PolyGamma

The polygamma function , the -th derivative of the digamma.

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PolyGamma(m, z)the polygamma at real .

SageMathpsi(m, z)SymPypolygamma(m, z)mpmathpsi(m, z)

PolyGamma(m, z)extends the same call to complex , for integer order .

SageMathpsi(m, z)SymPypolygamma(m, z)mpmathpsi(m, z)

Domain: Special functions

Details

is the digamma — see Digamma — and the trigamma.
For it is a Hurwitz zeta in disguise: . See HurwitzZeta.
So at every order is a zeta value: , giving and .
Half-integer arguments pick up the alternating zeta: .
Recurrence from : , so .
Poles at the nonpositive integers, of order — the poles of Gamma, differentiated.
Real is compute-engine's own; `@enumeratio/analytic` adds complex through the Euler–Maclaurin Hurwitz kernel, and the WGSL lowering for GPU plotting.

Examples