.. |newpage| raw:: latex \newpage .. |br| raw:: html
Hypergeometric Limit Function `\,_0F_1` =============================================================================== Confluent Hypergeometric Limit Function `{}_0F_1(b,x)` ------------------------------------------------------------------------------------- .. method:: ctx.hyperg_0f1(b, x) Returns `\displaystyle {}_0F_1(b,x) = \sum_{k=0}^{\infty} \frac{1}{(b)_k} \frac{x^k}{k!}`, the confluent hypergeometric limit function, where `b \ne 0,-1,-2,\ldots` See also Wikipedia :cite:p:`WikipediaFun90`, MathWorld :cite:p:`WolframFun90` BoostMath :cite:p:`BoostFun90`, :cite:t:`Ehrhardt2018` (3.8.6), Flint :cite:p:`FlintFun90`, Flint :cite:p:`FlintFun90a`, Mpmath :cite:p:`MpmathFun90`. If ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflintreal``, then `b, x \in \mathbb{R}` is exspected. If ``ctx`` is ``cmath53``, then `b \in \mathbb{R}` and `x \in \mathbb{C}` is exspected. If ``ctx`` is ``ctxflintcplx`` then `b, x \in \mathbb{C}` is accepted. Calls ``arb_hypgeom_0f1`` or ``acb_hypgeom_0f1``. Returns the hypergeometric function `{}_0F_1`. Gives the hypergeometric function `{}_0F_1`, sometimes known as the confluent limit function, defined as .. math:: {}_0F_1(a,z) = \sum_{k=0}^{\infty} \frac{1}{(a)_k} \frac{z^k}{k!}. |01a_TestHypergeom0F1_re| `\quad` |01b_TestHypergeom0F1_im| `\quad` |01c_TestHypergeom0F1_abs| .. |01a_TestHypergeom0F1_re| image:: ../_static/ExplicitSurfaces/Cplx0F1/01a_TestHypergeom0F1_re.3D.xml.jpg :width: 30 % .. |01b_TestHypergeom0F1_im| image:: ../_static/ExplicitSurfaces/Cplx0F1/01b_TestHypergeom0F1_im.3D.xml.jpg :width: 30 % .. |01c_TestHypergeom0F1_abs| image:: ../_static/ExplicitSurfaces/Cplx0F1/01c_TestHypergeom0F1_abs.3D.xml.jpg :width: 30 % **Left figure**: real part of the Hypergeometric function `{}_0F_1`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2. **Middle figure**: imaginary part of the Hypergeometric function `{}_0F_1`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2. **Right figure**: absolute value of the Hypergeometric function `{}_0F_1`, 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.Hyperg0F1(5.1,0.5) xreal('5.2359877559829887307E-1') >>> xreal.Hyperg0F1(15.2,0.5) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Hyperg0F1(5.1,0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Hyperg0F1(15.2,0.5) Gpr('5.3518479027559984754E-1') An example with real input: .. code-block:: pycon >>> from xlcalcnet import dec, mpm, gmp, fpm, apm >>> mpm.dps = 40; a= 10; x = 30 >>> \mathrm{d}x = dec.hyp0f1(a, x); mx = mpm.hyp0f1(a, x); gx = gmp.hyp0f1(a, x) >>> fx = fpm.hyp0f1(a, x); ax = apm.hyp0f1(a, x) >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax]) dec: 1.468615500968110064708314782694224995260E+1 mpm: 1.468615500968110064708314782694224995260e+1 gmp: 1.468615500968110064708314782694224995260E+01 fpm: 1.46861550096811E+01 apm: 1.468615500968110064708314782694224995260e+1 (1.251e-39%) An example with complex input: .. code-block:: pycon >>> from xlcalcnet import dec, mpm, gmp, fpm, apm >>> mpm.dps = 20; a= 10; z = '3 + 4j' >>> \mathrm{d}z = dec.hyp0f1(a, z); mz = mpm.hyp0f1(a, z); gz = gmp.hyp0f1(a, z) >>> fz = fpm.hyp0f1(a, z); az = apm.hyp0f1(a, z) >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True) dec: 1.2521260436915235828E+0 + 5.1371862393580499600E-1j mpm: 1.2521260436915235828e+0 + 5.1371862393580499600e-1j gmp: 1.2521260436915235828E+00 + 5.1371862393580499600E-01j fpm: 1.25212604369152E+00 + 5.13718623935805E-01j apm: 1.2521260436915235828e+0 (1.353e-19%) + 5.1371862393580499600e-1 (8.244e-20%)j |newpage| Regularized Confluent Hypergeometric Limit Function `{}_0\widetilde{F}_1(b;x)` ------------------------------------------------------------------------------------------------------ .. method:: ctx.hyperg_0f1r(b, x) Returns `\displaystyle {}_0\widetilde{F}_1(b;x) = \frac{1}{\Gamma(b)} {}_0F_1(b;x)`, the regularized confluent hypergeometric limit function, for `b \ne 0, -1, -2, \cdots`, or, `\\` if `b = 0, -1, -2, \cdots = -n`, the corresponding limit `\displaystyle {}_0\widetilde{F}_1(b;x) = x^{n+1} {}_0\widetilde{F}_1(n+2;x) = \frac{x^{n+1}}{\Gamma(n+2)} {}_0F_1(n+2;x)`. See also Wikipedia :cite:p:`WikipediaFun90`, MathWorld :cite:p:`WolframFun90a` BoostMath :cite:p:`BoostFun90`, :cite:t:`Ehrhardt2018` (3.8.7), Flint :cite:p:`FlintFun90`, Flint :cite:p:`FlintFun90a`, Mpmath :cite:p:`MpmathFun90`. If ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflintreal``, then `b, x \in \mathbb{R}` is exspected. If ``ctx`` is ``cmath53``, then `b \in \mathbb{R}` and `x \in \mathbb{C}` is exspected. If ``ctx`` is ``ctxflintcplx`` then `b, x \in \mathbb{C}` is accepted. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Hyperg0F1r(5.1,0.5) xreal('5.2359877559829887307E-1') >>> xreal.Hyperg0F1r(15.2,0.5) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Hyperg0F1r(5.1,0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Hyperg0F1r(15.2,0.5) Gpr('5.3518479027559984754E-1') An example with real input: .. code-block:: pycon >>> from xlcalcnet import dec, mpm, gmp, fpm, apm >>> mpm.dps = 40; a= 10; x = 30 >>> \mathrm{d}x = dec.hyp0f1r(a, x); mx = mpm.hyp0f1r(a, x); gx = gmp.hyp0f1r(a, x) >>> fx = fpm.hyp0f1r(a, x); ax = apm.hyp0f1r(a, x) >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax]) dec: 4.047110617747216889077146116331087398754E-5 mpm: 4.047110617747216889077146116331087398754e-5 gmp: 4.047110617747216889077146116331087398754E-05 fpm: 4.04711061774722E-05 apm: 4.047110617747216889077146116331087398754e-5 (1.731e-39%) An example with complex input: .. code-block:: pycon >>> from xlcalcnet import dec, mpm, gmp, fpm, apm >>> mpm.dps = 20; a= 10; z = '3 + 4j' >>> \mathrm{d}z = dec.hyp0f1r(a, z); mz = mpm.hyp0f1r(a, z); gz = gmp.hyp0f1r(a, z) >>> fz = fpm.hyp0f1r(a, z); az = apm.hyp0f1r(a, z) >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True) dec: 3.4505237094673820071E-6 + 1.4156708111105737324E-6j mpm: 3.4505237094673820071e-6 + 1.4156708111105737324e-6j gmp: 3.4505237094673820071E-06 + 1.4156708111105737324E-06j fpm: 3.45052370946738E-06 + 1.41567081111057E-06j apm: 3.4505237094673820071e-6 (9.364e-20%) + 1.4156708111105737324e-6 (1.141e-19%)j