.. |newpage| raw:: latex

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.. |br| raw:: html

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Hurwitz zeta and related functions
===============================================================================




.. _rst_mpm_hurwitz: 

Hurwitz zeta function, `\zeta(s,a)`
-------------------------------------------------------------------------------

.. method:: ctx.hurwitz_zeta(s, a)

    where ``ctx`` is ``math53``, ``ctxflint``.

    Returns the Hurwitz zeta function, defined as `\displaystyle \zeta(s,a) = \sum_{k=0}^\infty \frac{1}{(a+k)^s} \,`, `s>1, a \ne 0,-1,-2, \ldots`, and by analytic continuation for `s \ne 0`. The amath implementation requires `a>0`. If `a=1` then `\zeta(s)` is returned, and if `s=0` the result is `\frac{1}{2}-a`.

    See also   Wikipedia :cite:p:`WikipediaFun172`, MathWorld :cite:p:`WolframFun172`, NIST :cite:p:`DLMFun172`, :cite:t:`Ehrhardt2018` (3.6.6), Flint :cite:p:`FlintFun172`.


    .. math :: \zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right)}{n+1}.

    .. math :: \zeta\left(s,1\right)=\zeta\left(s\right).

    .. math :: \zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right).

    .. math :: \zeta'\left(0,a\right)=\log\Gamma\left(a\right)-\tfrac{1}{2}\log\left(2\pi\right).





    |02_0a_TestHurwitzZetaFlint_0_re| `\quad` |02_0b_TestHurwitzZetaFlint_0_im| `\quad` |02_0c_TestHurwitzZetaFlint_0_abs|

    .. |02_0a_TestHurwitzZetaFlint_0_re| image:: ../_static/ExplicitSurfaces/CplxLerch/02_0a_TestHurwitzZetaFlint_0_re.3D.xml.jpg
       :width: 30 %

    .. |02_0b_TestHurwitzZetaFlint_0_im| image:: ../_static/ExplicitSurfaces/CplxLerch/02_0b_TestHurwitzZetaFlint_0_im.3D.xml.jpg
       :width: 30 %

    .. |02_0c_TestHurwitzZetaFlint_0_abs| image:: ../_static/ExplicitSurfaces/CplxLerch/02_0c_TestHurwitzZetaFlint_0_abs.3D.xml.jpg
       :width: 30 %


   

    **3D wpf plot**: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function `z = \sin(x + iy)`, with `-6 \le x \le 6` (blue axis), `-6 \le y \le 6` (red axis), `-10 \le z \le 10` (green axis). Function values are :ref:`loglog-transformed <rst_mpm_loglog_transformation>`.







    |02_1a_TestHurwitzZetaFlint_1_re| `\quad` |02_1b_TestHurwitzZetaFlint_1_im| `\quad` |02_1c_TestHurwitzZetaFlint_1_abs|

    .. |02_1a_TestHurwitzZetaFlint_1_re| image:: ../_static/ExplicitSurfaces/CplxLerch/02_1a_TestHurwitzZetaFlint_1_re.3D.xml.jpg
       :width: 30 %

    .. |02_1b_TestHurwitzZetaFlint_1_im| image:: ../_static/ExplicitSurfaces/CplxLerch/02_1b_TestHurwitzZetaFlint_1_im.3D.xml.jpg
       :width: 30 %

    .. |02_1c_TestHurwitzZetaFlint_1_abs| image:: ../_static/ExplicitSurfaces/CplxLerch/02_1c_TestHurwitzZetaFlint_1_abs.3D.xml.jpg
       :width: 30 %

   

    **3D wpf plot**: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function `z = \sin(x + iy)`, with `-6 \le x \le 6` (blue axis), `-6 \le y \le 6` (red axis), `-10 \le z \le 10` (green axis). Function values are :ref:`loglog-transformed <rst_mpm_loglog_transformation>`.






    |02_2a_TestHurwitzZetaFlint_2_re| `\quad` |02_2b_TestHurwitzZetaFlint_2_im| `\quad` |02_2c_TestHurwitzZetaFlint_2_abs|

    .. |02_2a_TestHurwitzZetaFlint_2_re| image:: ../_static/ExplicitSurfaces/CplxLerch/02_2a_TestHurwitzZetaFlint_2_re.3D.xml.jpg
       :width: 30 %

