Source code for sympy.functions.special.error_functions

""" This module contains various functions that are special cases
    of incomplete gamma functions. It should probably be renamed. """

from sympy.core import Add, S, C, sympify, cacheit, pi, I
from sympy.core.function import Function, ArgumentIndexError
from sympy.functions.elementary.miscellaneous import sqrt, root
from sympy.functions.elementary.complexes import polar_lift
from sympy.functions.special.hyper import hyper, meijerg

# TODO series expansions
# TODO see the "Note:" in Ei

###############################################################################
################################ ERROR FUNCTION ###############################
###############################################################################

[docs]class erf(Function): """ The Gauss error function. This function is defined as: :math:`\\mathrm{erf}(x)=\\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} \\, \\mathrm{d}x` Or, in ASCII:: x / | | 2 | -t 2* | e dt | / 0 ------------- ____ \/ pi Examples ======== >>> from sympy import I, oo, erf >>> from sympy.abc import z Several special values are known: >>> erf(0) 0 >>> erf(oo) 1 >>> erf(-oo) -1 >>> erf(I*oo) oo*I >>> erf(-I*oo) -oo*I In general one can pull out factors of -1 and I from the argument: >>> erf(-z) -erf(z) The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erf(z)) erf(conjugate(z)) Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(erf(z), z) 2*exp(-z**2)/sqrt(pi) We can numerically evaluate the error function to arbitrary precision on the whole complex plane: >>> erf(4).evalf(30) 0.999999984582742099719981147840 >>> erf(-4*I).evalf(30) -1296959.73071763923152794095062*I References ========== .. [1] http://en.wikipedia.org/wiki/Error_function .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/Erf.html .. [4] http://functions.wolfram.com/GammaBetaErf/Erf """ nargs = 1 unbranched = True def fdiff(self, argindex=1): if argindex == 1: return 2*C.exp(-self.args[0]**2)/sqrt(S.Pi) else: raise ArgumentIndexError(self, argindex) @classmethod def eval(cls, 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 t = arg.extract_multiplicatively(S.ImaginaryUnit) if t == S.Infinity or t == S.NegativeInfinity: return arg if arg.could_extract_minus_sign(): 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) k = C.floor((n - 1)/S(2)) if len(previous_terms) > 2: return -previous_terms[-2] * x**2 * (n-2)/(n*k) else: return 2*(-1)**k * x**n/(n*C.factorial(k)*sqrt(S.Pi)) def _eval_conjugate(self): return self.func(self.args[0].conjugate()) def _eval_is_real(self): return self.args[0].is_real def _eval_rewrite_as_uppergamma(self, z): return sqrt(z**2)/z*(S.One - C.uppergamma(S.Half, z**2)/sqrt(S.Pi)) def _eval_rewrite_as_tractable(self, z): return S.One - _erfs(z)*C.exp(-z**2) 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 2*x/sqrt(pi) else: return self.func(arg) ############################################################################### #################### EXPONENTIAL INTEGRALS #################################### ###############################################################################
