Scorer functions#

Scorer function \(\mathrm{Gi}(x)\)#

math53.scorer_gi(x)#

Returns the Scorer function Gi, which gives a particular solution to the inhomogeneous Airy differential equation \(f''(x) - x f(x) = 1/\pi\).

See also: MathWorld [1128], Wikipedia [1510], Ehrhardt [309] (3.1.7.7), NIST [461], Mpmath [788].

The Scorer function Gi is defined as

\[\text{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin \left(xt - \frac{1}{3}t^3 \right) \mathrm{d}t.\]

Another particular solution is given by the Scorer Hi-function. The two functions are related as \(\mathrm{Gi}(z) + \mathrm{Hi}(z) = \mathrm{Bi}(z)\).

We also have

\[\text{Gi}(z) = \tfrac{1}{3} \text{Bi}(z) - \frac{ z^2}{2\pi} \: {}_1F_2\left(1; \tfrac{4}{3}, \tfrac{5}{3}; \tfrac{1}{9}z^3 \right)\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.AiryGi(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.AiryGi('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.AiryGi(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.AiryGi('0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; x = -3
>>> \mathrm{d}x = dec.scorergi(x); mx = mpm.scorergi(x); gx = gmp.scorergi(x)
>>> fx = fpm.scorergi(x); ax = apm.scorergi(x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -2.990547183713964238015818976238353474193E-1
mpm: -2.990547183713964238015818976238353474193e-1
gmp: -2.990547183713964238015818976238353474193E-01
fpm: -2.99054718371396E-01
apm: -2.990547183713964238015818976238353474192e-1 (-4.827e-37%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.scorergi(z); mz = mpm.scorergi(z); gz = gmp.scorergi(z)
>>> fz = fpm.scorergi(z); az = apm.scorergi(z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 8.4889448371208839089E-2             - 3.5862043476779870404E-2j
mpm: 8.4889448371208839089e-2             - 3.5862043476779870404e-2j
gmp: 8.4889448371208839089E-02            - 3.5862043476779870404E-02j
fpm: 8.48894483712088E-02                 - 3.58620434767799E-02j
apm: 8.4889448371208839123e-2 (6.78e-15%) - 3.5862043476779870473e-2 (-1.391e-14%)j

Scorer function \(\mathrm{Hi}(x)\)#

math53.scorer_hi(x)#

Returns the Scorer function Hi, which gives a particular solution to the inhomogeneous Airy differential equation \(f''(x) - x f(x) = 1/\pi\).

See also: MathWorld [1129], Wikipedia [1510], Ehrhardt [309] (3.1.7.8), Mpmath [789].

Returns the Scorer function Hi, defined as

\[\text{Hi}(x) = \frac{1}{\pi} \int_0^\infty \exp \left(xt - \frac{1}{3}t^3 \right) \mathrm{d}t\]

We also have

\[\text{Hi}(z) = \tfrac{2}{3} \text{Bi}(z) + \frac{ z^2}{2\pi} \: {}_1F_2\left(1; \tfrac{4}{3}, \tfrac{5}{3}; \tfrac{1}{9}z^3 \right)\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.AiryHi(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.AiryHi('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.AiryHi(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.AiryHi('0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; x = -3
>>> \mathrm{d}x = dec.scorerhi(x); mx = mpm.scorerhi(x); gx = gmp.scorerhi(x)
>>> fx = fpm.scorerhi(x); ax = apm.scorerhi(x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: 1.007650919964698805809370430513469907205E-1
mpm: 1.007650919964698805809370430513469907205e-1
gmp: 1.007650919964698805809370430513469907205E-01
fpm: 1.00765091996470E-01
apm: 1.007650919964698805809370430513469906431e-1 (1.334e-33%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.scorerhi(z); mz = mpm.scorerhi(z); gz = gmp.scorerhi(z)
>>> fz = fpm.scorerhi(z); az = apm.scorerhi(z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 9.5150834628338203598E-1              + 1.0872383260084919901E+0j
mpm: 9.5150834628338203598e-1              + 1.0872383260084919901e+0j
gmp: 9.5150834628338203598E-01             + 1.0872383260084919901E+00j
fpm: 9.51508346283382E-01                  + 1.08723832600849E+00j
apm: 9.5150834628338203594e-1 (6.049e-16%) + 1.0872383260084919901e+0 (4.588e-16%)j

First derivative of the Scorer function \(\mathrm{Gi}(x)\), \(\mathrm{Gi}'(x)\)#

math53.scorer_gi_prime(x)#

Returns \(\mathrm{Gi}'(x)\), the first derivative of the Airy (Scorer) function \(\mathrm{Gi}(x)\).

