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Error function, and related functions
===============================================================================




Generalized Dawson integral, `F(p, x)`
-------------------------------------------------------------------------------

.. method:: math53.dawson2(p, x)

    Returns the generalized Dawson integral `\displaystyle F(p, x) = e^{-x^p} \int_0^x e^{t^p} \mathrm{d}t, p \ge 0, x \ge 0`. See also :cite:t:`Ehrhardt2018` (3.3.2).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.Dawson2(1.5, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.Dawson2(1.5, '0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.Dawson2(1.5, 0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.Dawson2(1.5, '0.51')
        Gpr('5.3518479027559984754E-1')


        



Generalized error function, `\mathrm{erfg}(p, x)`
-------------------------------------------------------------------------------

.. method:: math53.erfg(p,x)

    Returns the generalized error function `\displaystyle \mathrm{erfg}(p, x) = \int_0^x e^{-t^p} \mathrm{d}t = \frac{1}{p}\gamma\left(\frac{1}{p}, x^p \right)`, where `\gamma(\cdot)` denotes the non-normalised lower incomplete gamma function. See also :cite:t:`Ehrhardt2018` (3.3.4).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.Erfg(1.5, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.Erfg(1.5, '0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.Erfg(1.5, 0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.Erfg(1.5, '0.51')
        Gpr('5.3518479027559984754E-1')




Expint3, `\mathrm{erfg}(3, x)`
-------------------------------------------------------------------------------

.. method:: math53.expint3(x)

    Returns `\displaystyle \mathrm{expint3}(p, x) = \mathrm{erfg}(3, x) = \int_0^x e^{-t^3} \mathrm{d}t = \frac{1}{3}\gamma\left(\frac{1}{3}, x^3 \right)`, where `\gamma(\cdot)` denotes the non-normalised lower incomplete gamma function. See also :cite:t:`Ehrhardt2018` (3.3.13).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.Expint3(0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.Expint3('0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.Expint3(0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.Expint3('0.51')
        Gpr('5.3518479027559984754E-1')



        


Exponentially scaled complementary error function, `\mathrm{erfcx}(x)`
-------------------------------------------------------------------------------

.. method:: math53.erfcx(x)


    Returns the exponentially scaled complementary error function `\displaystyle \mathrm{erfcx}(z) = \exp(z^2) \cdot \mathrm{erfc}(z) = w(iz)`. See also :cite:t:`Ehrhardt2018` (3.3.6).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.Erfce(0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.Erfce('0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.Erfce(0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.Erfce('0.51')
        Gpr('5.3518479027559984754E-1')







Scaled repeated integrals of  the complementary error function, `i^n \mathrm{erfc}(x)`
--------------------------------------------------------------------------------------------

.. method:: math53.inerfc(n, x)

    Returns the scaled repeated integrals of the complementary error function, defined for `n \ge -1` using the awkward but standard notation

    .. math::  i^n \mathrm{erfc}(x) = \int_x^{\infty} \mathrm{erfc}(t) \, \mathrm{d}t = \frac{2}{\sqrt{\pi}} \int_x^{\infty} \frac{(t-x)^n}{n!} e^{-t^2} \, \mathrm{d}t, \quad (n = 0,1,2,...).

    See also :cite:t:`Ehrhardt2018` (3.3.7),  NIST :cite:p:`DLMFun187` (eq. 7.18.10), MathWorld :cite:p:`WolframFun187`.


    These functions compute the scaled repeated integrals of complementary error function, defined for `n \geq -1` using the awkward but standard notation

    .. math :: i^n \text{erfc}(x) = \int_x^\infty i^{n-1} \text{erfc}(t)\mathrm{d}t = \frac{2}{\sqrt{\pi}} \int_x^\infty \frac{(t-x)^n}{n!} e^{-t^2} \mathrm{d}t, \quad (n=0,1,2,\ldots)


    .. math :: i^{-1} \text{erfc}(x) =  \frac{2}{\sqrt{\pi}} e^{-x^2}, \quad i^{0} \text{erfc}(x) =  \text{erfc}(x).


    .. math :: i^{n} \text{erfc}(x) =  -\frac{z}{n} i^{n-1} \text{erfc}(x) + \frac{1}{2n} i^{n-2} \text{erfc}(x.


