Coulomb, Whittaker and parabolic cylinder function

Regular Coulomb wave function \(F_{\ell}(\eta,x)\)

ctx.coulomb_f(l, eta, x)

where ctx is math53` or ctxflint.

Returns the Coulomb wave function F. See also Wikipedia [1486], MathWorld [1115], NIST [852], Ehrhardt [309] (3.1.10.4), Mpmath [773].

Calculates the regular Coulomb wave function

\[F_{\ell}(\eta,x) = \frac{1}{2i} \left(H^{(+)}_{\ell}(\eta,x) - H^{(-)}_{\ell}(\eta,x) \right),\]

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

\[y'' + \left(1 - \frac{2 \eta}{z} - \frac{\ell(\ell+1)}{z^2}\right) y = 0\]

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 Dzieciol et al. [308], Gaspard [357], Michel [436] and chapter 33 in NIST [852]. 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

>>> 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

>>> 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')

Irregular Coulomb wave function \(G_{\ell}(\eta,z)\)

ctx.coulomb_g(L, eta, x)

where ctx is math53` or ctxflint.

Returns the irregular Coulomb wave function. See also Wikipedia [1486], MathWorld [1115], NIST [852], Ehrhardt [309] (3.1.10.5), Mpmath [774].

Calculates the irregular Coulomb wave function

\[G_{\ell}(\eta,x) = \frac{1}{2} \left(H^{(+)}_{\ell}(\eta,x) - H^{(-)}_{\ell}(\eta,x) \right),\]

where \(H^{(+)}_{\ell}(\eta,x)\) and \(H^{(-)}_{\ell}(\eta,x)\) are irregular Coulomb wave functions.

An example with real input:

>>> 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:

>>> 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

Irregular Coulomb wave function \(H^{(+)}_{\ell}(\eta,x)\)

ctx.coulomb_hplus(L, eta, x)

where ctx is math53` or ctxflint.

Returns the irregular Coulomb wave function. See also Wikipedia [1486], MathWorld [1115], NIST [852], Ehrhardt [309] (3.1.10.5), Mpmath [774].

Calculates the irregular Coulomb wave function

\[H^{(+)}_{\ell}(\eta,x) = -2i(-2)^\ell e^{\pi \eta /2} e^{+i \sigma_{\ell}} x^{\ell+1} e^{+i x} U(\ell+1+i\eta, 2l+2,-2 i x),\]

where \(\sigma_{\ell} = \text{arg} \Gamma(\ell+1+i \eta)\) is called the Coulomb phase shift.

An example with real input:

>>> 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:

>>> 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

Irregular Coulomb wave function \(H^{(-)}_{\ell}(\eta,x)\)

ctx.coulomb_hminus(L, eta, x)

where ctx is math53` or ctxflint.

Returns the irregular Coulomb wave function. See also Wikipedia [1486], MathWorld [1115], NIST [852], Ehrhardt [309] (3.1.10.5), Mpmath [774].

Calculates the irregular Coulomb wave function

\[H^{(-)}_{\ell}(\eta,x) = +2i(-2)^\ell e^{\pi \eta /2} e^{+i \sigma_{\ell}} x^{\ell+1} e^{+i x} U(\ell+1+i\eta, 2l+2,-2 i x),\]

where \(\sigma_{\ell} = \text{arg} \Gamma(\ell+1+i \eta)\) is called the Coulomb phase shift.

An example with real input:

>>> 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:

>>> 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

Whittaker function \(M_{\kappa, \mu}(x)\)

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 [1130], Wikipedia [1515], Ehrhardt [309] (3.8.9), NIST [198], Mpmath [790].

\[M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} \,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z)\]

Evaluates the Whittaker function \(M(k,m,z)\), which gives a solution to the Whittaker differential equation

\[\frac{d^2f}{\mathrm{d}z^2} + \left(-\frac{1}{4}+\frac{k}{z}+ \frac{(\frac{1}{4}-m^2)}{z^2}\right) f = 0.\]

A second solution is given by whitw().

The Whittaker functions are defined in Abramowitz & Stegun, section 13.1. They are alternate forms of the confluent hypergeometric functions \(\,_1F_1\) and \(U\):

\[ \begin{align}\begin{aligned}M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} \,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z)\\W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} U(\tfrac{1}{2}+m-k, 1+2m, z).\end{aligned}\end{align} \]

An example in Python

>>> 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

>>> 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:

>>> 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:

>>> 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

Whittaker function \(W_{\kappa, \mu}(x)\)

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 [1131], Wikipedia [1515], Ehrhardt [309] (3.8.10), NIST [198], Mpmath [791].

\[W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} U(\tfrac{1}{2}+m-k, 1+2m, z).\]

Evaluates the Whittaker function \(W(k,m,z)\), which gives a second solution to the Whittaker differential equation.

