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Coulomb, Whittaker and parabolic cylinder function
==============================================================================================
Regular Coulomb wave function `F_{\ell}(\eta,x)`
-------------------------------------------------------------------------------
.. method:: ctx.coulomb_f(l, eta, x)
where ``ctx`` is ``math53``` or ``ctxflint``.
Returns the Coulomb wave function F. See also Wikipedia :cite:p:`WikipediaFun1052`, MathWorld :cite:p:`WolframFun1052`, NIST :cite:p:`DLMFun1052`, :cite:t:`Ehrhardt2018` (3.1.10.4), Mpmath :cite:p:`MpmathFun1053`.
Calculates the regular Coulomb wave function
.. math :: F_{\ell}(\eta,x) = \frac{1}{2i} \left(H^{(+)}_{\ell}(\eta,x) - H^{(-)}_{\ell}(\eta,x) \right),
where `H^{(+)}_{\ell}(\eta,x)` and `H^{(-)}_{\ell}(\eta,x)` are irregular Coulomb wave functions.
Coulomb wave functions are solutions of the Coulomb wave equation
.. math ::
y'' + \left(1 - \frac{2 \eta}{z} - \frac{\ell(\ell+1)}{z^2}\right) y = 0
which is the radial Schrödinger equation for a charged particle in a
Coulomb potential `1/z`, where `\ell` is the orbital angular momentum and
`\eta` is the Sommerfeld parameter.
The standard solutions are named `F_{\ell}(\eta,z)` (regular
at the origin `z = 0`) and `G_{\ell}(\eta,z)` (irregular at the origin).
The irregular solutions
`H^{\pm}_{\ell}(\eta,z) = G_{\ell}(\eta,z) \pm i F_{\ell}(\eta,z)`
are also used. The redundant functions `H^{\pm}` are provided explicitly since taking
the linear combination of *F* and *G* suffers from cancellation in
parts of the complex plane.
Coulomb wave functions are special cases of confluent hypergeometric functions. The normalization constants and connection formulas are discussed in :cite:t:`Dzieciol1999`, :cite:t:`Gaspard2018`, :cite:t:`Michel2007` and chapter 33 in NIST :cite:p:`DLMFun1052`. In this implementation, we define the analytic continuations of all the functions so that the branch cut with respect to *z* is placed on the negative real axis.
An example in Python
.. code-block:: pycon
>>> from xlcalcnet import xreal
>>> xreal.CoulombF(3, 0.5, 2.25)
xreal('5.2359877559829887307E-1')
>>> xreal.CoulombF(3, 0.5, 8.25)
xreal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from xlcalcnet import Gpr
>>> Gpr.CoulombF(3, 0.5, 2.25)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CoulombF(3, 0.5, 8.25)
Gpr('5.3518479027559984754E-1')
|newpage|
.. _rst_mpm_coulombg:
Irregular Coulomb wave function `G_{\ell}(\eta,z)`
-------------------------------------------------------------------------------
.. method:: ctx.coulomb_g(L, eta, x)
where ``ctx`` is ``math53``` or ``ctxflint``.
Returns the irregular Coulomb wave function. See also Wikipedia :cite:p:`WikipediaFun1052`, MathWorld :cite:p:`WolframFun1052`, NIST :cite:p:`DLMFun1052`, :cite:t:`Ehrhardt2018` (3.1.10.5), Mpmath :cite:p:`MpmathFun1054`.
Calculates the irregular Coulomb wave function
.. math :: G_{\ell}(\eta,x) = \frac{1}{2} \left(H^{(+)}_{\ell}(\eta,x) - H^{(-)}_{\ell}(\eta,x) \right),
where `H^{(+)}_{\ell}(\eta,x)` and `H^{(-)}_{\ell}(\eta,x)` are irregular Coulomb wave functions.