[docs]class Ei(Function): r""" The classical exponential integral. For the use in SymPy, this function is defined as .. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!} + \log(x) + \gamma, where :math:`\gamma` is the Euler-Mascheroni constant. If :math:`x` is a polar number, this defines an analytic function on the riemann surface of the logarithm. Otherwise this defines an analytic function in the cut plane :math:`\mathbb{C} \setminus (-\infty, 0]`. **Background** The name 'exponential integral' comes from the following statement: .. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t If the integral is interpreted as a Cauchy principal value, this statement holds for :math:`x > 0` and :math:`\operatorname{Ei}(x)` as defined above. Note that we carefully avoided defining :math:`\operatorname{Ei}(x)` for negative real x. This is because above integral formula does not hold for any polar lift of such :math:`x`, indeed all branches of :math:`\operatorname{Ei}(x)` above the negative reals are imaginary. However, the following statement holds for all :math:`x \in \mathbb{R}^*`: .. math:: \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t = \frac{\operatorname{Ei}\left(|x|e^{i \arg(x)}\right) + \operatorname{Ei}\left(|x|e^{- i \arg(x)}\right)}{2}, where the integral is again understood to be a principal value if :math:`x > 0`, and :math:`|x|e^{i \arg(x)}`, :math:`|x|e^{- i \arg(x)}` denote two conjugate polar lifts of :math:`x`. See Also ======== expint, sympy.functions.special.gamma_functions.uppergamma References ========== - Abramowitz & Stegun, section 5: http://www.math.sfu.ca/~cbm/aands/page_228.htm - http://en.wikipedia.org/wiki/Exponential_integral Examples ======== >>> from sympy import Ei, polar_lift, exp_polar, I, pi >>> from sympy.abc import x The exponential integral in SymPy is strictly undefined for negative values of the argument. For convenience, exponential integrals with negative arguments are immediately converted into an expression that agrees with the classical integral definition: >>> Ei(-1) -I*pi + Ei(exp_polar(I*pi)) This yields a real value: >>> Ei(-1).n(chop=True) -0.219383934395520 On the other hand the analytic continuation is not real: >>> Ei(polar_lift(-1)).n(chop=True) -0.21938393439552 + 3.14159265358979*I The exponential integral has a logarithmic branch point at the origin: >>> Ei(x*exp_polar(2*I*pi)) Ei(x) + 2*I*pi Differentiation is supported: >>> Ei(x).diff(x) exp(x)/x The exponential integral is related to many other special functions. For example: >>> from sympy import uppergamma, expint, Shi >>> Ei(x).rewrite(expint) -expint(1, x*exp_polar(I*pi)) - I*pi >>> Ei(x).rewrite(Shi) Chi(x) + Shi(x) """ nargs = 1 @classmethod def eval(cls, z): if not z.is_polar and z.is_negative: # Note: is this a good idea? return Ei(polar_lift(z)) - pi*I nz, n = z.extract_branch_factor() if n: return Ei(nz) + 2*I*pi*n def fdiff(self, argindex=1): from sympy import unpolarify arg = unpolarify(self.args[0]) if argindex == 1: return C.exp(arg)/arg else: raise ArgumentIndexError(self, argindex) def _eval_evalf(self, prec): if (self.args[0]/polar_lift(-1)).is_positive: return Function._eval_evalf(self, prec) + (I*pi)._eval_evalf(prec) return Function._eval_evalf(self, prec) def _eval_rewrite_as_uppergamma(self, z): from sympy import uppergamma # XXX this does not currently work usefully because uppergamma # immediately turns into expint return -uppergamma(0, polar_lift(-1)*z) - I*pi def _eval_rewrite_as_expint(self, z): return -expint(1, polar_lift(-1)*z) - I*pi def _eval_rewrite_as_Si(self, z): return Shi(z) + Chi(z) _eval_rewrite_as_Ci = _eval_rewrite_as_Si _eval_rewrite_as_Chi = _eval_rewrite_as_Si _eval_rewrite_as_Shi = _eval_rewrite_as_Si