    .. |02_2b_TestHurwitzZetaFlint_2_im| image:: ../_static/ExplicitSurfaces/CplxLerch/02_2b_TestHurwitzZetaFlint_2_im.3D.xml.jpg
       :width: 30 %

    .. |02_2c_TestHurwitzZetaFlint_2_abs| image:: ../_static/ExplicitSurfaces/CplxLerch/02_2c_TestHurwitzZetaFlint_2_abs.3D.xml.jpg
       :width: 30 %


   

    **3D wpf plot**: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function `z = \sin(x + iy)`, with `-6 \le x \le 6` (blue axis), `-6 \le y \le 6` (red axis), `-10 \le z \le 10` (green axis). Function values are :ref:`loglog-transformed <rst_mpm_loglog_transformation>`.







    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import ereal
        >>> ereal.HurwitzZeta(2,5)
        ereal('5.2359877559829887307E-1')
        >>> ereal.HurwitzZeta(2,'51')
        ereal('5.3518479027559984754E-1')




    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; s = '10'; a = '1.5'
        >>> \mathrm{d}x = dec.hurwitz(s, a); mx = mpm.hurwitz(s, a); gx = gmp.hurwitz(s, a)
        >>> fx = fpm.hurwitz(s, a); ax = apm.hurwitz(s, a)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.745035575790129990031595502635439715798E-2
        mpm:  1.745035575790129990031595502635439715798e-2
        gmp:  1.745035575790129990031595502635439715798E-02
        fpm:  1.74503557579013E-02
        apm:  1.745035575790129990031595502635439715798e-2 (1.028e-39%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; s = '10 + 5j'; a = '5.0 + 3j'
        >>> \mathrm{d}z = dec.hurwitz(s, a); mz = mpm.hurwitz(s, a); gz = gmp.hurwitz(s, a)
        >>> fz = fpm.hurwitz(s, a); az = apm.hurwitz(s, a)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: -2.9834537360028672178E-8               - 3.9761878565611985695E-7j
        mpm: -2.9834537360028672178e-8               - 3.9761878565611985695e-7j
        gmp: -2.9834537360028672178E-08              - 3.9761878565611985695E-07j
        fpm: -2.98345373600287E-08                   - 3.97618785656120E-07j
        apm: -2.9834537360028672177e-8 (-8.461e-20%) - 3.9761878565611985695e-7 (-5.079e-20%)j








|newpage|

Generalized harmonic number function, `H_x^{(r)}`
-------------------------------------------------------------------------------

.. method:: math53.harmonic2(x, s)

    Returns the generalized harmonic function  `H_x^{(r)} = \zeta(r) - \zeta(r,x+1)` for `r \ne 1` and `H_x^{(r)} = H_x` for `r = 1`. 

    See also:  Wikipedia :cite:p:`WikipediaFun127`, MathWorld :cite:p:`WolframFun127b`, :cite:t:`Ehrhardt2018` (3.6.21).




    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import ereal
        >>> ereal.Harmonic2(5, 10.5)
        ereal('5.2359877559829887307E-1')
        >>> ereal.Harmonic2(5, '10.1')
        ereal('5.3518479027559984754E-1')






    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '10'; r = '1.5'
        >>> \mathrm{d}x = dec.harmonic2(x, r); mx = mpm.harmonic2(x, r); gx = gmp.harmonic2(x, r)
        >>> fx = fpm.harmonic2(x, r); ax = apm.harmonic2(x, r)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.995336493345601714521693592714339476261E+0
        mpm:  1.995336493345601714521693592714339476261e+0
        gmp:  1.995336493345601714521693592714339476261E+00
        fpm:  1.99533649334560E+00
        apm:  1.995336493345601714521693592714339476261e+0 (1.668e-38%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '10 + 5j'; r = '5.0 + 3j'
        >>> \mathrm{d}z = dec.harmonic2(z, r); mz = mpm.harmonic2(z, r); gz = gmp.harmonic2(z, r)
        >>> fz = fpm.harmonic2(z, r); az = apm.harmonic2(z, r)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: 9.8046716104400925422E-1              - 2.5426375920749034566E-2j
        mpm: 9.8046716104400925422e-1              - 2.5426375920749034566e-2j
        gmp: 9.8046716104400925422E-01             - 2.5426375920749034566E-02j
        fpm: 9.80467161044009E-01                  - 2.54263759207490E-02j
        apm: 9.8046716104400925422e-1 (8.639e-20%) - 2.5426375920749034566e-2 (-1.041e-19%)j





|newpage|

.. _rst_mpm_bernoulli: 

Bernoulli numbers, `B_n`
-------------------------------------------------------------------------------

.. method:: ctx.bernoulli(n)

    where ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflint``.

