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    Polygamma Function

    Usage

    PolyGamma[z] gives the digamma function ψ (z). PolyGamma[n, z] gives the n ^th derivative of the digamma function ψ^(n) (z).

    Basic examples

    Evaluate the digamma function: In[1]:=PolyGamma[5] Out[1]=25/12 - EulerGamma Evaluate the pentagamma function: In[2]:=PolyGamma[3, 5] Out[2]=6 (-22369/20736 + π^4/90) Evaluate the second derivative of the gamma function: In[1]:=D[Gamma[z], {z, 2}] Out[1]=Gamma[z] (PolyGamma[0, z])^2 + Gamma[z] PolyGamma[1, z] Plot the digamma function over a subset of the reals: In[1]:=Plot[PolyGamma[x], {x, -3, 3}] Out[1]=

    Attributes

    Listable | NumericFunction | Protected | ReadProtected

    Relationships with other entities

    Gamma | LogGamma | EulerGamma | HarmonicNumber | QPolyGamma | HurwitzZeta

    Typical ranks of usage in programs

    489th most common (1 in 23700 symbols)

    1395th most common (1 in 924000 symbols)

    1106th most common (1 in 21500 symbols)

    History

    introduced in Version 1 (June 1988) last modified in Version 13.1 (June 2022)

    Timeline

    Timeline

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