Psi
the Digamma and Polygamma functions
Calling Sequence
Parameters
Description
Examples
References
Psi(x)
Ψx
Psi(n,x)
Ψn,x
x
-
expression
n
Psi(x) is the digamma function,
Ψx=ⅆⅆxlnΓx=ⅆⅆxΓxΓx
Psi(n, x) is the nth polygamma function, which is the nth derivative of the digamma function when n is a nonnegative integer.
You can enter the command Psi using either the 1-D or 2-D calling sequence.
If n is an integer greater than one, Psi(n) + gamma is a rational number. (gamma is Euler's constant.) For small values of n, Psi(n) computes as a sum of gamma and a rational number. To perform this computation for larger values of n, use expand.
Ψn,x=ⅆnⅆxnΨx
Ψ0,x=Ψx
Psi(n, x) is extended to complex n, including negative integer indices, by the balanced polygamma formula of Espinosa and Moll
Ψw,z=ζ1w+1,z+γ+Ψ−wζ0w+1,zΓ−w
where Ζ is the Hurwitz zeta function.
Ψ2
1−γ
Ψ1,2
−1+π26
Ψ3.5+4.7I
1.717883835+1.001470255I
Ψ7,−2.2+3.3I
−0.02713341434+0.003825068416I
Ψ−2,1.543
−0.7957394716
Ψ1.342+I,3.5233
−0.6988919005−0.7978763419I
Ψ50
138812566871391350266313099044504245996706400−γ
Ψ51
Evaluating Psi(51) directly is faster than expanding and then evaluating.
expandΨ51
−γ+139432375772240549607593099044504245996706400
evalf
3.921989673
evalfΨ51
Unlike the negapolygamma of Gosper, the balanced polygamma at n=−1 differs from lnGAMMA by a constant
convertlnGAMMAx,Ψ−Ψ−1,x
ln2π2
Espinosa, O., and Moll, V. "A Generalized Polygamma Function." Integral Transforms and Special Functions, (April 2004): 101-115.
See Also
expand
GAMMA
initialfunctions
Zeta
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