An example with real input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; l = '10'; eta = '3.0'; x = '5.0'
>>> \mathrm{d}x = dec.coulombg(l, eta, x); mx = mpm.coulombg(l, eta, x); gx = gmp.coulombg(l, eta, x)
>>> fx = fpm.coulombg(l, eta, x); ax = apm.coulombg(l, eta, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 4.961542114057365010741557537693681681810E+3
mpm: 4.961542114057365010741557537693681681810e+3
gmp: 4.961542114057365010741557537693681681810E+03
fpm: 4.96154211405737E+03
apm: 4.961542114057365010741557537693681681810e+3 (3.791e-39%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; l = '10'; eta = '3 + 4j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.coulombg(l, eta, z); mz = mpm.coulombg(l, eta, z); gz = gmp.coulombg(l, eta, z)
>>> fz = fpm.coulombg(l, eta, z); az = apm.coulombg(l, eta, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 1.2323201237533547519E+3 - 1.0269675160562813715E+2j
mpm: 1.2323201237533547519e+3 - 1.0269675160562813715e+2j
gmp: 1.2323201237533547519E+03 - 1.0269675160562813715E+02j
fpm: 1.23232012375335E+03 - 1.02696751605628E+02j
apm: 1.2323201237533547519e+3 (4.223e-19%) - 1.0269675160562813715e+2 (-3.537e-18%)j
|newpage|
Irregular Coulomb wave function `H^{(+)}_{\ell}(\eta,x)`
-------------------------------------------------------------------------------
.. method:: ctx.coulomb_hplus(L, eta, x)
where ``ctx`` is ``math53``` or ``ctxflint``.
Returns the irregular Coulomb wave function. See also Wikipedia :cite:p:`WikipediaFun1052`, MathWorld :cite:p:`WolframFun1052`, NIST :cite:p:`DLMFun1052`, :cite:t:`Ehrhardt2018` (3.1.10.5), Mpmath :cite:p:`MpmathFun1054`.
Calculates the irregular Coulomb wave function
.. math :: H^{(+)}_{\ell}(\eta,x) = -2i(-2)^\ell e^{\pi \eta /2} e^{+i \sigma_{\ell}} x^{\ell+1} e^{+i x} U(\ell+1+i\eta, 2l+2,-2 i x),
where `\sigma_{\ell} = \text{arg} \Gamma(\ell+1+i \eta)` is called the Coulomb phase shift.
An example with real input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; l = '10'; eta = '3.0'; x = '5.0'
>>> \mathrm{d}x = dec.coulombg(l, eta, x); mx = mpm.coulombg(l, eta, x); gx = gmp.coulombg(l, eta, x)
>>> fx = fpm.coulombg(l, eta, x); ax = apm.coulombg(l, eta, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 4.961542114057365010741557537693681681810E+3
mpm: 4.961542114057365010741557537693681681810e+3
gmp: 4.961542114057365010741557537693681681810E+03
fpm: 4.96154211405737E+03
apm: 4.961542114057365010741557537693681681810e+3 (3.791e-39%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; l = '10'; eta = '3 + 4j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.coulombg(l, eta, z); mz = mpm.coulombg(l, eta, z); gz = gmp.coulombg(l, eta, z)
>>> fz = fpm.coulombg(l, eta, z); az = apm.coulombg(l, eta, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 1.2323201237533547519E+3 - 1.0269675160562813715E+2j
mpm: 1.2323201237533547519e+3 - 1.0269675160562813715e+2j
gmp: 1.2323201237533547519E+03 - 1.0269675160562813715E+02j
fpm: 1.23232012375335E+03 - 1.02696751605628E+02j
apm: 1.2323201237533547519e+3 (4.223e-19%) - 1.0269675160562813715e+2 (-3.537e-18%)j
|newpage|
Irregular Coulomb wave function `H^{(-)}_{\ell}(\eta,x)`
-------------------------------------------------------------------------------
.. method:: ctx.coulomb_hminus(L, eta, x)
where ``ctx`` is ``math53``` or ``ctxflint``.
Returns the irregular Coulomb wave function. See also Wikipedia :cite:p:`WikipediaFun1052`, MathWorld :cite:p:`WolframFun1052`, NIST :cite:p:`DLMFun1052`, :cite:t:`Ehrhardt2018` (3.1.10.5), Mpmath :cite:p:`MpmathFun1054`.