See also: MathWorld [1128], Wikipedia [1510], Ehrhardt [309] (3.1.7.7), NIST [461], Mpmath [788].

The function is calculated as

\[\text{Gi}'(z) = \tfrac{1}{3} \text{Bi}'(z) - \frac{1}{40\pi} \left[ 40 z \cdot {}_1F_2\left(1; \tfrac{4}{3}, \tfrac{5}{3}; \tfrac{1}{9}z^3 \right) + 3 z^4 \cdot {}_1F_2\left(2; \tfrac{7}{3}, \tfrac{8}{3}; \tfrac{1}{9}z^3 \right) \right ]\]

We also have \(\mathrm{Gi}'(x) = \mathrm{Bi}'(x) - \mathrm{Hi}'(x)\).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.AiryGi(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.AiryGi('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.AiryGi(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.AiryGi('0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; x = -3
>>> \mathrm{d}x = dec.scorergi(x); mx = mpm.scorergi(x); gx = gmp.scorergi(x)
>>> fx = fpm.scorergi(x); ax = apm.scorergi(x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -2.990547183713964238015818976238353474193E-1
mpm: -2.990547183713964238015818976238353474193e-1
gmp: -2.990547183713964238015818976238353474193E-01
fpm: -2.99054718371396E-01
apm: -2.990547183713964238015818976238353474192e-1 (-4.827e-37%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.scorergi(z); mz = mpm.scorergi(z); gz = gmp.scorergi(z)
>>> fz = fpm.scorergi(z); az = apm.scorergi(z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 8.4889448371208839089E-2             - 3.5862043476779870404E-2j
mpm: 8.4889448371208839089e-2             - 3.5862043476779870404e-2j
gmp: 8.4889448371208839089E-02            - 3.5862043476779870404E-02j
fpm: 8.48894483712088E-02                 - 3.58620434767799E-02j
apm: 8.4889448371208839123e-2 (6.78e-15%) - 3.5862043476779870473e-2 (-1.391e-14%)j

First derivative of the Scorer function \(\mathrm{Hi}(x)\), \(\mathrm{Hi}'(x)\)#

math53.scorer_hi_prime(x)#

Returns \(\mathrm{Hi}'(x)\), the first derivative of the Airy (Scorer) function \(\mathrm{Hi}(x)\).

See also: MathWorld [1129], Wikipedia [1510], Ehrhardt [309] (3.1.7.8), Mpmath [789].

The function is calculated as

\[\text{Hi}'(z) = \tfrac{2}{3} \text{Bi}'(z) + \frac{1}{40\pi} \left[ 40 z \cdot {}_1F_2\left(1; \tfrac{4}{3}, \tfrac{5}{3}; \tfrac{1}{9}z^3 \right) + 3 z^4 \cdot {}_1F_2\left(2; \tfrac{7}{3}, \tfrac{8}{3}; \tfrac{1}{9}z^3 \right) \right ]\]

It can also be represented as an integral:

\[\text{Hi}'(x) = \frac{1}{\pi} \int_0^\infty t \exp \left(xt - \tfrac{1}{3}t^3 \right) \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.AiryHi(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.AiryHi('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.AiryHi(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.AiryHi('0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; x = -3
>>> \mathrm{d}x = dec.scorerhi(x); mx = mpm.scorerhi(x); gx = gmp.scorerhi(x)
>>> fx = fpm.scorerhi(x); ax = apm.scorerhi(x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: 1.007650919964698805809370430513469907205E-1
mpm: 1.007650919964698805809370430513469907205e-1
gmp: 1.007650919964698805809370430513469907205E-01
fpm: 1.00765091996470E-01
apm: 1.007650919964698805809370430513469906431e-1 (1.334e-33%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.scorerhi(z); mz = mpm.scorerhi(z); gz = gmp.scorerhi(z)
>>> fz = fpm.scorerhi(z); az = apm.scorerhi(z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 9.5150834628338203598E-1              + 1.0872383260084919901E+0j
mpm: 9.5150834628338203598e-1              + 1.0872383260084919901e+0j
gmp: 9.5150834628338203598E-01             + 1.0872383260084919901E+00j
fpm: 9.51508346283382E-01                  + 1.08723832600849E+00j
apm: 9.5150834628338203594e-1 (6.049e-16%) + 1.0872383260084919901e+0 (4.588e-16%)j

Scorer functions

Contents

Scorer functions#

Scorer function \(\mathrm{Gi}(x)\)#

Scorer function \(\mathrm{Hi}(x)\)#

First derivative of the Scorer function \(\mathrm{Gi}(x)\), \(\mathrm{Gi}'(x)\)#

First derivative of the Scorer function \(\mathrm{Hi}(x)\), \(\mathrm{Hi}'(x)\)#