    Note ctxboost.BernoulliB2n(n)

    Returns the Bernoulli numbers `B_n`, which are defined by their generating function `\displaystyle \frac{1}{e^t-1} \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}`, `|t < 2\pi|`. If `n<0` or if `n>2` is odd, the result is `0`, and `B_1=-1/2`.

    See also   Wikipedia :cite:p:`WikipediaFun80`, MathWorld :cite:p:`WolframFun80`, NIST :cite:p:`DLMFun80`,  BoostMath :cite:p:`BoostFun80`, :cite:t:`Ehrhardt2018` (3.10.2), Mpmath :cite:p:`MpmathFun80`. 







    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import ereal
        >>> ereal.Bernoulli(8)
        ereal('5.2359877559829887307E-1')
        >>> ereal.Bernoulli(14)
        ereal('5.3518479027559984754E-1')




    An example:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; n = '16'
        >>> \mathrm{d}x = dec.bernoulli(n); mx = mpm.bernoulli(n); gx = gmp.bernoulli(n)
        >>> fx = fpm.bernoulli(n); ax = apm.bernoulli(n)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  -7.092156862745098039215686274509803921569E+0
        mpm:  -7.092156862745098039215686274509803921569e+0
        gmp:  -7.092156862745098039215686274509803921569E+00
        fpm:  -7.09215686274510E+00
        apm:  -7.092156862745098039215686274509803921569e+0 (-1.295e-39%)






|newpage|

Bernoulli polynomials, `B_n(x)`
-------------------------------------------------------------------------------

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

    Returns `\displaystyle B_n(x) = \sum_{n=0}^{\infty} \binom{n}{k} B_k x^{n-k}`, the Bernoulli polynomial of degree `n \ge 0`.

    See also  Wikipedia :cite:p:`WikipediaFun81`, MathWorld :cite:p:`WolframFun81`, NIST :cite:p:`DLMFun80`, :cite:t:`Ehrhardt2018` (3.10.3), Flint :cite:p:`FlintFun81`, Mpmath :cite:p:`MpmathFun81`. 


    See the Amath documentation (p. 146) for connection formula to Hurwitz Zeta.



    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import ereal
        >>> ereal.Bernpoly(3,4)
        ereal('5.2359877559829887307E-1')
        >>> ereal.Bernpoly(13,14)
        ereal('5.3518479027559984754E-1')




    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; n = '16'; x = '1.5'
        >>> \mathrm{d}x = dec.bernpoly(n, x); mx = mpm.bernpoly(n, x); gx = gmp.bernpoly(n, x)
        >>> fx = fpm.bernpoly(n, x); ax = apm.bernpoly(n, x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  7.092428708543964460784313725490196078431E+0
        mpm:  7.092428708543964460784313725490196078431e+0
        gmp:  7.092428708543964460784313725490196078431E+00
        fpm:  7.09242870854397E+00
        apm:  7.092428708543964460784313725490196078432e+0 (1.457e-37%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; n = '16'; z = '10 + 5j'
        >>> \mathrm{d}z = dec.bernpoly(n, z); mz = mpm.bernpoly(n, z); gz = gmp.bernpoly(n, z)
        >>> fz = fpm.bernpoly(n, z); az = apm.bernpoly(n, z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: 9.1935042603632624118E+14              + 2.9617209867479000000E+16j
        mpm: 9.1935042603632624118e+14              + 2.9617209867479000000e+16j
        gmp: 9.1935042603632624118E+14              + 2.9617209867479000000E+16j
        fpm: 9.19350426036326E+14                   + 2.96172098674790E+16j
        apm: 9.1935042603632624118e+14 (1.089e-18%) + 2.9617209867479000000e+16 (1.03e-19%)j




|newpage|

Euler numbers
-------------------------------------------------------------------------------

.. method:: ctx.eulernum(n)

    where ``ctx`` is ``math53``, ``ctxflint``.