[docs]class expint(Function): r""" Generalized exponential integral. This function is defined as .. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z), where `\Gamma(1 - \nu, z)` is the upper incomplete gamma function (``uppergamma``). Hence for :math:`z` with positive real part we have .. math:: \operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t, which explains the name. The representation as an incomplete gamma function provides an analytic continuation for :math:`\operatorname{E}_\nu(z)`. If :math:`\nu` is a non-positive integer the exponential integral is thus an unbranched function of :math:`z`, otherwise there is a branch point at the origin. Refer to the incomplete gamma function documentation for details of the branching behavior. See Also ======== E1: The classical case, returns expint(1, z). Ei: Another related function called exponential integral. sympy.functions.special.gamma_functions.uppergamma References ========== - http://dlmf.nist.gov/8.19 - http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/ - http://en.wikipedia.org/wiki/Exponential_integral Examples ======== >>> from sympy import expint, S >>> from sympy.abc import nu, z Differentiation is supported. Differentiation with respect to z explains further the name: for integral orders, the exponential integral is an iterated integral of the exponential function. >>> expint(nu, z).diff(z) -expint(nu - 1, z) Differentiation with respect to nu has no classical expression: >>> expint(nu, z).diff(nu) -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, -nu + 1), ()), z) At non-postive integer orders, the exponential integral reduces to the exponential function: >>> expint(0, z) exp(-z)/z >>> expint(-1, z) exp(-z)/z + exp(-z)/z**2 At half-integers it reduces to error functions: >>> expint(S(1)/2, z) -sqrt(pi)*erf(sqrt(z))/sqrt(z) + sqrt(pi)/sqrt(z) At positive integer orders it can be rewritten in terms of exponentials and expint(1, z). Use expand_func() to do this: >>> from sympy import expand_func >>> expand_func(expint(5, z)) z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24 The generalised exponential integral is essentially equivalent to the incomplete gamma function: >>> from sympy import uppergamma >>> expint(nu, z).rewrite(uppergamma) z**(nu - 1)*uppergamma(-nu + 1, z) As such it is branched at the origin: >>> from sympy import exp_polar, pi, I >>> expint(4, z*exp_polar(2*pi*I)) I*pi*z**3/3 + expint(4, z) >>> expint(nu, z*exp_polar(2*pi*I)) z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(-nu + 1) + expint(nu, z) """ nargs = 2 @classmethod def eval(cls, nu, z): from sympy import (unpolarify, expand_mul, uppergamma, exp, gamma, factorial) nu2 = unpolarify(nu) if nu != nu2: return expint(nu2, z) if nu.is_Integer and nu <= 0 or (not nu.is_Integer and (2*nu).is_Integer): return unpolarify(expand_mul(z**(nu - 1)*uppergamma(1 - nu, z))) # Extract branching information. This can be deduced from what is # explained in lowergamma.eval(). z, n = z.extract_branch_factor() if n == 0: return if nu.is_integer: if (nu > 0) is not True: return return expint(nu, z) \ - 2*pi*I*n*(-1)**(nu-1)/factorial(nu-1)*unpolarify(z)**(nu - 1) else: return (exp(2*I*pi*nu*n) - 1)*z**(nu-1)*gamma(1 - nu) + expint(nu, z) def fdiff(self, argindex): from sympy import meijerg nu, z = self.args if argindex == 1: return -z**(nu - 1)*meijerg([], [1, 1], [0, 0, 1 - nu], [], z) elif argindex == 2: return -expint(nu - 1, z) else: raise ArgumentIndexError(self, argindex) def _eval_rewrite_as_uppergamma(self, nu, z): from sympy import uppergamma return z**(nu - 1)*uppergamma(1 - nu, z) def _eval_rewrite_as_Ei(self, nu, z): from sympy import exp_polar, unpolarify, exp, factorial if nu == 1: return -Ei(z*exp_polar(-I*pi)) - I*pi elif nu.is_Integer and nu > 1: # DLMF, 8.19.7 x = -unpolarify(z) return x**(nu-1)/factorial(nu - 1)*E1(z).rewrite(Ei) + \ exp(x)/factorial(nu - 1) * \ Add(*[factorial(nu - k - 2)*x**k for k in range(nu - 1)]) else: return self def _eval_expand_func(self, **hints): return self.rewrite(Ei).rewrite(expint, **hints) def _eval_rewrite_as_Si(self, nu, z): if nu != 1: return self return Shi(z) - Chi(z) _eval_rewrite_as_Ci = _eval_rewrite_as_Si _eval_rewrite_as_Chi = _eval_rewrite_as_Si _eval_rewrite_as_Shi = _eval_rewrite_as_Si
