sin 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 + cos(im)**2 return (sinh(re)*cosh(re)/denom, sin(im)*cos(im)/denom) def _eval_rewrite_as_tractable(self, arg): neg_exp, pos_exp = exp(-arg), exp(arg) return (pos_exp - neg_exp)/(pos_exp + neg_exp) def _eval_rewrite_as_exp(self, arg): neg_exp, pos_exp = exp(-arg), 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): 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
[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): from sympy import cot 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.ComplexInfinity 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 * cot(-i_coeff) return -S.ImaginaryUnit * 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): 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 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): from sympy import cos, sin 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 + sin(im)**2 return (sinh(re)*cosh(re)/denom, -sin(im)*cos(im)/denom) def _eval_rewrite_as_tractable(self, arg): neg_exp, pos_exp = exp(-arg), exp(arg) return (pos_exp + neg_exp)/(pos_exp - neg_exp) def _eval_rewrite_as_exp(self, arg): neg_exp, pos_exp = exp(-arg), 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): 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)
class ReciprocalHyperbolicFunction(HyperbolicFunction): """Base class for reciprocal functions of hyperbolic functions. """ #To be defined in class _reciprocal_of = None _is_even = None _is_odd = 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) 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 _eval_rewrite_as_exp(self, arg): return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg) def _eval_rewrite_as_tractable(self, arg): return self._rewrite_reciprocal("_eval_rewrite_as_tractable", arg) def _eval_rewrite_as_tanh(self, arg): return self._rewrite_reciprocal("_eval_rewrite_as_tanh", arg) def _eval_rewrite_as_coth(self, arg): return self._rewrite_reciprocal("_eval_rewrite_as_coth", arg) def as_real_imag(self, deep = True, **hints): return (1 / self._reciprocal_of(self.args[0])).as_real_imag(deep, **hints) def _eval_conjugate(self): return self.func(self.args[0].conjugate()) def _eval_expand_complex(self, deep=True, **hints): re_part, im_part = self.as_real_imag(deep=True, **hints) return re_part + S.ImaginaryUnit*im_part def _eval_as_leading_term(self, x): return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x) def _eval_is_real(self): return self._reciprocal_of(self.args[0]).is_real def _eval_is_finite(self): return (1/self._reciprocal_of(self.args[0])).is_finite
[docs]class csch(ReciprocalHyperbolicFunction): r""" The hyperbolic cosecant function, `\frac{2}{e^x - e^{-x}}` * csch(x) -> Returns the hyperbolic cosecant of x See Also ======== sinh, cosh, tanh, sech, asinh, acosh """ _reciprocal_of = sinh _is_odd = True
[docs] def fdiff(self, argindex=1): """ Returns the first derivative of this function """ if argindex == 1: return -coth(self.args[0]) * csch(self.args[0]) else: raise ArgumentIndexError(self, argindex)
@staticmethod @cacheit
[docs] def taylor_term(n, x, *previous_terms): """ Returns the next term in the Taylor series expansion """ 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 2 * (1 - 2**n) * B/F * x**n
def _eval_rewrite_as_cosh(self, arg): return S.ImaginaryUnit / cosh(arg + S.ImaginaryUnit * S.Pi / 2) def _sage_(self): import sage.all as sage return sage.csch(self.args[0]._sage_())
[docs]class sech(ReciprocalHyperbolicFunction): r""" The hyperbolic secant function, `\frac{2}{e^x + e^{-x}}` * sech(x) -> Returns the hyperbolic secant of x See Also ======== sinh, cosh, tanh, coth, csch, asinh, acosh """ _reciprocal_of = cosh _is_even = True def fdiff(self, argindex=1): if argindex == 1: return - tanh(self.args[0])*sech(self.args[0]) else: raise ArgumentIndexError(self, argindex) @staticmethod @cacheit def taylor_term(n, x, *previous_terms): from sympy.functions.combinatorial.numbers import euler if n < 0 or n % 2 == 1: return S.Zero else: x = sympify(x) return euler(n) / factorial(n) * x**(n) def _eval_rewrite_as_sinh(self, arg): return S.ImaginaryUnit / sinh(arg + S.ImaginaryUnit * S.Pi /2) def _sage_(self): import sage.all as sage return sage.sech(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): from sympy import asin arg = sympify(arg) if arg.is_Number: if arg is S.NaN: return S.NaN elif arg is S.Infinity: return S.Infinity elif arg is S.NegativeInfinity: return S.NegativeInfinity elif arg is S.Zero: return S.Zero elif arg is S.One: return log(sqrt(2) + 1) elif arg is S.NegativeOne: return log(sqrt(2) - 1) elif arg.is_negative: return -cls(-arg) else: if arg is S.ComplexInfinity: return S.ComplexInfinity i_coeff = arg.as_coefficient(S.ImaginaryUnit) if i_coeff is not None: return S.ImaginaryUnit * asin(i_coeff) else: if _coeff_isneg(arg): return -cls(-arg) @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 (-1)**k * 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_log(self, x): """ Rewrites asinh as log function. """ return log(x + sqrt(x**2 + 1))