    Returns the Euler numbers `E_n`. See also Wikipedia :cite:p:`WikipediaFun120`, MathWorld :cite:p:`WolframFun120`, Flint :cite:p:`FlintFun113`, :cite:t:`Ehrhardt2018` (3.10.8), Mpmath :cite:p:`MpmathFun120`. 


    The Euler numbers `E_n` can be defined as

    .. math:: E_n = \frac{4^n \beta(n+1)}{\zeta(n)} \frac{2B_n}{\pi}




    An example:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; n = '16'
        >>> \mathrm{d}x = dec.eulernum(n); mx = mpm.eulernum(n); gx = gmp.eulernum(n)
        >>> fx = fpm.eulernum(n); ax = apm.eulernum(n)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.939151214500000000000000000000000000000E+10
        mpm:  1.939151214500000000000000000000000000000e+10
        gmp:  1.939151214500000000000000000000000000000E+10
        fpm:  1.93915121450000E+10
        apm:  1.939151214500000000000000000000000000000e+10 (0.0%)




|newpage|

Euler polynomials, `E_n(x)`
-------------------------------------------------------------------------------

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

    Returns `\displaystyle E_n(x) = \frac{2}{n+1} \left( B_{n+1}(x)-2^{n+1}B_{n+1}\left(\frac{x}{2}\right) \right)`, the Euler polynomial of degree `n \ge 0`. Special values include the Euler numbers `E_n = 2^n E_n(1/2)`. 

    See also  Wikipedia :cite:p:`WikipediaFun121`, MathWorld :cite:p:`WolframFun121`, NIST :cite:p:`DLMFun80`, :cite:t:`Ehrhardt2018` (3.10.9), Mpmath :cite:p:`MpmathFun121`. 



    .. math:: E_{n-1}\left(x\right)=\frac{2}{n}\left(B_{n}\left(x\right)-2^{n}B_{n}\left(\tfrac{1}{2}x\right)\right),


    .. math:: E_{n-1}\left(x\right)=\frac{2^{n}}{n}\left(B_{n}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)-B_{n}\left(\tfrac{1}{2}x\right)\right).




    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import ereal
        >>> ereal.Eulerpoly(3, 4)
        ereal('5.2359877559829887307E-1')
        >>> ereal.Eulerpoly(3, 12)
        ereal('5.3518479027559984754E-1')




    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; n = '16'; x = '1.5'
        >>> \mathrm{d}x = dec.eulerpoly(n, x); mx = mpm.eulerpoly(n, x); gx = gmp.eulerpoly(n, x)
        >>> fx = fpm.eulerpoly(n, x); ax = apm.eulerpoly(n, x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  -2.958909933929443359375000000000000000000E+5
        mpm:  -2.958909933929443359375000000000000000000e+5
        gmp:  -2.958909933929443359375000000000000000000E+05
        fpm:  -2.95890993392944E+05
        apm:  -2.958909933929443359375000000000000000000e+5 (-3.254e-38%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; n = '16'; z = '10 + 5j'
        >>> \mathrm{d}z = dec.eulerpoly(n, z); mz = mpm.eulerpoly(n, z); gz = gmp.eulerpoly(n, z)
        >>> fz = fpm.eulerpoly(n, z); az = apm.eulerpoly(n, z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: -2.7730057139461850000E+15               + 2.6625051419981595000E+16j
        mpm: -2.7730057139461850000e+15               + 2.6625051419981595000e+16j
        gmp: -2.7730057139461850000E+15               + 2.6625051419981595000E+16j
        fpm: -2.77300571394618E+15                    + 2.66250514199816E+16j
        apm: -2.7730057139461850000e+15 (-6.053e-18%) + 2.6625051419981595000e+16 (7.45e-19%)j



|newpage|

.. _rst_mpm_barnesg: 

Barnes G-function
-------------------------------------------------------------------------------

.. method:: ctxflint.barnes_g(z)


    Returns the Barnes G-function of *z*. See also Wikipedia :cite:p:`WikipediaFun131`, MathWorld :cite:p:`WolframFun131`, NIST :cite:p:`DLMFun131`, :cite:t:`Whittaker1927`, Mpmath :cite:p:`MpmathFun131`.