Calculates the irregular Coulomb wave function
.. math :: H^{(-)}_{\ell}(\eta,x) = +2i(-2)^\ell e^{\pi \eta /2} e^{+i \sigma_{\ell}} x^{\ell+1} e^{+i x} U(\ell+1+i\eta, 2l+2,-2 i x),
where `\sigma_{\ell} = \text{arg} \Gamma(\ell+1+i \eta)` is called the Coulomb phase shift.
An example with real input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; l = '10'; eta = '3.0'; x = '5.0'
>>> \mathrm{d}x = dec.coulombg(l, eta, x); mx = mpm.coulombg(l, eta, x); gx = gmp.coulombg(l, eta, x)
>>> fx = fpm.coulombg(l, eta, x); ax = apm.coulombg(l, eta, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 4.961542114057365010741557537693681681810E+3
mpm: 4.961542114057365010741557537693681681810e+3
gmp: 4.961542114057365010741557537693681681810E+03
fpm: 4.96154211405737E+03
apm: 4.961542114057365010741557537693681681810e+3 (3.791e-39%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; l = '10'; eta = '3 + 4j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.coulombg(l, eta, z); mz = mpm.coulombg(l, eta, z); gz = gmp.coulombg(l, eta, z)
>>> fz = fpm.coulombg(l, eta, z); az = apm.coulombg(l, eta, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 1.2323201237533547519E+3 - 1.0269675160562813715E+2j
mpm: 1.2323201237533547519e+3 - 1.0269675160562813715e+2j
gmp: 1.2323201237533547519E+03 - 1.0269675160562813715E+02j
fpm: 1.23232012375335E+03 - 1.02696751605628E+02j
apm: 1.2323201237533547519e+3 (4.223e-19%) - 1.0269675160562813715e+2 (-3.537e-18%)j
|newpage|
.. _rst_mpm_whitm:
Whittaker function `M_{\kappa, \mu}(x)`
-------------------------------------------------------------------------------
.. method:: math53.whittaker_m(k, m, x)
Returns `M_{\kappa, \mu}(x) = e^{-\frac{1}{2}x} x^{\frac{1}{2}+\mu} M(\tfrac{1}{2}+\mu-\kappa, 1+2\mu, x)`, the Whittaker function M.
See also MathWorld :cite:p:`WolframFun1055a`, Wikipedia :cite:p:`WikipediaFun1055`, :cite:t:`Ehrhardt2018` (3.8.9), NIST :cite:p:`DLMFun1055`, Mpmath :cite:p:`MpmathFun1055`.
.. math :: M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} \,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z)
Evaluates the Whittaker function `M(k,m,z)`, which gives a solution
to the Whittaker differential equation
.. math ::
\frac{d^2f}{\mathrm{d}z^2} + \left(-\frac{1}{4}+\frac{k}{z}+
\frac{(\frac{1}{4}-m^2)}{z^2}\right) f = 0.
A second solution is given by :ref:`whitw() `.
The Whittaker functions are defined in Abramowitz & Stegun, section 13.1.
They are alternate forms of the confluent hypergeometric functions
`\,_1F_1` and `U`:
.. math ::
M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m}
\,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z)
W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m}
U(\tfrac{1}{2}+m-k, 1+2m, z).
