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 ereal
>>> ereal.CoulombF(3, 0.5, 2.25)
ereal('5.2359877559829887307E-1')
>>> ereal.CoulombF(3, 0.5, 8.25)
ereal('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().

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.WhittakerM(5.1,0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.WhittakerM(15.2,0.5)
ereal('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 ereal
>>> ereal.WhittakerW(5.1,0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.WhittakerW(15.2,0.5)
ereal('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 \(D_n(z) = U(-n-1/2, z)\) (see pcfu()).

See also: MathWorld [293], Wikipedia [1507], NIST [848], Mpmath [784], Ehrhardt [309] (3.8.11.1).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.CylinderD(5.1,0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.CylinderD(15.2,0.5)
ereal('5.3518479027559984754E-1')
>>> ereal.CylinderD(15.2,0.5)
ereal('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 \(U(a, x)\).

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

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

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

\[u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}} \,_1F_1\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}} \,_1F_1\left(-\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),\]

\[u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}} \,_1F_1\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}} \,_1F_1\left(-\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).\]

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

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 ereal
>>> ereal.CylinderU(5.1,0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.CylinderU(15.2,0.5)
ereal('5.3518479027559984754E-1')
>>> ereal.CylinderU(15.2,0.5)
ereal('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 \(\displaystyle V(a,x)\), which can be represented in terms of pcfu() as

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

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

In DAMath, this is restricted to \(2a \in \mathbb{Z}\).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.CylinderV(5.1,0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.CylinderV(15.2,0.5)
ereal('5.3518479027559984754E-1')
>>> ereal.CylinderV(15.2,0.5)
ereal('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

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

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

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

The function can also be represented as

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

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