[docs]def E1(z): """ Classical case of the generalized exponential integral. This is equivalent to ``expint(1, z)``. """ return expint(1, z) ############################################################################### #################### TRIGONOMETRIC INTEGRALS ################################## ###############################################################################
class TrigonometricIntegral(Function): """ Base class for trigonometric integrals. """ nargs = 1 @classmethod def eval(cls, z): if z == 0: return cls._atzero nz = z.extract_multiplicatively(polar_lift(I)) if nz is None and cls._trigfunc(0) == 0: nz = z.extract_multiplicatively(I) if nz is not None: return cls._Ifactor(nz, 1) nz = z.extract_multiplicatively(polar_lift(-I)) if nz is not None: return cls._Ifactor(nz, -1) nz = z.extract_multiplicatively(polar_lift(-1)) if nz is None and cls._trigfunc(0) == 0: nz = z.extract_multiplicatively(-1) if nz is not None: return cls._minusfactor(nz) nz, n = z.extract_branch_factor() if n == 0 and nz == z: return return 2*pi*I*n*cls._trigfunc(0) + cls(nz) def fdiff(self, argindex=1): from sympy import unpolarify arg = unpolarify(self.args[0]) if argindex == 1: return self._trigfunc(arg)/arg def _eval_rewrite_as_Ei(self, z): return self._eval_rewrite_as_expint(z).rewrite(Ei) def _eval_rewrite_as_uppergamma(self, z): from sympy import uppergamma return self._eval_rewrite_as_expint(z).rewrite(uppergamma) def _eval_nseries(self, x, n, logx): # NOTE this is fairly inefficient from sympy import log, EulerGamma, Pow n += 1 if self.args[0].subs(x, 0) != 0: return super(TrigonometricIntegral, self)._eval_nseries(x, n, logx) baseseries = self._trigfunc(x)._eval_nseries(x, n, logx) if self._trigfunc(0) != 0: baseseries -= 1 baseseries = baseseries.replace(Pow, lambda t, n: t**n/n) if self._trigfunc(0) != 0: baseseries += EulerGamma + log(x) return baseseries.subs(x, self.args[0])._eval_nseries(x, n, logx)
[docs]class Si(TrigonometricIntegral): r""" Sine integral. This function is defined by .. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t. It is an entire function. See Also ======== Ci: Cosine integral. Shi: Sinh integral. Chi: Cosh integral. expint: The generalised exponential integral. References ========== - http://en.wikipedia.org/wiki/Trigonometric_integral Examples ======== >>> from sympy import Si >>> from sympy.abc import z The sine integral is an antiderivative of sin(z)/z: >>> Si(z).diff(z) sin(z)/z It is unbranched: >>> from sympy import exp_polar, I, pi >>> Si(z*exp_polar(2*I*pi)) Si(z) Sine integral behaves much like ordinary sine under multiplication by I: >>> Si(I*z) I*Shi(z) >>> Si(-z) -Si(z) It can also be expressed in terms of exponential integrals, but beware that the latter is branched: >>> from sympy import expint >>> Si(z).rewrite(expint) -I*(-expint(1, z*exp_polar(-I*pi/2))/2 + expint(1, z*exp_polar(I*pi/2))/2) + pi/2 """ _trigfunc = C.sin _atzero = S(0) @classmethod def _minusfactor(cls, z): return -Si(z) @classmethod def _Ifactor(cls, z, sign): return I*Shi(z)*sign def _eval_rewrite_as_expint(self, z): # XXX should we polarify z? return pi/2 + (E1(polar_lift(I)*z) - E1(polar_lift(-I)*z))/2/I