An example in Python
.. code-block:: pycon
>>> from xlcalcnet import xreal
>>> xreal.WhittakerM(5.1,0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.WhittakerM(15.2,0.5)
xreal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from xlcalcnet import Gpr
>>> Gpr.WhittakerM(5.1,0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.WhittakerM(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; k = '10'; m = '3.0'; x = '5.0'
>>> \mathrm{d}x = dec.whitm(k, m, x); mx = mpm.whitm(k, m, x); gx = gmp.whitm(k, m, x)
>>> fx = fpm.whitm(k, m, x); ax = apm.whitm(k, m, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 5.037724223377349504794983027900485241922E-2
mpm: 5.037724223377349504794983027900485241922e-2
mpm: 5.037724223377349504794983027900485241922e-2
fpm: 5.03772422337738E-02
apm: 5.037724223377349504794983027900485241947e-2 (1.976e-36%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; k = '10'; m = '3 + 4j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.whitm(k, m, z); mz = mpm.whitm(k, m, z); gz = gmp.whitm(k, m, z)
>>> fz = fpm.whitm(k, m, z); az = apm.whitm(k, m, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 4.9746800935982291618E-1 - 5.5703088408623297578E-1j
mpm: 4.9746800935982291618e-1 - 5.5703088408623297578e-1j
mpm: 4.9746800935982291618e-1 - 5.5703088408623297578e-1j
fpm: 4.97468009359823E-01 - 5.57030884086233E-01j
apm: 4.9746800935982291618e-1 (8.939e-19%) - 5.5703088408623297578e-1 (-6.082e-19%)j
|newpage|
.. _rst_mpm_whitw:
Whittaker function `W_{\kappa, \mu}(x)`
-------------------------------------------------------------------------------
.. method:: math53.whittaker_w(k, m, x)
Returns `W_{\kappa, \mu}(x) = e^{-\frac{1}{2}x} x^{\frac{1}{2}+\mu} U(\tfrac{1}{2}+\mu-\kappa, 1+2\mu, x)`, the Whittaker function W.
See also MathWorld :cite:p:`WolframFun1055b`, Wikipedia :cite:p:`WikipediaFun1055`, :cite:t:`Ehrhardt2018` (3.8.10), NIST :cite:p:`DLMFun1055`, Mpmath :cite:p:`MpmathFun1056`.
.. math :: W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} U(\tfrac{1}{2}+m-k, 1+2m, z).
Evaluates the Whittaker function `W(k,m,z)`, which gives a second
solution to the Whittaker differential equation.
An example in Python
.. code-block:: pycon
>>> from xlcalcnet import xreal
>>> xreal.WhittakerW(5.1,0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.WhittakerW(15.2,0.5)
xreal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from xlcalcnet import Gpr
>>> Gpr.WhittakerW(5.1,0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.WhittakerW(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; k = '10'; m = '3.0'; x = '5.0'
>>> \mathrm{d}x = dec.whitw(k, m, x); mx = mpm.whitw(k, m, x); gx = gmp.whitw(k, m, x)
>>> fx = fpm.whitw(k, m, x); ax = apm.whitw(k, m, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: -9.206869244108589870287530029194217374270E+5
mpm: -9.206869244108589870287530029194217374270e+5
mpm: -9.206869244108589870287530029194217374270e+5
fpm: -9.20686924410859E+05
apm: -9.206869244108589870287530029194217374270e+5 (-7.191e-39%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; k = '10'; m = '3 + 4j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.whitw(k, m, z); mz = mpm.whitw(k, m, z); gz = gmp.whitw(k, m, z)
>>> fz = fpm.whitw(k, m, z); az = apm.whitw(k, m, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 7.5723638666590182121E+7 - 1.6673471905439061612E+8j
mpm: 7.5723638666590182121e+7 - 1.6673471905439061612e+8j
mpm: 7.5723638666590182121e+7 - 1.6673471905439061612e+8j
fpm: 7.57236386665902E+07 - 1.66734719054391E+08j
apm: 7.5723638666590182120e+7 (6.756e-19%) - 1.6673471905439061612e+8 (-4.091e-19%)j
|newpage|
Parabolic cylinder function `D_{\nu}(x)`
-------------------------------------------------------------------------------
.. method:: math53.cylinder_d(nu, x)
Returns Whittaker’s parabolic cylinder function `\displaystyle D_{\nu}(x) = 2^{-\nu/2} e^{-x^2/4} U\left(-\frac{\nu}{2}, \frac{1}{2}, \frac{x^2}{2} \right)`, for `x \ge 0`.
