from sympy.core.logic import FuzzyBool from sympy.core import S, sympify, cacheit, pi, I, Rational from sympy.core.add import Add from sympy.core.function import Function, ArgumentIndexError from sympy.functions.combinatorial.factorials import factorial, RisingFactorial from sympy.functions.elementary.exponential import exp, log, match_real_imag from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.integers import floor from sympy.core.logic import fuzzy_or, fuzzy_and def _rewrite_hyperbolics_as_exp(expr): expr = sympify(expr) return expr.xreplace({h: h.rewrite(exp) for h in expr.atoms(HyperbolicFunction)}) ############################################################################### ########################### HYPERBOLIC FUNCTIONS ############################## ############################################################################### class HyperbolicFunction(Function): """ Base class for hyperbolic functions. See Also ======== sinh, cosh, tanh, coth """ unbranched = True def _peeloff_ipi(arg): r""" Split ARG into two parts, a "rest" and a multiple of $I\pi$. This assumes ARG to be an ``Add``. The multiple of $I\pi$ returned in the second position is always a ``Rational``. Examples ======== >>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel >>> from sympy import pi, I >>> from sympy.abc import x, y >>> peel(x + I*pi/2) (x, 1/2) >>> peel(x + I*2*pi/3 + I*pi*y) (x + I*pi*y + I*pi/6, 1/2) """ ipi = S.Pi*S.ImaginaryUnit for a in Add.make_args(arg): if a == ipi: K = S.One break elif a.is_Mul: K, p = a.as_two_terms() if p == ipi and K.is_Rational: break else: return arg, S.Zero m1 = (K % S.Half) m2 = K - m1 return arg - m2*ipi, m2 class sinh(HyperbolicFunction): r""" ``sinh(x)`` is the hyperbolic sine of ``x``. The hyperbolic sine function is $\frac{e^x - e^{-x}}{2}$. Examples ======== >>> from sympy import sinh >>> from sympy.abc import x >>> sinh(x) sinh(x) See Also ======== cosh, tanh, asinh """ 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) def inverse(self, argindex=1): """ Returns the inverse of this function. """ return asinh @classmethod def eval(cls, arg): from sympy.functions.elementary.trigonometric 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_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 arg.could_extract_minus_sign(): return -cls(-arg) if arg.is_Add: x, m = _peeloff_ipi(arg) if m: m = m*S.Pi*S.ImaginaryUnit return sinh(m)*cosh(x) + cosh(m)*sinh(x) if arg.is_zero: return S.Zero 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 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()) def as_real_imag(self, deep=True, **hints): """ Returns this function as a complex coordinate. """ from sympy.functions.elementary.trigonometric import (cos, sin) if self.args[0].is_extended_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, limitvar=None, **kwargs): return (exp(arg) - exp(-arg)) / 2 def _eval_rewrite_as_exp(self, arg, **kwargs): return (exp(arg) - exp(-arg)) / 2 def _eval_rewrite_as_cosh(self, arg, **kwargs): return -S.ImaginaryUnit*cosh(arg + S.Pi*S.ImaginaryUnit/2) def _eval_rewrite_as_tanh(self, arg, **kwargs): tanh_half = tanh(S.Half*arg) return 2*tanh_half/(1 - tanh_half**2) def _eval_rewrite_as_coth(self, arg, **kwargs): coth_half = coth(S.Half*arg) return 2*coth_half/(coth_half**2 - 1) def _eval_as_leading_term(self, x, logx=None, cdir=0): arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir) arg0 = arg.subs(x, 0) if arg0 is S.NaN: arg0 = arg.limit(x, 0, dir='-' if cdir.is_negative else '+') if arg0.is_zero: return arg elif arg0.is_finite: return self.func(arg0) else: return self def _eval_is_real(self): arg = self.args[0] if arg.is_real: return True # if `im` is of the form n*pi # else, check if it is a number re, im = arg.as_real_imag() return (im%pi).is_zero def _eval_is_extended_real(self): if self.args[0].is_extended_real: return True def _eval_is_positive(self): if self.args[0].is_extended_real: return self.args[0].is_positive def _eval_is_negative(self): if self.args[0].is_extended_real: return self.args[0].is_negative def _eval_is_finite(self): arg = self.args[0] return arg.is_finite def _eval_is_zero(self): rest, ipi_mult = _peeloff_ipi(self.args[0]) if rest.is_zero: return ipi_mult.is_integer class cosh(HyperbolicFunction): r""" ``cosh(x)`` is the hyperbolic cosine of ``x``. The hyperbolic cosine function is $\frac{e^x + e^{-x}}{2}$. Examples ======== >>> from sympy import cosh >>> from sympy.abc import x >>> cosh(x) cosh(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.functions.elementary.trigonometric 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_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 arg.could_extract_minus_sign(): return cls(-arg) if arg.is_Add: x, m = _peeloff_ipi(arg) if m: m = m*S.Pi*S.ImaginaryUnit return cosh(m)*cosh(x) + sinh(m)*sinh(x) if arg.is_zero: return S.One 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.functions.elementary.trigonometric import (cos, sin) if self.args[0].is_extended_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, limitvar=None, **kwargs): return (exp(arg) + exp(-arg)) / 2 def _eval_rewrite_as_exp(self, arg, **kwargs): return (exp(arg) + exp(-arg)) / 2 def _eval_rewrite_as_sinh(self, arg, **kwargs): return -S.ImaginaryUnit*sinh(arg + S.Pi*S.ImaginaryUnit/2) def _eval_rewrite_as_tanh(self, arg, **kwargs): tanh_half = tanh(S.Half*arg)**2 return (1 + tanh_half)/(1 - tanh_half) def _eval_rewrite_as_coth(self, arg, **kwargs): coth_half = coth(S.Half*arg)**2 return (coth_half + 1)/(coth_half - 1) def _eval_as_leading_term(self, x, logx=None, cdir=0): arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir) arg0 = arg.subs(x, 0) if arg0 is S.NaN: arg0 = arg.limit(x, 0, dir='-' if cdir.is_negative else '+') if arg0.is_zero: return S.One elif arg0.is_finite: return self.func(arg0) else: return self def _eval_is_real(self): arg = self.args[0] # `cosh(x)` is real for real OR purely imaginary `x` if arg.is_real or arg.is_imaginary: return True # cosh(a+ib) = cos(b)*cosh(a) + i*sin(b)*sinh(a) # the imaginary part can be an expression like n*pi # if not, check if the imaginary part is a number re, im = arg.as_real_imag() return (im%pi).is_zero def _eval_is_positive(self): # cosh(x+I*y) = cos(y)*cosh(x) + I*sin(y)*sinh(x) # cosh(z) is positive iff it is real and the real part is positive. # So we need sin(y)*sinh(x) = 0 which gives x=0 or y=n*pi # Case 1 (y=n*pi): cosh(z) = (-1)**n * cosh(x) -> positive for n even # Case 2 (x=0): cosh(z) = cos(y) -> positive when cos(y) is positive z = self.args[0] x, y = z.as_real_imag() ymod = y % (2*pi) yzero = ymod.is_zero # shortcut if ymod is zero if yzero: return True xzero = x.is_zero # shortcut x is not zero if xzero is False: return yzero return fuzzy_or([ # Case 1: yzero, # Case 2: fuzzy_and([ xzero, fuzzy_or([ymod < pi/2, ymod > 3*pi/2]) ]) ]) def _eval_is_nonnegative(self): z = self.args[0] x, y = z.as_real_imag() ymod = y % (2*pi) yzero = ymod.is_zero # shortcut if ymod is zero if yzero: return True xzero = x.is_zero # shortcut x is not zero if xzero is False: return yzero return fuzzy_or([ # Case 1: yzero, # Case 2: fuzzy_and([ xzero, fuzzy_or([ymod <= pi/2, ymod >= 3*pi/2]) ]) ]) def _eval_is_finite(self): arg = self.args[0] return arg.is_finite def _eval_is_zero(self): rest, ipi_mult = _peeloff_ipi(self.args[0]) if ipi_mult and rest.is_zero: return (ipi_mult - S.Half).is_integer class tanh(HyperbolicFunction): r""" ``tanh(x)`` is the hyperbolic tangent of ``x``. The hyperbolic tangent function is $\frac{\sinh(x)}{\cosh(x)}$. Examples ======== >>> from sympy import tanh >>> from sympy.abc import x >>> tanh(x) tanh(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) def inverse(self, argindex=1): """ Returns the inverse of this function. """ return atanh @classmethod def eval(cls, arg): from sympy.functions.elementary.trigonometric 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_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 i_coeff.could_extract_minus_sign(): return -S.ImaginaryUnit * tan(-i_coeff) return S.ImaginaryUnit * tan(i_coeff) else: if arg.could_extract_minus_sign(): return -cls(-arg) if arg.is_Add: x, m = _peeloff_ipi(arg) if m: tanhm = tanh(m*S.Pi*S.ImaginaryUnit) if tanhm is S.ComplexInfinity: return coth(x) else: # tanhm == 0 return tanh(x) if arg.is_zero: return S.Zero 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.functions.combinatorial.numbers 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.functions.elementary.trigonometric import (cos, sin) if self.args[0].is_extended_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_expand_trig(self, **hints): arg = self.args[0] if arg.is_Add: from sympy.polys.specialpolys import symmetric_poly n = len(arg.args) TX = [tanh(x, evaluate=False)._eval_expand_trig() for x in arg.args] p = [0, 0] # [den, num] for i in range(n + 1): p[i % 2] += symmetric_poly(i, TX) return p[1]/p[0] elif arg.is_Mul: from sympy.functions.combinatorial.numbers import nC coeff, terms = arg.as_coeff_Mul() if coeff.is_Integer and coeff > 1: n = [] d = [] T = tanh(terms) for k in range(1, coeff + 1, 2): n.append(nC(range(coeff), k)*T**k) for k in range(0, coeff + 1, 2): d.append(nC(range(coeff), k)*T**k) return Add(*n)/Add(*d) return tanh(arg) def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs): neg_exp, pos_exp = exp(-arg), exp(arg) return (pos_exp - neg_exp)/(pos_exp + neg_exp) def _eval_rewrite_as_exp(self, arg, **kwargs): neg_exp, pos_exp = exp(-arg), exp(arg) return (pos_exp - neg_exp)/(pos_exp + neg_exp) def _eval_rewrite_as_sinh(self, arg, **kwargs): return S.ImaginaryUnit*sinh(arg)/sinh(S.Pi*S.ImaginaryUnit/2 - arg) def _eval_rewrite_as_cosh(self, arg, **kwargs): return S.ImaginaryUnit*cosh(S.Pi*S.ImaginaryUnit/2 - arg)/cosh(arg) def _eval_rewrite_as_coth(self, arg, **kwargs): return 1/coth(arg) def _eval_as_leading_term(self, x, logx=None, cdir=0): from sympy.series.order 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): arg = self.args[0] if arg.is_real: return True re, im = arg.as_real_imag() # if denom = 0, tanh(arg) = zoo if re == 0 and im % pi == pi/2: return None # check if im is of the form n*pi/2 to make sin(2*im) = 0 # if not, im could be a number, return False in that case return (im % (pi/2)).is_zero def _eval_is_extended_real(self): if self.args[0].is_extended_real: return True def _eval_is_positive(self): if self.args[0].is_extended_real: return self.args[0].is_positive def _eval_is_negative(self): if self.args[0].is_extended_real: return self.args[0].is_negative def _eval_is_finite(self): from sympy.functions.elementary.trigonometric import cos arg = self.args[0] re, im = arg.as_real_imag() denom = cos(im)**2 + sinh(re)**2 if denom == 0: return False elif denom.is_number: return True if arg.is_extended_real: return True def _eval_is_zero(self): arg = self.args[0] if arg.is_zero: return True class coth(HyperbolicFunction): r""" ``coth(x)`` is the hyperbolic cotangent of ``x``. The hyperbolic cotangent function is $\frac{\cosh(x)}{\sinh(x)}$. Examples ======== >>> from