[docs]class Ci(TrigonometricIntegral): r""" Cosine integral. This function is defined for positive :math:`x` by .. math:: \operatorname{Ci}(x) = \gamma + \log{x} + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t, where :math:`\gamma` is the Euler-Mascheroni constant. We have .. math:: \operatorname{Ci}(z) = -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right) + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2} which holds for all polar :math:`z` and thus provides an analytic continuation to the Riemann surface of the logarithm. The formula also holds as stated for :math:`z \in \mathbb{C}` with :math:`Re(z) > 0`. By lifting to the principal branch we obtain an analytic function on the cut complex plane. See Also ======== Si: Sine integral. Shi: Sinh integral. Chi: Cosh integral. expint: The generalised exponential integral. References ========== - http://en.wikipedia.org/wiki/Trigonometric_integral Examples ======== >>> from sympy import Ci >>> from sympy.abc import z The cosine integral is a primitive of cos(z)/z: >>> Ci(z).diff(z) cos(z)/z It has a logarithmic branch point at the origin: >>> from sympy import exp_polar, I, pi >>> Ci(z*exp_polar(2*I*pi)) Ci(z) + 2*I*pi Cosine integral behaves somewhat like ordinary cos under multiplication by I: >>> from sympy import polar_lift >>> Ci(polar_lift(I)*z) Chi(z) + I*pi/2 >>> Ci(polar_lift(-1)*z) Ci(z) + I*pi It can also be expressed in terms of exponential integrals: >>> from sympy import expint >>> Ci(z).rewrite(expint) -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2 """ _trigfunc = C.cos _atzero = S.ComplexInfinity @classmethod def _minusfactor(cls, z): return Ci(z) + I*pi @classmethod def _Ifactor(cls, z, sign): return Chi(z) + I*pi/2*sign def _eval_rewrite_as_expint(self, z): return -(E1(polar_lift(I)*z) + E1(polar_lift(-I)*z))/2
[docs]class Shi(TrigonometricIntegral): r""" Sinh integral. This function is defined by .. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t. It is an entire function. See Also ======== Si: Sine integral. Ci: Cosine integral. Chi: Cosh integral. expint: The generalised exponential integral. References ========== - http://en.wikipedia.org/wiki/Trigonometric_integral Examples ======== >>> from sympy import Shi >>> from sympy.abc import z The Sinh integral is a primitive of sinh(z)/z: >>> Shi(z).diff(z) sinh(z)/z It is unbranched: >>> from sympy import exp_polar, I, pi >>> Shi(z*exp_polar(2*I*pi)) Shi(z) Sinh integral behaves much like ordinary sinh under multiplication by I: >>> Shi(I*z) I*Si(z) >>> Shi(-z) -Shi(z) It can also be expressed in terms of exponential integrals, but beware that the latter is branched: >>> from sympy import expint >>> Shi(z).rewrite(expint) expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2 """ _trigfunc = C.sinh _atzero = S(0) @classmethod def _minusfactor(cls, z): return -Shi(z) @classmethod def _Ifactor(cls, z, sign): return I*Si(z)*sign def _eval_rewrite_as_expint(self, z): from sympy import exp_polar # XXX should we polarify z? return (E1(z)-E1(exp_polar(I*pi)*z))/2 - I*pi/2
[docs]class Chi(TrigonometricIntegral): r""" Cosh integral. This function is defined for positive :math:`x` by .. math:: \operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t, where :math:`\gamma` is the Euler-Mascheroni constant. We have .. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2}, which holds for all polar :math:`z` and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane. See Also ======== Si: Sine integral. Ci: Cosine integral. Shi: Sinh integral. expint: The generalised exponential integral. References ========== - http://en.wikipedia.org/wiki/Trigonometric_integral Examples ======== >>> from sympy import Chi >>> from sympy.abc import z The cosh integral is a primitive of cosh(z)/z: >>> Chi(z).diff(z) cosh(z)/z It has a logarithmic branch point at the origin: >>> from sympy import exp_polar, I, pi >>> Chi(z*exp_polar(2*I*pi)) Chi(z) + 2*I*pi Cosh integral behaves somewhat like ordinary cosh under multiplication by I: >>> from sympy import polar_lift >>> Chi(polar_lift(I)*z) Ci(z) + I*pi/2 >>> Chi(polar_lift(-1)*z) Chi(z) + I*pi It can also be expressed in terms of exponential integrals: >>> from sympy import expint >>> Chi(z).rewrite(expint) -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2 """ _trigfunc = C.cosh _atzero = S.ComplexInfinity @classmethod def _minusfactor(cls, z): return Chi(z) + I*pi @classmethod def _Ifactor(cls, z, sign): return Ci(z) + I*pi/2*sign def _eval_rewrite_as_expint(self, z): from sympy import exp_polar return -I*pi/2 - (E1(z) + E1(exp_polar(I*pi)*z))/2 ############################################################################### #################### FRESNEL INTEGRALS ######################################## ###############################################################################
class FresnelIntegral(Function): """ Base class for the Fresnel integrals.""" nargs = 1 unbranched = True @classmethod def eval(cls, z): # Value at zero if z is S.Zero: return S(0) # Try to pull out factors of -1 and I prefact = S.One newarg = z changed = False nz = newarg.extract_multiplicatively(-1) if nz is not None: prefact = -prefact newarg = nz changed = True nz = newarg.extract_multiplicatively(I) if nz is not None: prefact = cls._sign*I*prefact newarg = nz changed = True if changed: return prefact*cls(newarg) # Values at positive infinities signs # if any were extracted automatically if z is S.Infinity: return S.Half elif z is I*S.Infinity: return cls._sign*I*S.Half def fdiff(self, argindex=1): if argindex == 1: return self._trigfunc(S.Half*pi*self.args[0]**2) else: raise ArgumentIndexError(self, argindex) def _eval_is_real(self): return self.args[0].is_real 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 (re, im) def as_real_imag(self, deep=True, **hints): # Fresnel S # http://functions.wolfram.com/06.32.19.0003.01 # http://functions.wolfram.com/06.32.19.0006.01 # Fresnel C # http://functions.wolfram.com/06.33.19.0003.01 # http://functions.wolfram.com/06.33.19.0006.01 x, y = self._as_real_imag(deep=deep, **hints) sq = -y**2/x**2 re = S.Half*(self.func(x+x*sqrt(sq))+self.func(x-x*sqrt(sq))) im = x/(2*y) * sqrt(sq) * (self.func(x-x*sqrt(sq)) - self.func(x+x*sqrt(sq))) return (re, im) class fresnels(FresnelIntegral): r""" Fresnel integral S. This function is defined by .. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import I, oo, fresnels >>> from sympy.abc import z Several special values are known: >>> fresnels(0) 0 >>> fresnels(oo) 1/2 >>> fresnels(-oo) -1/2 >>> fresnels(I*oo) -I/2 >>> fresnels(-I*oo) I/2 In general one can pull out factors of -1 and I from the argument: >>> fresnels(-z) -fresnels(z) >>> fresnels(I*z) -I*fresnels(z) The Fresnel S integral obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(fresnels(z)) fresnels(conjugate(z)) Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(fresnels(z), z) sin(pi*z**2/2) Defining the Fresnel functions via an integral >>> from sympy import integrate, pi, sin, gamma, expand_func >>> integrate(sin(pi*z**2/2), z) 3*fresnels(z)*gamma(3/4)/(4*gamma(7/4)) >>> expand_func(integrate(sin(pi*z**2/2), z)) fresnels(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnels(2).evalf(30) 0.343415678363698242195300815958 >>> fresnels(-2*I).evalf(30) 0.343415678363698242195300815958*I See Also ======== fresnelc References ========== .. [1] http://en.wikipedia.org/wiki/Fresnel_integral .