For `x<0` and `\nu \notin \mathbb{N}` we have `\displaystyle D_{\nu}(-x) = D_{\nu}(x) - \frac{2^{(\nu+1)/2} \nu \Gamma(\nu/2)\sin(\pi\nu/2)}{\sqrt{\pi}} x e^{-x^2/4} {}_1F_1\left(\frac{1-\nu}{2}, \frac{3}{2}, \frac{x^2}{2} \right)`,
and for `\nu = n \in \mathbb{N}` the relation `D_n(-x) = (-1)^n D_n(x)` is applied.
See also: MathWorld :cite:p:`WolframFun1057a`, Wikipedia :cite:p:`WikipediaFun1057`, :cite:t:`Ehrhardt2018` (3.8.11.1).
Returns the parabolic cylinder function D. See also Wikipedia :cite:p:`WikipediaFun1057`, MathWorld :cite:p:`WolframFun1057a`, NIST :cite:p:`DLMFun1057`, Mpmath :cite:p:`MpmathFun1057`.
.. math :: D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).
Gives the parabolic cylinder function in Whittaker's notation
`D_n(z) = U(-n-1/2, z)` (see :ref:`pcfu() `).
It solves the differential equation
.. math ::
y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0.
and can be represented in terms of Hermite polynomials
.. math ::
D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).
!!! NEED TO COVER CASE n is integer!!!
An example in Python
.. code-block:: pycon
>>> from xlcalcnet import xreal
>>> xreal.CylinderD(5.1,0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.CylinderD(15.2,0.5)
xreal('5.3518479027559984754E-1')
>>> xreal.CylinderD(15.2,0.5)
xreal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from xlcalcnet import Gpr
>>> Gpr.CylinderD(5.1,0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CylinderD(15.2,0.5)
Gpr('5.3518479027559984754E-1')
>>> Gpr.CylinderD(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; n = '10.1'; x = '5.0'
>>> \mathrm{d}x = dec.pcfd(n, x); mx = mpm.pcfd(n, x); gx = gmp.pcfd(n, x)
>>> fx = fpm.pcfd(n, x); ax = apm.pcfd(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 2.989637576862617956820755116429869259351E+2
mpm: 2.989637576862617956820755116429869259351e+2
mpm: 2.989637576862617956820755116429869259352e+2
fpm: 2.98963757686262E+02
apm: 2.989637576862617956820755116429869259352e+2 (1.914e-36%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10.1'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.pcfd(n, z); mz = mpm.pcfd(n, z); gz = gmp.pcfd(n, z)
>>> fz = fpm.pcfd(n, z); az = apm.pcfd(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 5.7176371555791286919E+5 - 4.3671006346521104926E+5j
mpm: 5.7176371555791286919e+5 - 4.3671006346521104926e+5j
mpm: 5.7176371555791286919e+5 - 4.3671006346521104926e+5j
fpm: 5.71763715557912E+05 - 4.36710063465211E+05j
apm: 5.7176371555791286919e+5 (1.577e-17%) - 4.3671006346521104926e+5 (-1.825e-17%)j
|newpage|
.. _rst_mpm_pcfu:
Parabolic cylinder function `U(a, x)`
-------------------------------------------------------------------------------
.. method:: math53.cylinder_u(a, x)
Returns the parabolic cylinder function `U(a, x) = D_{-a-\frac{1}{2}}(x)`.
See also: MathWorld :cite:p:`WolframFun1057`, Wikipedia :cite:p:`WikipediaFun1057`, :cite:t:`Ehrhardt2018` (3.8.11.2), NIST :cite:p:`DLMFun1057`, Mpmath :cite:p:`MpmathFun1058`.
The function is defined, for arbitrary `z`, as
.. math :: U\left(a,z\right)=U\left(a,0\right)u_{1}(a,z)+U'\left(a,0\right)u_{2}(a,z),
.. math :: u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),
.. math :: u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).
