# Source code for sympy.functions.elementary.hyperbolic

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 """ nargs = 1
[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 """ nargs = 1 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 """ nargs = 1 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 """ nargs = 1 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 """ nargs = 1 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 """ nargs = 1 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 """ nargs = 1 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 """ nargs = 1 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_())