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS """ _trigfunc = C.sin _sign = -S.One @staticmethod @cacheit def taylor_term(n, x, *previous_terms): if n < 0: return S.Zero else: x = sympify(x) if len(previous_terms) > 1: p = previous_terms[-1] return (-pi**2*x**4*(4*n - 1)/(8*n*(2*n + 1)*(4*n + 3))) * p else: return x**3 * (-x**4)**n * (S(2)**(-2*n - 1)*pi**(2*n + 1)) / ((4*n + 3)*C.factorial(2*n + 1)) def _eval_rewrite_as_erf(self, z): return (S.One + I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z)) def _eval_rewrite_as_hyper(self, z): return pi*z**3/6 * hyper([S(3)/4], [S(3)/2, S(7)/4], -pi**2*z**4/16) def _eval_rewrite_as_meijerg(self, z): return (pi*z**(S(9)/4) / (sqrt(2)*(z**2)**(S(3)/4)*(-z)**(S(3)/4)) * meijerg([],[1],[S(3)/4],[S(1)/4,0],-pi**2*z**4/16)) class fresnelc(FresnelIntegral): r""" Fresnel integral C. This function is defined by .. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import I, oo, fresnelc >>> from sympy.abc import z Several special values are known: >>> fresnelc(0) 0 >>> fresnelc(oo) 1/2 >>> fresnelc(-oo) -1/2 >>> fresnelc(I*oo) I/2 >>> fresnelc(-I*oo) -I/2 In general one can pull out factors of -1 and I from the argument: >>> fresnelc(-z) -fresnelc(z) >>> fresnelc(I*z) I*fresnelc(z) The Fresnel C integral obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(fresnelc(z)) fresnelc(conjugate(z)) Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(fresnelc(z), z) cos(pi*z**2/2) Defining the Fresnel functions via an integral >>> from sympy import integrate, pi, cos, gamma, expand_func >>> integrate(cos(pi*z**2/2), z) fresnelc(z)*gamma(1/4)/(4*gamma(5/4)) >>> expand_func(integrate(cos(pi*z**2/2), z)) fresnelc(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnelc(2).evalf(30) 0.488253406075340754500223503357 >>> fresnelc(-2*I).evalf(30) -0.488253406075340754500223503357*I See Also ======== fresnels References ========== .. [1] http://en.wikipedia.org/wiki/Fresnel_integral .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC """ _trigfunc = C.cos _sign = S.One @staticmethod @cacheit def taylor_term(n, x, *previous_terms): if n < 0: return S.Zero else: x = sympify(x) if len(previous_terms) > 1: p = previous_terms[-1] return (-pi**2*x**4*(4*n - 3)/(8*n*(2*n - 1)*(4*n + 1))) * p else: return x * (-x**4)**n * (S(2)**(-2*n)*pi**(2*n)) / ((4*n + 1)*C.factorial(2*n)) def _eval_rewrite_as_erf(self, z): return (S.One - I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z)) def _eval_rewrite_as_hyper(self, z): return z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16) def _eval_rewrite_as_meijerg(self, z): return (pi*z**(S(3)/4) / (sqrt(2)*root(z**2,4)*root(-z,4)) * meijerg([],[1],[S(1)/4],[S(3)/4,0],-pi**2*z**4/16)) ############################################################################### #################### HELPER FUNCTIONS ######################################### ############################################################################### class _erfs(Function): """ Helper function to make the :math:`erf(z)` function tractable for the Gruntz algorithm. """ nargs = 1 def _eval_aseries(self, n, args0, x, logx): if args0[0] != S.Infinity: return super(_erfs, self)._eval_aseries(n, args0, x, logx) z = self.args[0] l = [ 1/sqrt(S.Pi) * C.factorial(2*k)*(-S(4))**(-k)/C.factorial(k) * (1/z)**(2*k+1) for k in range(0,n) ] o = C.Order(1/z**(2*n+1), x) # It is very inefficient to first add the order and then do the nseries return (Add(*l))._eval_nseries(x, n, logx) + o def fdiff(self, argindex=1): if argindex == 1: z = self.args[0] return -2/sqrt(S.Pi)+2*z*_erfs(z) else: raise ArgumentIndexError(self, argindex) def _eval_rewrite_as_intractable(self, z): return (S.One-erf(z))*C.exp(z**2)