sympy import coth >>> from sympy.abc import x >>> coth(x) coth(x) See Also ======== sinh, cosh, acoth """ def fdiff(self, argindex=1): if argindex == 1: return -1/sinh(self.args[0])**2 else: raise ArgumentIndexError(self, argindex) def inverse(self, argindex=1): """ Returns the inverse of this function. """ return acoth @classmethod def eval(cls, arg): from sympy.functions.elementary.trigonometric 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_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 i_coeff.could_extract_minus_sign(): return S.ImaginaryUnit * cot(-i_coeff) return -S.ImaginaryUnit * cot(i_coeff) else: if arg.could_extract_minus_sign(): return -cls(-arg) if arg.is_Add: x, m = _peeloff_ipi(arg) if m: cothm = coth(m*S.Pi*S.ImaginaryUnit) if cothm is S.ComplexInfinity: return coth(x) else: # cothm == 0 return tanh(x) if arg.is_zero: return S.ComplexInfinity 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.functions.combinatorial.numbers 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.functions.elementary.trigonometric import (cos, sin) if self.args[0].is_extended_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, limitvar=None, **kwargs): neg_exp, pos_exp = exp(-arg), exp(arg) return (pos_exp + neg_exp)/(pos_exp - neg_exp) def _eval_rewrite_as_exp(self, arg, **kwargs): neg_exp, pos_exp = exp(-arg), exp(arg) return (pos_exp + neg_exp)/(pos_exp - neg_exp) def _eval_rewrite_as_sinh(self, arg, **kwargs): return -S.ImaginaryUnit*sinh(S.Pi*S.ImaginaryUnit/2 - arg)/sinh(arg) def _eval_rewrite_as_cosh(self, arg, **kwargs): return -S.ImaginaryUnit*cosh(arg)/cosh(S.Pi*S.ImaginaryUnit/2 - arg) def _eval_rewrite_as_tanh(self, arg, **kwargs): return 1/tanh(arg) def _eval_is_positive(self): if self.args[0].is_extended_real: return self.args[0].is_positive def _eval_is_negative(self): if self.args[0].is_extended_real: return self.args[0].is_negative def _eval_as_leading_term(self, x, logx=None, cdir=0): from sympy.series.order 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_expand_trig(self, **hints): arg = self.args[0] if arg.is_Add: from sympy.polys.specialpolys import symmetric_poly CX = [coth(x, evaluate=False)._eval_expand_trig() for x in arg.args] p = [[], []] n = len(arg.args) for i in range(n, -1, -1): p[(n - i) % 2].append(symmetric_poly(i, CX)) return Add(*p[0])/Add(*p[1]) elif arg.is_Mul: from sympy.functions.combinatorial.factorials import binomial coeff, x = arg.as_coeff_Mul(rational=True) if coeff.is_Integer and coeff > 1: c = coth(x, evaluate=False) p = [[], []] for i in range(coeff, -1, -1): p[(coeff - i) % 2].append(binomial(coeff, i)*c**i) return Add(*p[0])/Add(*p[1]) return coth(arg) class ReciprocalHyperbolicFunction(HyperbolicFunction): """Base class for reciprocal functions of hyperbolic functions. """ #To be defined in class _reciprocal_of = None _is_even = None # type: FuzzyBool _is_odd = None # type: FuzzyBool @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 is not 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 is not None else t def _rewrite_reciprocal(self, method_name, arg): # Special handling for rewrite functions. If reciprocal rewrite returns # unmodified expression, then return None t = self._call_reciprocal(method_name, arg) if t is not None and t != self._reciprocal_of(arg): return 1/t def _eval_rewrite_as_exp(self, arg, **kwargs): return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg) def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs): return self._rewrite_reciprocal("_eval_rewrite_as_tractable", arg) def _eval_rewrite_as_tanh(self, arg, **kwargs): return self._rewrite_reciprocal("_eval_rewrite_as_tanh", arg) def _eval_rewrite_as_coth(self, arg, **kwargs): 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_expand_trig(self, **hints): return self._calculate_reciprocal("_eval_expand_trig", **hints) def _eval_as_leading_term(self, x, logx=None, cdir=0): return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x) def _eval_is_extended_real(self): return self._reciprocal_of(self.args[0]).is_extended_real def _eval_is_finite(self): return (1/self._reciprocal_of(self.args[0])).is_finite class csch(ReciprocalHyperbolicFunction): r""" ``csch(x)`` is the hyperbolic cosecant of ``x``. The hyperbolic cosecant function is $\frac{2}{e^x - e^{-x}}$ Examples ======== >>> from sympy import csch >>> from sympy.abc import x >>> csch(x) csch(x) See Also ======== sinh, cosh, tanh, sech, asinh, acosh """ _reciprocal_of = sinh _is_odd = True 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 def taylor_term(n, x, *previous_terms): """ Returns the next term in the Taylor series expansion """ from sympy.functions.combinatorial.numbers 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, **kwargs): return S.ImaginaryUnit / cosh(arg + S.ImaginaryUnit * S.Pi / 2) def _eval_is_positive(self): if self.args[0].is_extended_real: return self.args[0].is_positive def _eval_is_negative(self): if self.args[0].is_extended_real: return self.args[0].is_negative class sech(ReciprocalHyperbolicFunction): r""" ``sech(x)`` is the hyperbolic secant of ``x``. The hyperbolic secant function is $\frac{2}{e^x + e^{-x}}$ Examples ======== >>> from sympy import sech >>> from sympy.abc import x >>> sech(x) sech(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, **kwargs): return S.ImaginaryUnit / sinh(arg + S.ImaginaryUnit * S.Pi /2) def _eval_is_positive(self): if self.args[0].is_extended_real: return True ############################################################################### ############################# HYPERBOLIC INVERSES ############################# ############################################################################### class InverseHyperbolicFunction(Function): """Base class for inverse hyperbolic functions.""" pass class asinh(InverseHyperbolicFunction): """ ``asinh(x)`` is the inverse hyperbolic sine of ``x``. The inverse hyperbolic sine function. Examples ======== >>> from sympy import asinh >>> from sympy.abc import x >>> asinh(x).diff(x) 1/sqrt(x**2 + 1) >>> asinh(1) log(1 + sqrt(2)) 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.functions.elementary.trigonometric 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_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 if arg.is_zero: return S.Zero i_coeff = arg.as_coefficient(S.ImaginaryUnit) if i_coeff is not None: return S.ImaginaryUnit * asin(i_coeff) else: if arg.could_extract_minus_sign(): return -cls(-arg) if isinstance(arg, sinh) and arg.args[0].is_number: z = arg.args[0] if z.is_real: return z r, i = match_real_imag(z) if r is not None and i is not None: f = floor((i + pi/2)/pi) m = z - I*pi*f even = f.is_even if even is True: return m elif even is False: return -m @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 S.NegativeOne**k * R / F * x**n / n def _eval_as_leading_term(self, x, logx=None, cdir=0): from sympy.series.order 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, **kwargs): return log(x + sqrt(x**2 + 1)) def inverse(self, argindex=1): """ Returns the inverse of this function. """ return sinh def _eval_is_zero(self): return self.args[0].is_zero class acosh(InverseHyperbolicFunction): """ ``acosh(x)`` is the inverse hyperbolic cosine of ``x``. The inverse hyperbolic cosine function. Examples ======== >>> from sympy import acosh >>> from sympy.abc import x >>> acosh(x).diff(x) 