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Gauss Hypergeometric Function `\,_2F_1`
===============================================================================



.. _rst_mpm_hyp2f1: 

Gauss Hypergeometric Function, `{}_2F_1(a,b;c;x)`
--------------------------------------------------------------------------------

.. method:: ctx.hyperg_2f1(a, b, c, x)

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

    Note: math53.hyperg2F1(a, b, c, x)


    Returns `\displaystyle  \,_2F_1(a,b,c,x) = \sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{x^k}{k!}`, the Gauss hypergeometric function, defined for `|x| < 1`.

    See also  Wikipedia :cite:p:`WikipediaFun92`, MathWorld :cite:p:`WolframFun92`, NIST :cite:p:`DLMFun92`, :cite:t:`Ehrhardt2018` (3.8.1), BoostMath :cite:p:`BoostFun92`, Flint :cite:p:`FlintFun92`, Flint :cite:p:`FlintFun92a`, Mpmath :cite:p:`MpmathFun92`. 


    Except for special cases it is required that `-c \ne \mathbb{N}`. For `x > 1` the function is generally complex and not implemented in AMath; but if `a` or `b` is a non-positive integer, then `\,_2F_1(a,b,c,x)` becomes a polynomial in `x` and there is no restriction on `x`. 

    Special values are `\,_2F_1(0,b,c,x) = \,_2F_1(a,0,c,x) = \,_2F_1(a,b,c,0) = 1` and, if `c-a-b>0`, `\displaystyle  \,_2F_1(a,b,c,1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}`.

    In AMath the analytic continuation is done using one or two linear transformations: For all `x < 1` :cite:t:`Abramowitz1970` equations (15.3.3-5) are used, and for `0 < x < 1` if `c` and `c - a - b` are no integers  :cite:t:`Abramowitz1970` equation (15.3.6). For `c = a+b \pm m, (m = 0, 1, \ldots)` the (complicated) formulas from :cite:t:`Abramowitz1970` (15.3.10-12) are implemented. If `a = -m` is a negative integer (or `b` with `a` and `b` swapped, or both and `a \ge b)` the limiting cases for the (polynomial) transformations are implemented as in  :cite:t:`Abramowitz1970` equations (15.8.6/7).


    This calls ``arb_hypgeom_2f1`` or ``acb_hypgeom_2f1``

    Returns the Gauss hypergeometric function `{}_2F_1(a, b; c; z)`.

    The Gauss hypergeometric function `{}_2F_1` is defined for `| z | < 1` by the series 


    .. math:: {}_2F_1(a,b;c;z) = \sum_{k=0}^\infty\frac{(a)_k(b)_k}{(c)_k}\cdot\frac{z^k}{k!}




    |01a_TestHypergeom2F1_re| `\quad` |01b_TestHypergeom2F1_im| `\quad` |01c_TestHypergeom2F1_abs|

    .. |01a_TestHypergeom2F1_re| image:: ../_static/ExplicitSurfaces/CplxpFq/01a_TestHypergeom2F1_re.3D.xml.jpg
       :width: 30 %

    .. |01b_TestHypergeom2F1_im| image:: ../_static/ExplicitSurfaces/CplxpFq/01b_TestHypergeom2F1_im.3D.xml.jpg
       :width: 30 %

    .. |01c_TestHypergeom2F1_abs| image:: ../_static/ExplicitSurfaces/CplxpFq/01c_TestHypergeom2F1_abs.3D.xml.jpg
       :width: 30 %



    **Left figure**: real part of the Gauss Hypergeometric Function, `{}_2F_1(a,b;c;x)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


    **Middle figure**: imaginary part of the Gauss Hypergeometric Function, `{}_2F_1(a,b;c;x)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


    **Right figure**:  absolute value of the Gauss Hypergeometric Function, `{}_2F_1(a,b;c;x)`, with color-coded phase. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.







    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.Hyperg2F1(3,4,5,0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.Hyperg2F1(13,14,15,0.5)
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.Hyperg2F1(3,4,5,0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.Hyperg2F1(13,14,15,0.5)
        Gpr('5.3518479027559984754E-1')




    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; a = '11.0'; b = '12.0'; c = '32.0'; x = '0.3'
        >>> \mathrm{d}x = dec.hyp2f1(a, b, c, x); mx = mpm.hyp2f1(a, b, c, x); gx = gmp.hyp2f1(a, b, c, x)
        >>> fx = fpm.hyp2f1(a, b, c, x); ax = apm.hyp2f1(a, b, c, x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  3.927580225263206566696488844725497448841E+0
        mpm:  3.927580225263206566696488844725497448841e+0
        gmp:  3.927580225263206566696488844725497448841E+00
        fpm:  3.92758022526321E+00
        apm:  3.927580225263206566696488844725497448841e+0 (2.338e-39%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; a = '11.0 + 2.0j'; b = '12.0 + 3.0j'; c = '42.0 + 3.0j';z = '3.0 + 4.0j'
        >>> \mathrm{d}z = dec.hyp2f1(a, b, c, z); mz = mpm.hyp2f1(a, b, c, z); gz = gmp.hyp2f1(a, b, c, z)
        >>> fz = fpm.hyp2f1(a, b, c, z); az = apm.hyp2f1(a, b, c, z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: -1.0355451917135984282E-3              - 1.4654778601071805787E-3j
        mpm: -1.0355451917135984282e-3              - 1.4654778601071805787e-3j
        gmp: -1.0355451917135984282E-03             - 1.4654778601071805787E-03j
        fpm: -1.03554519171360E-03                  - 1.46547786010718E-03j
        apm: -1.0355451917135951286e-3 (-1.197e-9%) - 1.4654778601071774453e-3 (-8.461e-10%)j