    Evaluates the Barnes G-function, which generalizes the superfactorial (:ref:`superfac() <rst_mpm_superfac>`) and by extension also the hyperfactorial (:ref:`hyperfac() <rst_mpm_hyperfac>`) to the complex numbers in an analogous way to how the gamma function generalizes the ordinary factorial.

    The Barnes G-function may be defined in terms of a Weierstrass product:

    .. math ::

        G(z+1) = (2\pi)^{z/2} e^{-[z(z+1)+\gamma z^2]/2}
        \prod_{n=1}^\infty
        \left[\left(1+\frac{z}{n}\right)^ne^{-z+z^2/(2n)}\right]

    For positive integers `n`, we have have relation to superfactorials `G(n) = \mathrm{sf}(n-2) = 0! \cdot 1! \cdots (n-2)!`.





    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '1.5'
        >>> \mathrm{d}x = dec.barnesg(x); mx = mpm.barnesg(x); gx = gmp.barnesg(x)
        >>> fx = fpm.barnesg(x); ax = apm.barnesg(x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.069222649266412949543008878697891604653E+0
        mpm:  1.069222649266412949543008878697891604653e+0
        gmp:  1.069222649266412949543008878697891604653E+00
        fpm:  1.06922264926641E+00
        apm:  1.069222649266412949543008878697891604653e+0 (2.147e-39%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '10 + 5j'
        >>> \mathrm{d}z = dec.barnesg(z); mz = mpm.barnesg(z); gz = gmp.barnesg(z)
        >>> fz = fpm.barnesg(z); az = apm.barnesg(z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: 4.0615260490805827996E+3              - 1.5756073051910381056E+3j
        mpm: 4.0615260490805827996e+3              - 1.5756073051910381056e+3j
        gmp: 4.0615260490805827996E+03             - 1.5756073051910381056E+03j
        fpm: 4.06152604908058E+03                  - 1.57560730519104E+03j
        apm: 4.0615260490805827996e+3 (8.969e-19%) - 1.5756073051910381056e+3 (-2.808e-18%)j





|newpage|

Logarithm of Barnes G function 
-------------------------------------------------------------------------------

.. method:: ctx.logbarnes_g(x)

    where ``ctx`` is ``math53`` or ``ctxflint``.

    Returns `\log G(z)`, the logarithm of Barnes `G` function, with \log G(z) = (z-1) \log \Gamma(z) - \zeta'(-1,z) + \zeta'(-1).

    See also:  Wikipedia :cite:p:`WikipediaFun131`, MathWorld :cite:p:`WolframFun131`, MathWorld :cite:p:`WolframFun131a`, NIST :cite:p:`DLMFun131`, :cite:t:`Ehrhardt2018` (3.5.6.9).


    Computes Barnes *G*-function or the logarithmic Barnes *G*-function, respectively. The logarithmic version has branch cuts on the negative real axis and is continuous elsewhere in the complex plane, in analogy with the logarithmic gamma function. The functional equation

    .. math ::

        \log G(z+1) = \log \Gamma(z) + \log G(z).

    holds for all *z*.



    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import ereal
        >>> ereal.LogBarnesG(7.1)
        ereal('5.2359877559829887307E-1')
        >>> ereal.LogBarnesG('4.51')
        ereal('5.3518479027559984754E-1')





    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '1.5'
        >>> \mathrm{d}x = dec.Logbarnesg(x); mx = mpm.Logbarnesg(x); gx = gmp.Logbarnesg(x)
        >>> fx = fpm.Logbarnesg(x); ax = apm.Logbarnesg(x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  6.693188843500470427402868586818440410225E-2
        mpm:  6.693188843500470427402868586818440410225e-2
        gmp:  6.693188843500470427402868586818440410225E-02
        fpm:  6.69318884350047E-02
        apm:  6.693188843500470427402868586818440410225e-2 (2.144e-39%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '10 + 5j'
        >>> \mathrm{d}z = dec.Logbarnesg(z); mz = mpm.Logbarnesg(z); gz = gmp.Logbarnesg(z)
        >>> fz = fpm.Logbarnesg(z); az = apm.Logbarnesg(z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: 8.3794095077852315352E+0              - 3.7006227106476637064E-1j
        mpm: 8.3794095077852315352e+0              - 3.7006227106476637064e-1j
        gmp: 8.3794095077852315352E+00             - 3.7006227106476637064E-01j
        fpm: 8.37940950778523E+00                  - 3.70062271064766E-01j
        apm: 8.3794095077852315352e+0 (1.617e-19%) + 5.6178605493551511922e+1 (4.825e-20%)j



|newpage|

.. _rst_mpm_hyperfac: 

Hyperfactorial
-------------------------------------------------------------------------------

.. method:: ctxflint.hyperfactorial(z)


    Returns the hyperfactorial of *z*. See also Wikipedia :cite:p:`WikipediaFun130`, MathWorld :cite:p:`WolframFun130`, :cite:t:`OEISFun130`, Mpmath :cite:p:`MpmathFun130`.