An example in Python

>>> 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

>>> 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:

>>> 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:

>>> 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

Parabolic cylinder function \(D_{\nu}(x)\)

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 [293], Wikipedia [1507], Ehrhardt [309] (3.8.11.1).

Returns the parabolic cylinder function D. See also Wikipedia [1507], MathWorld [293], NIST [848], Mpmath [784].

\[D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).\]

Gives the parabolic cylinder function in Whittaker’s notation \(D_n(z) = U(-n-1/2, z)\) (see pcfu()). It solves the differential equation

\[y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0.\]

and can be represented in terms of Hermite polynomials

\[D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).\]

!!! NEED TO COVER CASE n is integer!!!

An example in Python

>>> 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

>>> 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:

>>> 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:

>>> 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

Parabolic cylinder function \(U(a, x)\)

math53.cylinder_u(a, x)

Returns the parabolic cylinder function \(U(a, x) = D_{-a-\frac{1}{2}}(x)\).

See also: MathWorld [1127], Wikipedia [1507], Ehrhardt [309] (3.8.11.2), NIST [848], Mpmath [785].

The function is defined, for arbitrary \(z\), as

\[U\left(a,z\right)=U\left(a,0\right)u_{1}(a,z)+U'\left(a,0\right)u_{2}(a,z),\]

\[u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),\]

\[u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).\]

\[U\left(a,0\right)=\frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}, \quad \text{and } U'\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)},\]

\[e^{-\frac{1}{4}z^2} U(a,z) = U(a,0) \,_1F_1\left(-\tfrac{a}{2}+\tfrac{1}{4}; \tfrac{1}{2}; -\tfrac{1}{2}z^2\right) + U'(a,0) z \,_1F_1\left(-\tfrac{a}{2}+\tfrac{3}{4}; \tfrac{3}{2}; -\tfrac{1}{2}z^2\right), \quad \text{where}\]

\[U\left(a,0\right)=\frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}, \quad \text{and } U'\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)},\]

For \(\Re(z) > 0\), the function may be in terms of the confluent U-function (see hyperu()) by

\[U(a,z) = 2^{-\frac{1}{4}-\frac{a}{2}} e^{-\frac{1}{4} z^2} U\left(\frac{a}{2}+\frac{1}{4}, \frac{1}{2}, \frac{1}{2}z^2\right)\]

An example in Python

>>> 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

>>> 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:

>>> 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:

>>> 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

Parabolic cylinder function \(V(a,x)\)

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 [1127], Wikipedia [1507], Ehrhardt [309] (3.8.11.3), NIST [848], Mpmath [786].

\[V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.\]

Gives the parabolic cylinder function \(V(a,z)\), which can be represented in terms of pcfu() as

\[V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.\]

An example in Python

>>> 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

>>> 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:

>>> 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:

>>> 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

Parabolic cylinder function \(W(a,x)\)

ctxflint.cylinder_w(a, z)

Returns the parabolic cylinder function \(W(a,z)\). See also Wikipedia [1507], MathWorld [1127], NIST [848], Mpmath [787].

The function is defined as

\[W\left(a,x\right)=W\left(a,0\right)w_{1}(a,x)+W'\left(a,0\right)w_{2}(a,x).\]

\[W\left(a,0\right)=2^{-\frac{3}{4}}\left|\frac{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}ia\right)}{\Gamma\left(\tfrac{3}{4}+\tfrac{1}{2}ia\right)}\right|^{\frac{1}{2}}, \quad W'\left(a,0\right)=-2^{-\frac{1}{4}}\left|\frac{\Gamma\left(\tfrac{3}{4}+\tfrac{1}{2}ia\right)}{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}ia\right)}\right|^{\frac{1}{2}}.\]

\[w_{1}(a,x)=e^{-\frac{1}{4}ix^{2}} \,_1F_1\left(\tfrac{1}{4}-\tfrac{1}{2}ia,\tfrac{1}{2},\tfrac{1}{2}ix^{2}\right), \quad w_{2}(a,x)=xe^{-\frac{1}{4}ix^{2}} \,_1F_1\left(\tfrac{3}{4}-\tfrac{1}{2}ia,\tfrac{3}{2},\tfrac{1}{2}ix^{2}\right).\]

\[W\left(a,x\right) = \sqrt{k/2}\,e^{\frac{1}{4}\pi a} \left[e^{i\rho} U\left(ia,xe^{-\pi i/4}\right) + e^{-i\rho} U\left(-ia,xe^{\pi i/4}\right) \right].\]

An example with real input:

>>> 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):

>>> 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