.. math :: U\left(a,0\right)=\frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}, \quad \text{and } U'\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)},
.. math :: e^{-\frac{1}{4}z^2} U(a,z) = U(a,0) \,_1F_1\left(-\tfrac{a}{2}+\tfrac{1}{4}; \tfrac{1}{2}; -\tfrac{1}{2}z^2\right) + U'(a,0) z \,_1F_1\left(-\tfrac{a}{2}+\tfrac{3}{4}; \tfrac{3}{2}; -\tfrac{1}{2}z^2\right), \quad \text{where}
.. math :: U\left(a,0\right)=\frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}, \quad \text{and } U'\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)},
For `\Re(z) > 0`, the function may be in terms of the confluent U-function (see :ref:`hyperu() `) by
.. math :: U(a,z) = 2^{-\frac{1}{4}-\frac{a}{2}} e^{-\frac{1}{4} z^2} U\left(\frac{a}{2}+\frac{1}{4}, \frac{1}{2}, \frac{1}{2}z^2\right)
An example in Python
.. code-block:: pycon
>>> from xlcalcnet import xreal
>>> xreal.CylinderU(5.1,0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.CylinderU(15.2,0.5)
xreal('5.3518479027559984754E-1')
>>> xreal.CylinderU(15.2,0.5)
xreal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from xlcalcnet import Gpr
>>> Gpr.CylinderU(5.1,0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CylinderU(15.2,0.5)
Gpr('5.3518479027559984754E-1')
>>> Gpr.CylinderU(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 = '5.0'
>>> \mathrm{d}x = dec.pcfu(a, x); mx = mpm.pcfu(a, x); gx = gmp.pcfu(a, x)
>>> fx = fpm.pcfu(a, x); ax = apm.pcfu(a, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 1.549656896287937648701131850232398032454E-11
mpm: 1.549656896287937648701131850232398032454e-11
mpm: 1.549656896287937648701131850232398032454e-11
fpm: 1.54965689628794E-11
apm: 1.549656896287937648701131850232398032454e-11 (1.078e-39%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; a = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.pcfu(a, z); mz = mpm.pcfu(a, z); gz = gmp.pcfu(a, z)
>>> fz = fpm.pcfu(a, z); az = apm.pcfu(a, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 5.6244468909587524688E-11 + 3.2113324546772489474E-11j
mpm: 5.6244468909587524688e-11 + 3.2113324546772489474e-11j
mpm: 5.6244468909587524688e-11 + 3.2113324546772489474e-11j
fpm: 5.62444689095875E-11 + 3.21133245467725E-11j
apm: 5.6244468909587524688e-11 (4.383e-20%) + 3.2113324546772489474e-11 (7.677e-20%)j
|newpage|
Parabolic cylinder function `V(a,x)`
-------------------------------------------------------------------------------
.. method:: math53.cylinder_v(a, x)
Returns the parabolic cylinder function `\displaystyle V(a,x) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-x)-\sin(\pi a) U(a,x)}{\pi}`, where a is restricted to `2a \in \mathbb{Z}`.
See also: MathWorld :cite:p:`WolframFun1057`, Wikipedia :cite:p:`WikipediaFun1057`, :cite:t:`Ehrhardt2018` (3.8.11.3), NIST :cite:p:`DLMFun1057`, Mpmath :cite:p:`MpmathFun1059`.
.. math :: V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.
Gives the parabolic cylinder function `V(a,z)`, which can be
represented in terms of :ref:`pcfu() ` as
.. math ::
V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.