    Computes the hyperfactorial, defined for integers as the product

    .. math ::  H(n) = \prod_{k=1}^n k^k.

    The hyperfactorial satisfies the recurrence formula `H(z) = z^z H(z-1)`. It can be defined more generally in terms of the Barnes G-function (see :ref:`barnesg() <rst_mpm_barnesg>`) and the gamma function by the formula

    .. math ::   H(z) = \frac{\Gamma(z+1)^z}{G(z+1)}.



    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '1.5'
        >>> \mathrm{d}x = dec.hyperfac(x); mx = mpm.hyperfac(x); gx = gmp.hyperfac(x)
        >>> fx = fpm.hyperfac(x); ax = apm.hyperfac(x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.617488527948946062817035460231792759012E+0
        mpm:  1.617488527948946062817035460231792759012e+0
        gmp:  1.617488527948946062817035460231792759012E+00
        fpm:  1.61748852794895E+00
        apm:  1.617488527948946062817035460231792771778e+0 (2.059e-33%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '10.5 + 1j'
        >>> \mathrm{d}z = dec.hyperfac(z); mz = mpm.hyperfac(z); gz = gmp.hyperfac(z)
        >>> fz = fpm.hyperfac(z); az = apm.hyperfac(z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: 4.6669290555030590228E+48              + 1.2961711093304818907E+49j
        mpm: 4.6669290555030590228e+48              + 1.2961711093304818907e+49j
        gmp: 4.6669290555030590228E+48              + 1.2961711093304818907E+49j
        fpm: 4.66692905550306E+48                   + 1.29617110933048E+49j
        apm: 4.6669290555030590228e+48 (5.257e-17%) + 1.2961711093304818907e+49 (3.064e-17%)j





|newpage|

.. _rst_mpm_superfac: 

Superfactorial
-------------------------------------------------------------------------------

.. method:: ctxflint.superfactorial(z)


    Returns the Superfactorial of *z*. See also Wikipedia :cite:p:`WikipediaFun129`, MathWorld :cite:p:`WolframFun129`, :cite:t:`OEISFun129`, Mpmath :cite:p:`MpmathFun129`.

    Computes the superfactorial, defined as the product of consecutive factorials

    .. math ::  \mathrm{sf}(n) = \prod_{k=1}^n k!

    For general complex `z`, `\mathrm{sf}(z)` is defined in terms of the Barnes G-function (see :ref:`barnesg() <rst_mpm_barnesg>`).

    .. math ::  \mathrm{sf}(z) = G(z+2)


    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '1.5'
        >>> \mathrm{d}x = dec.superfac(x); mx = mpm.superfac(x); gx = gmp.superfac(x)
        >>> fx = fpm.superfac(x); ax = apm.superfac(x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.259648257495192144086307951096069825563E+0
        mpm:  1.259648257495192144086307951096069825563e+0
        gmp:  1.259648257495192144086307951096069825563E+00
        fpm:  1.25964825749519E+00
        apm:  1.259648257495192144086307951096069825563e+0 (9.113e-40%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '10.5 + 1j'
        >>> \mathrm{d}z = dec.superfac(z); mz = mpm.superfac(z); gz = gmp.superfac(z)
        >>> fz = fpm.superfac(z); az = apm.superfac(z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
        dec: 1.8638249390462723001E+30             - 8.9432323302755030353E+30j
        mpm: 1.8638249390462723001e+30             - 8.9432323302755030353e+30j
        gmp: 1.8638249390462723001E+30             - 8.9432323302755030353E+30j
        fpm: 1.86382493904627E+30                  - 8.94323233027550E+30j
        apm: 1.8638249390462723001e+30 (2.42e-18%) - 8.9432323302755030353e+30 (-1.345e-18%)j