[docs] def inverse(self, argindex=1): """ Returns the inverse of this function. """ return sinh
[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: log(S.ImaginaryUnit*(1 + sqrt(2))), -S.ImaginaryUnit: log(-S.ImaginaryUnit*(1 + sqrt(2))), S.Half: S.Pi/3, -S.Half: 2*S.Pi/3, sqrt(2)/2: S.Pi/4, -sqrt(2)/2: 3*S.Pi/4, 1/sqrt(2): S.Pi/4, -1/sqrt(2): 3*S.Pi/4, sqrt(3)/2: S.Pi/6, -sqrt(3)/2: 5*S.Pi/6, (sqrt(3) - 1)/sqrt(2**3): 5*S.Pi/12, -(sqrt(3) - 1)/sqrt(2**3): 7*S.Pi/12, sqrt(2 + sqrt(2))/2: S.Pi/8, -sqrt(2 + sqrt(2))/2: 7*S.Pi/8, sqrt(2 - sqrt(2))/2: 3*S.Pi/8, -sqrt(2 - sqrt(2))/2: 5*S.Pi/8, (1 + sqrt(3))/(2*sqrt(2)): S.Pi/12, -(1 + sqrt(3))/(2*sqrt(2)): 11*S.Pi/12, (sqrt(5) + 1)/4: S.Pi/5, -(sqrt(5) + 1)/4: 4*S.Pi/5 } if arg in cst_table: if arg.is_real: return cst_table[arg]*S.ImaginaryUnit return cst_table[arg] if arg.is_infinite: return S.Infinity @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 = RisingFactorial(S.Half, k) F = factorial(k) return -R / F * S.ImaginaryUnit * 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 S.ImaginaryUnit*S.Pi/2 else: return self.func(arg)
[docs] def inverse(self, argindex=1): """ Returns the inverse of this function. """ return cosh
[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): from sympy import atan 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 * atan(arg) elif arg is S.NegativeInfinity: return S.ImaginaryUnit * atan(-arg) elif arg.is_negative: return -cls(-arg) else: if arg is S.ComplexInfinity: return S.NaN i_coeff = arg.as_coefficient(S.ImaginaryUnit) if i_coeff is not None: return S.ImaginaryUnit * atan(i_coeff) else: if _coeff_isneg(arg): return -cls(-arg) @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): 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)
[docs] def inverse(self, argindex=1): """ Returns the inverse of this function. """ return tanh
[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): from sympy import acot 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 * 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): 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.ImaginaryUnit*S.Pi/2 else: return self.func(arg)
[docs] def inverse(self, argindex=1): """ Returns the inverse of this function. """ return coth
[docs]class asech(Function): """ The inverse hyperbolic secant function. * asech(x) -> Returns the inverse hyperbolic secant of x Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(-x**2 + 1)) >>> asech(1).diff(x) 0 >>> asech(1) 0 >>> asech(S(2)) I*pi/3 >>> asech(-sqrt(2)) 3*I*pi/4 >>> asech((sqrt(6) - sqrt(2))) I*pi/12 See Also ======== asinh, atanh, cosh, acoth References ========== .. [1] http://en.wikipedia.org/wiki/Hyperbolic_function .. [2] http://dlmf.nist.gov/4.37 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/ """ def fdiff(self, argindex=1): if argindex == 1: z = self.args[0] return -1/(z*sqrt(1 - z**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.Pi*S.ImaginaryUnit / 2 elif arg is S.NegativeInfinity: return S.Pi*S.ImaginaryUnit / 2 elif arg is S.Zero: return S.Infinity 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: - (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)), -S.ImaginaryUnit: (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)), (sqrt(6) - sqrt(2)): S.Pi / 12, (sqrt(2) - sqrt(6)): 11*S.Pi / 12, sqrt(2 - 2/sqrt(5)): S.Pi / 10, -sqrt(2 - 2/sqrt(5)): 9*S.Pi / 10, 2 / sqrt(2 + sqrt(2)): S.Pi / 8, -2 / sqrt(2 + sqrt(2)): 7*S.Pi / 8, 2 / sqrt(3): S.Pi / 6, -2 / sqrt(3): 5*S.Pi / 6, (sqrt(5) - 1): S.Pi / 5, (1 - sqrt(5)): 4*S.Pi / 5, sqrt(2): S.Pi / 4, -sqrt(2): 3*S.Pi / 4, sqrt(2 + 2/sqrt(5)): 3*S.Pi / 10, -sqrt(2 + 2/sqrt(5)): 7*S.Pi / 10, S(2): S.Pi / 3, -S(2): 2*S.Pi / 3, sqrt(2*(2 + sqrt(2))): 3*S.Pi / 8, -sqrt(2*(2 + sqrt(2))): 5*S.Pi / 8, (1 + sqrt(5)): 2*S.Pi / 5, (-1 - sqrt(5)): 3*S.Pi / 5, (sqrt(6) + sqrt(2)): 5*S.Pi / 12, (-sqrt(6) - sqrt(2)): 7*S.Pi / 12, } 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.NaN @staticmethod @cacheit def expansion_term(n, x, *previous_terms): if n == 0: return 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)**2 // (n // 2)**2 * x**2 / 4 else: k = n // 2 R = RisingFactorial(S.Half , k) * n F = factorial(k) * n // 2 * n // 2 return -1 * R / F * x**n / 4
[docs] def inverse(self, argindex=1): """ Returns the inverse of this function. """ return sech
def _eval_rewrite_as_log(self, arg): return log(1/arg + sqrt(1/arg**2 - 1))