An example in Python
.. code-block:: pycon
>>> from xlcalcnet import xreal
>>> xreal.CylinderV(5.1,0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.CylinderV(15.2,0.5)
xreal('5.3518479027559984754E-1')
>>> xreal.CylinderV(15.2,0.5)
xreal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from xlcalcnet import Gpr
>>> Gpr.CylinderV(5.1,0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CylinderV(15.2,0.5)
Gpr('5.3518479027559984754E-1')
>>> Gpr.CylinderV(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 = '5.0'
>>> \mathrm{d}x = dec.pcfv(a, x); mx = mpm.pcfv(a, x); gx = gmp.pcfv(a, x)
>>> fx = fpm.pcfv(a, x); ax = apm.pcfv(a, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 6.386215506514216377692001055435111083031E+9
mpm: 6.386215506514216377692001055435111083031e+9
mpm: 6.386215506514216377692001055435111083031e+9
fpm: 6.38621550651422E+09
apm: 6.386215506514216377692001055435111083031e+9 (5.404e-39%)
An example with complex input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; a = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.pcfv(a, z); mz = mpm.pcfv(a, z); gz = gmp.pcfv(a, z)
>>> fz = fpm.pcfv(a, z); az = apm.pcfv(a, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 1.1155789211452870283E+9 - 1.0701503668604253775E+9j
mpm: 1.1155789211452870283e+9 - 1.0701503668604253775e+9j
mpm: 1.1155789211452870283e+9 - 1.0701503668604253775e+9j
fpm: 1.11557892114529E+09 - 1.07015036686043E+09j
apm: 1.1155789211452870283e+9 (8.968e-19%) - 1.0701503668604253775e+9 (-8.499e-19%)j
|newpage|
.. _rst_mpm_pcfw:
Parabolic cylinder function `W(a,x)`
-------------------------------------------------------------------------------
.. method:: ctxflint.cylinder_w(a, z)
Returns the parabolic cylinder function `W(a,z)`. See also Wikipedia :cite:p:`WikipediaFun1057`, MathWorld :cite:p:`WolframFun1057`, NIST :cite:p:`DLMFun1057`, Mpmath :cite:p:`MpmathFun1060`.
The function is defined as
.. math :: W\left(a,x\right)=W\left(a,0\right)w_{1}(a,x)+W'\left(a,0\right)w_{2}(a,x).
.. math :: W\left(a,0\right)=2^{-\frac{3}{4}}\left|\frac{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}ia\right)}{\Gamma\left(\tfrac{3}{4}+\tfrac{1}{2}ia\right)}\right|^{\frac{1}{2}}, \quad W'\left(a,0\right)=-2^{-\frac{1}{4}}\left|\frac{\Gamma\left(\tfrac{3}{4}+\tfrac{1}{2}ia\right)}{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}ia\right)}\right|^{\frac{1}{2}}.
.. math :: w_{1}(a,x)=e^{-\frac{1}{4}ix^{2}} \,_1F_1\left(\tfrac{1}{4}-\tfrac{1}{2}ia,\tfrac{1}{2},\tfrac{1}{2}ix^{2}\right), \quad w_{2}(a,x)=xe^{-\frac{1}{4}ix^{2}} \,_1F_1\left(\tfrac{3}{4}-\tfrac{1}{2}ia,\tfrac{3}{2},\tfrac{1}{2}ix^{2}\right).
.. math ::
W\left(a,x\right) = \sqrt{k/2}\,e^{\frac{1}{4}\pi a} \left[e^{i\rho} U\left(ia,xe^{-\pi i/4}\right) + e^{-i\rho} U\left(-ia,xe^{\pi i/4}\right) \right].
An example with real input:
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; a = '10'; x = '5.0'
>>> \mathrm{d}x = dec.pcfw(a, x); mx = mpm.pcfw(a, x); gx = gmp.pcfw(a, x)
>>> fx = fpm.pcfw(a, x); ax = apm.pcfw(a, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec: 4.662395750711469900575353811000346598048E-7
mpm: 4.662395750711469900575353811000346598048e-7
mpm: 4.662395750711469900575353811000346598048e-7
fpm: 4.66239575071147E-07
apm: 4.662395750711469899862962927504081215235e-7 (5.87e-40%)
An example with complex input (this is an example where all digits returned by mpmath are wrong because of insufficient precision):
.. code-block:: pycon
>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; a = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.pcfw(a, z); mz = mpm.pcfw(a, z); gz = gmp.pcfw(a, z)
>>> fz = fpm.pcfw(a, z); az = apm.pcfw(a, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 8.8066753933700124107E-7 - 6.3128500751345654949E-7j
mpm: 8.8066753933700124107e-7 - 6.3128500751345654949e-7j
mpm: 8.8066753933700124107e-7 - 6.3128500751345654949e-7j
fpm: 8.80667539337001E-07 - 6.31285007513457E-07j
apm: 6.9582517470815592536e-8 (7.256e-20%) - 4.9878527483184454344e-8 (-5.061e-20%)j