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PolyGamma

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

PolyGamma[z]
    gives the digamma function psi(z) (rewritten as PolyGamma[0, z]).
PolyGamma[n, z]
    gives the n-th derivative of the digamma function, psi^(n)(z).
Positive-integer arguments reduce to exact values: psi(m) to a rational
minus EulerGamma, and psi^(n)(m) for odd n to a rational plus a rational
multiple of Pi^(n+1); even orders stay symbolic. Non-positive integer
arguments give ComplexInfinity. Inexact real and complex arguments evaluate
numerically at machine or arbitrary (MPFR) precision. PolyGamma[-1, z] gives
LogGamma[z]. Listable.

Examples

No verified examples yet for this function.

Implementation notes

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

The digamma function at positive integers reduces to an exact rational minus Euler's constant:

In[1]:= PolyGamma[1]
Out[1]= -EulerGamma

In[2]:= PolyGamma[5]
Out[2]= 25/12 - EulerGamma

Higher derivatives psi^(n) at integer points give exact closed forms. The trigamma at 1 is the Basel constant, and the odd-order values are rational multiples of even powers of Pi:

In[1]:= PolyGamma[1, 1]
Out[1]= 1/6 Pi^2

In[2]:= PolyGamma[3, 1]
Out[2]= 1/15 Pi^4

The numeric paths cover arbitrary precision and complex arguments — here the tetragamma value psi^(2)(1+i) to 30 digits:

In[1]:= N[PolyGamma[0, 3/2], 40]
Out[1]= 0.036489973978576520559023667001244432806843

In[2]:= N[PolyGamma[2, 1 + I], 30]
Out[2]= 0.3685529315879351717366345429807 + 0.7666528503450662124026953776316*I

The order -1 is the special "integral of psi" case, the log-gamma function:

In[1]:= PolyGamma[-1, z]
Out[1]= LogGamma[z]

Notes

PolyGamma[z] is the digamma function ψ(z), stored internally as PolyGamma[0, z]; PolyGamma[n, z] is its n-th derivative ψ⁽ⁿ⁾(z). Positive integer arguments reduce exactly: ψ(m) to a rational minus EulerGamma, and ψ⁽ⁿ⁾(m) for odd n to a rational plus a rational multiple of Pi^(n+1) (even orders are left symbolic). Non-positive integer arguments give ComplexInfinity (the poles). Inexact real and complex arguments evaluate numerically at machine or MPFR precision, and PolyGamma[-1, z] returns LogGamma[z]. Listable.