Polygamma, and related functions

Pentagamma function \(\psi'''(x)\)

math53.pentagamma(x)

Returns the pentagamma function \(\psi'''(x), \quad x \ne 0, -1, -2, \ldots\) The function returns the Hurwitz zeta value \(6\zeta(4,x)\) if \(x\) is positive; for \(x<0\) it returns \(\psi'''(x) = -2\pi^4\left(1+ 4\cot^2(\pi x)+3\cot^4(\pi x)\right) - 6\zeta(4,1-x)\).

See also Wikipedia [1373], MathWorld [966], NIST [25], BoostMath [115], Ehrhardt [309] (3.5.6.5).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.Pentagamma(7)
xreal('5.2359877559829887307E-1')
>>> xreal.Pentagamma('4.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Pentagamma(7)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Pentagamma('4.51')
Gpr('5.3518479027559984754E-1')

Tetragamma function, \(\psi''(x)\)

math53.tetragamma(x)

Returns the tetragamma function \(\psi''(x), \quad x \ne 0, -1, -2, \ldots\) The function returns the Hurwitz zeta value \(-2\zeta(3,x)\) if \(x\) is positive; for \(x<0\) it returns \(\psi''(x) = -2\pi^3 \cot(\pi x)\left(1+\cot^2(\pi x)\right) - 2\zeta(3,1-x)\).

See also Wikipedia [1373], MathWorld [966], NIST [25], BoostMath [115], Ehrhardt [309] (3.5.6.4).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.Tetragamma(7)
xreal('5.2359877559829887307E-1')
>>> xreal.Tetragamma('4.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Tetragamma(7)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Tetragamma('4.51')
Gpr('5.3518479027559984754E-1')

Auxiliary function \(\psi^{*}(x)\)

math53.psistar(x)

Returns \(\psi^{*}(x) = \psi(x) - \log(x), x > 0\). The function is useful when computing differences of \(\psi\) functions, because it can avoid cancellation for larger \(x\) values. See also Ehrhardt [309] (3.5.6.2).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.PsiStar(7)
xreal('5.2359877559829887307E-1')
>>> xreal.PsiStar('4.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.PsiStar(7)
Gpr('5.2359877559829887307E-1')
>>> Gpr.PsiStar('4.51')
Gpr('5.3518479027559984754E-1')

Inverse digamma function \(\psi^{-1}(y)\)

math53.psi_inv(y)

Returns \(\psi^{-1}(y)\), the functional inverse of of the digamma function, i.e. it returns \(x\) with \(\psi(x) = y\).

See also Wikipedia [1381], MathWorld [990], NIST [25], BoostMath [123], Ehrhardt [309] (3.5.6.7).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.PsiInv(7.1)
xreal('5.2359877559829887307E-1')
>>> xreal.PsiInv('4.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.PsiInv(7.1)
Gpr('5.2359877559829887307E-1')
>>> Gpr.PsiInv('4.51')
Gpr('5.3518479027559984754E-1')

Bateman function \(G(x)\)

math53.bateman_g(x)

Returns the Bateman function \(G(x) = \psi((x+1)/2) - \psi(x/2); x \ne 0,-1,-2,\ldots\) See also Ehrhardt [309] (3.5.6.8).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BatemanG(7.1)
xreal('5.2359877559829887307E-1')
>>> xreal.BatemanG('4.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BatemanG(7.1)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BatemanG('4.51')
Gpr('5.3518479027559984754E-1')