1/sqrt(x**2 - 1) >>> acosh(1) 0 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_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, Rational(-1, 2): S.Pi*Rational(2, 3), sqrt(2)/2: S.Pi/4, -sqrt(2)/2: S.Pi*Rational(3, 4), 1/sqrt(2): S.Pi/4, -1/sqrt(2): S.Pi*Rational(3, 4), sqrt(3)/2: S.Pi/6, -sqrt(3)/2: S.Pi*Rational(5, 6), (sqrt(3) - 1)/sqrt(2**3): S.Pi*Rational(5, 12), -(sqrt(3) - 1)/sqrt(2**3): S.Pi*Rational(7, 12), sqrt(2 + sqrt(2))/2: S.Pi/8, -sqrt(2 + sqrt(2))/2: S.Pi*Rational(7, 8), sqrt(2 - sqrt(2))/2: S.Pi*Rational(3, 8), -sqrt(2 - sqrt(2))/2: S.Pi*Rational(5, 8), (1 + sqrt(3))/(2*sqrt(2)): S.Pi/12, -(1 + sqrt(3))/(2*sqrt(2)): S.Pi*Rational(11, 12), (sqrt(5) + 1)/4: S.Pi/5, -(sqrt(5) + 1)/4: S.Pi*Rational(4, 5) } if arg in cst_table: if arg.is_extended_real: return cst_table[arg]*S.ImaginaryUnit return cst_table[arg] if arg is S.ComplexInfinity: return S.ComplexInfinity if arg == S.ImaginaryUnit*S.Infinity: return S.Infinity + S.ImaginaryUnit*S.Pi/2 if arg == -S.ImaginaryUnit*S.Infinity: return S.Infinity - S.ImaginaryUnit*S.Pi/2 if arg.is_zero: return S.Pi*S.ImaginaryUnit*S.Half if isinstance(arg, cosh) and arg.args[0].is_number: z = arg.args[0] if z.is_real: from sympy.functions.elementary.complexes import Abs return Abs(z) r, i = match_real_imag(z) if r is not None and i is not None: f = floor(i/pi) m = z - I*pi*f even = f.is_even if even is True: if r.is_nonnegative: return m elif r.is_negative: return -m elif even is False: m -= I*pi if r.is_nonpositive: return -m elif r.is_positive: return m @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, logx=None, cdir=0): from sympy.series.order 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) def _eval_rewrite_as_log(self, x, **kwargs): return log(x + sqrt(x + 1) * sqrt(x - 1)) def inverse(self, argindex=1): """ Returns the inverse of this function. """ return cosh def _eval_is_zero(self): if (self.args[0] - 1).is_zero: return True class atanh(InverseHyperbolicFunction): """ ``atanh(x)`` is the inverse hyperbolic tangent of ``x``. The inverse hyperbolic tangent function. Examples ======== >>> from sympy import atanh >>> from sympy.abc import x >>> atanh(x).diff(x) 1/(1 - x**2) 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.functions.elementary.trigonometric import atan arg = sympify(arg) if arg.is_Number: if arg is S.NaN: return S.NaN elif arg.is_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: from sympy.calculus.accumulationbounds import AccumBounds return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2) i_coeff = arg.as_coefficient(S.ImaginaryUnit) if i_coeff is not None: return S.ImaginaryUnit * atan(i_coeff) else: if arg.could_extract_minus_sign(): return -cls(-arg) if arg.is_zero: return S.Zero if isinstance(arg, tanh) and arg.args[0].is_number: z = arg.args[0] if z.is_real: return z r, i = match_real_imag(z) if r is not None and i is not None: f = floor(2*i/pi) even = f.is_even m = z - I*f*pi/2 if even is True: return m elif even is False: return m - I*pi/2 @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, logx=None, cdir=0): from sympy.series.order 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, **kwargs): return (log(1 + x) - log(1 - x)) / 2 def _eval_is_zero(self): if self.args[0].is_zero: return True def _eval_is_imaginary(self): return self.args[0].is_imaginary def inverse(self, argindex=1): """ Returns the inverse of this function. """ return tanh class acoth(InverseHyperbolicFunction): """ ``acoth(x)`` is the inverse hyperbolic cotangent of ``x``. The inverse hyperbolic cotangent function. Examples ======== >>> from sympy import acoth >>> from sympy.abc import x >>> acoth(x).diff(x) 1/(1 - x**2) See Also ======== asinh, acosh, coth """ 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.functions.elementary.trigonometric 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_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 S.Zero i_coeff = arg.as_coefficient(S.ImaginaryUnit) if i_coeff is not None: return -S.ImaginaryUnit * acot(i_coeff) else: if arg.could_extract_minus_sign(): return -cls(-arg) if arg.is_zero: return S.Pi*S.ImaginaryUnit*S.Half @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, logx=None, cdir=0): from sympy.series.order 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) def _eval_rewrite_as_log(self, x, **kwargs): return (log(1 + 1/x) - log(1 - 1/x)) / 2 def inverse(self, argindex=1): """ Returns the inverse of this function. """ return coth class asech(InverseHyperbolicFunction): """ ``asech(x)`` is the inverse hyperbolic secant of ``x``. The inverse hyperbolic secant function. Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(1 - x**2)) >>> 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] https://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_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_extended_real: return cst_table[arg]*S.ImaginaryUnit return cst_table[arg] if arg is S.ComplexInfinity: from sympy.calculus.accumulationbounds import AccumBounds return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2) if arg.is_zero: return S.Infinity @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 def inverse(self, argindex=1): """ Returns the inverse of this function. """ return sech def _eval_rewrite_as_log(self, arg, **kwargs): return log(1/arg + sqrt(1/arg - 1) * sqrt(1/arg + 1)) class acsch(InverseHyperbolicFunction): """ ``acsch(x)`` is the inverse hyperbolic cosecant of ``x``. The inverse hyperbolic cosecant function. Examples ======== >>> from sympy import acsch, sqrt, S >>> from sympy.abc import x >>> acsch(x).diff(x) -1/(x**2*sqrt(1 + x**(-2))) >>> acsch(1).diff(x) 0 >>> acsch(1) log(1 + sqrt(2)) >>> acsch(S.ImaginaryUnit) -I*pi/2 >>> acsch(-2*S.ImaginaryUnit) I*pi/6 >>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2))) -5*I*pi/12 See Also ======== asinh References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] http://dlmf.nist.gov/4.37 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/ """ def fdiff(self, argindex=1): if argindex == 1: z = self.args[0] return -1/(z**2*sqrt(1 + 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.Zero elif arg is S.NegativeInfinity: return S.Zero elif arg.is_zero: return S.ComplexInfinity elif arg is S.One: return log(1 + sqrt(2)) elif arg is S.NegativeOne: return - log(1 + sqrt(2)) if arg.is_number: cst_table = { S.ImaginaryUnit: -S.Pi / 2, S.ImaginaryUnit*(sqrt(2) + sqrt(6)): -S.Pi / 12, S.ImaginaryUnit*(1 + sqrt(5)): -S.Pi / 10, S.ImaginaryUnit*2 / sqrt(2 - sqrt(2)): -S.Pi / 8, S.ImaginaryUnit*2: -S.Pi / 6, S.ImaginaryUnit*sqrt(2 + 2/sqrt(5)): -S.Pi / 5, S.ImaginaryUnit*sqrt(2): -S.Pi / 4, S.ImaginaryUnit*(sqrt(5)-1): -3*S.Pi / 10, S.ImaginaryUnit*2 / sqrt(3): -S.Pi / 3, S.ImaginaryUnit*2 / sqrt(2 + sqrt(2)): -3*S.Pi / 8, S.ImaginaryUnit*sqrt(2 - 2/sqrt(5)): -2*S.Pi / 5, S.ImaginaryUnit*(sqrt(6) - sqrt(2)): -5*S.Pi / 12, S(2): -S.ImaginaryUnit*log((1+sqrt(5))/2), } if arg in cst_table: return cst_table[arg]*S.ImaginaryUnit if arg is S.ComplexInfinity: return S.Zero if arg.is_zero: return S.ComplexInfinity if arg.could_extract_minus_sign(): return -cls(-arg) def inverse(self, argindex=1): """ Returns the inverse of this function. """ return csch def _eval_rewrite_as_log(self, arg, **kwargs): return log(1/arg + sqrt(1/arg**2 + 1)) def _eval_is_zero(self): return self.args[0].is_infinite