    We also have

    .. math :: i^{n} \text{erfc}(x) =  \frac{e^{-z^2}}{2^n \sqrt{\pi}}  U \left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^2  \right)  




    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.InErfc(3, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.InErfc(3, '0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.InErfc(3, 0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.InErfc(3, '0.51')
        Gpr('5.3518479027559984754E-1')



    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; n = '4'; x = '5.0'
        >>> \mathrm{d}x = dec.inerfc(n, x); mx = mpm.inerfc(n, x); gx = gmp.inerfc(n, x)
        >>> fx = fpm.inerfc(n, x); ax = apm.inerfc(n, x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  8.631401306269731883125812071651943647995E-6
        mpm:  8.631401306269731883125812071651943647995e-6
        gmp:  8.631401306269731883125812071651943647995E-06
        fpm:  8.63140130626973E-06
        apm:  8.631401306269731883125812071651943657349e-6 (1.945e-32%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; n = '4'; z = '5.0 + 3.0j'
        >>> \mathrm{d}z = dec.inerfc(n, z); mz = mpm.inerfc(n, z); gz = gmp.inerfc(n, z)
        >>> fz = fpm.inerfc(n, z); az = apm.inerfc(n, z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: -3.8136067248438755019E-6               - 2.7044284620493310438E-6j
        mpm: -3.8136067248438755019e-6               - 2.7044284620493310438e-6j
        gmp: -3.8136067248438755019E-06              - 2.7044284620493310438E-06j
        fpm: -3.81360672484388E-06                   - 2.70442846204933E-06j
        apm: -3.8136067248438754005e-6 (-2.848e-13%) - 2.7044284620493310996e-6 (-3.025e-13%)j



        

Fresnel auxiliary function `\mathrm{f}(x)`
-------------------------------------------------------------------------------

.. method:: math53.fresnel_f(x) 

    Returns the Fresnel auxiliary function `\displaystyle \mathrm{f}\left(x\right)=\left(\tfrac{1}{2}-S\left(x\right)\right)\cos\left(\tfrac{1}{2}\pi x^{2}\right)-\left(\tfrac{1}{2}-C\left(x\right)\right)\sin\left(\tfrac{1}{2}\pi x^{2}\right)`.

    See also  Wikipedia :cite:p:`WikipediaFun182`, MathWorld :cite:p:`WolframFun182b`, NIST :cite:p:`DLMFun182`, :cite:t:`Ehrhardt2018` (3.3.15).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.FresnelF(0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.FresnelF('0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.FresnelF(0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.FresnelF('0.51')
        Gpr('5.3518479027559984754E-1')






Fresnel auxiliary function `\mathrm{g}(x)`
-------------------------------------------------------------------------------

.. method:: math53.fresnel_g(x)

    Returns the Fresnel auxiliary function `\displaystyle \mathrm{g}\left(x\right)=\left(\tfrac{1}{2}-C\left(x\right)\right)\cos\left(\tfrac{1}{2}\pi x^{2}\right)+\left(\tfrac{1}{2}-S\left(x\right)\right)\sin\left(\tfrac{1}{2}\pi x^{2}\right)`.

    See also  Wikipedia :cite:p:`WikipediaFun182`, MathWorld :cite:p:`WolframFun182b`, NIST :cite:p:`DLMFun182`, :cite:t:`Ehrhardt2018` (3.3.15).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.FresnelG(0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.FresnelG('0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.FresnelG(0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.FresnelG('0.51')
        Gpr('5.3518479027559984754E-1')






Goodwin-Staton integral, `G(x)`
-------------------------------------------------------------------------------

.. method:: math53.goodwin_staton(x)

    Returns the Goodwin-Staton integral  `\displaystyle G(x) = \int_0^{\infty} \frac{e^{-t^2}}{t+x} \, \mathrm{d}t = \sqrt{\pi} F(x) - \tfrac{1}{2} e^{-x^2} \mathrm{Ei}(x^2), \quad x \ne 0`.

    See also Wikipedia :cite:p:`WikipediaFun183`, NIST :cite:p:`DLMFun183`, :cite:t:`Ehrhardt2018` (3.3.16).


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.GoodwinStaton(0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.GoodwinStaton('0.51')
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.GoodwinStaton(0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.GoodwinStaton('0.51')
        Gpr('5.3518479027559984754E-1')