|newpage|

Regularized Hypergeometric Function, `{}_2\widetilde{F}_1(a,b;c;x)`
--------------------------------------------------------------------------------------------------------

.. method:: ctx.hyperg_2f1r(a, b, c, x)

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

    Note: math53.hyperg2F1r(a, b, c, x)

    Returns `\displaystyle {}_2\widetilde{F}_1(a,b;c;x)  = \frac{1}{\Gamma(c)} {}_2F_1(a,b;c;x)`, the regularized Gauss hypergeometric function, for `c \ne 0, -1, -2, \cdots`, or, `\\` if `c = 0, -1, -2, \cdots = -m`, the corresponding limit `\displaystyle {}_2\widetilde{F}_1(a,b;-m;x)  =  \frac{(a)_{m+1} (b)_{m+1}}{(m+1)!} x^{m+1} {}_2F_1(a+m+1,b+m+1;m+2;x)`.

    See also  Wikipedia :cite:p:`WikipediaFun92`, MathWorld :cite:p:`WolframFun92a`, NIST :cite:p:`DLMFun92`, :cite:t:`Ehrhardt2018` (3.8.2), BoostMath :cite:p:`BoostFun92`, Flint :cite:p:`FlintFun92`, Flint :cite:p:`FlintFun92a`, Mpmath :cite:p:`MpmathFun92`. 


    This calls ``arb_hypgeom_2f1`` or ``acb_hypgeom_2f1`` with regularized  set.


    Returns the regularized Gauss hypergeometric function `{}_2\widetilde{F}_1(a,b;c;z)`.

    The regularized Gauss hypergeometric function `{}_2\widetilde{F}_1(a,b;c;z)` for unrestricted `c`, is defined by

    .. math:: {}_2\widetilde{F}_1(a,b;c;z)  = \frac{1}{\Gamma(c)} {}_2F_1(a,b;c;z) = \boldsymbol{F}(a,b;c;z), \quad \quad (c \neq    0, -1, -2, \cdots)

    and by the corresponding limit if `c = 0, -1, -2, \cdots, = -n`. 





    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.Hyperg2F1r(3,4,5,0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.Hyperg2F1r(13,14,15,0.5)
        xreal('5.3518479027559984754E-1')


    An example in Visual Basic 

    .. code-block:: pycon

        >>> from xlcalcnet import Gpr
        >>> Gpr.Hyperg2F1r(3,4,5,0.5)
        Gpr('5.2359877559829887307E-1')
        >>> Gpr.Hyperg2F1r(13,14,15,0.5)
        Gpr('5.3518479027559984754E-1')


    An example with real input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; a = '11.0'; b = '12.0'; c = '32.0'; x = '0.3'
        >>> \mathrm{d}x = dec.hyp2f1r(a, b, c, x); mx = mpm.hyp2f1r(a, b, c, x); gx = gmp.hyp2f1r(a, b, c, x)
        >>> fx = fpm.hyp2f1r(a, b, c, x); ax = apm.hyp2f1r(a, b, c, x)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  4.776428664652992475010160932750460985859E-34
        mpm:  4.776428664652992475010160932750460985859e-34
        gmp:  4.776428664652992475010160932750460985859E-34
        fpm:  4.77642866465299E-34
        apm:  4.776428664652992475010160932750460985859e-34 (1.851e-39%)


    An example with complex input:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; a = '11.0 + 2.0j'; b = '12.0 + 3.0j'; c = '42.0 + 3.0j';z = '3.0 + 4.0j'
        >>> \mathrm{d}z = dec.hyp2f1r(a, b, c, z); mz = mpm.hyp2f1r(a, b, c, z); gz = gmp.hyp2f1r(a, b, c, z)
        >>> fz = fpm.hyp2f1r(a, b, c, z); az = apm.hyp2f1r(a, b, c, z)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: 4.1676257256292296278E-53             - 4.2855390713740090672E-53j
        mpm: 4.1676257256292296278e-53             - 4.2855390713740090672e-53j
        gmp: 4.1676257256292296278E-53             - 4.2855390713740090672E-53j
        fpm: 4.16762572562923E-53                  - 4.28553907137401E-53j
        apm: 4.1676257256292213791e-53 (8.47e-10%) - 4.2855390713739963494e-53 (-8.237e-10%)j