PolyGamma(m, z) at a complex z; Part of #340

[enumeratio]

Part of #340

PolyGamma(m, z) now evaluates at a complex z for every integer order. For m ≥ 1 it uses ψ⁽ᵐ⁾(z) = (−1)^(m+1) m! ζ(m+1, z) (DLMF 5.15.2) on the complex HurwitzZeta kernel; the digamma gets its own shift and asymptotic series. A non-positive integer z is still the pole at every order, and real z is unchanged.

At a high order and large negative Re(z), the Euler–Maclaurin sum cancels past what doubles can hold. There it stays unevaluated rather than answer, using a guard on the ratio of the largest term to the result. Checked against mpmath polygamma for m ≤ 8 over |z| ≤ 50 in every quadrant, near the poles and between them: no answered value off by more than 4e-13.

Also: the compiled JavaScript PolyGamma is now marked real-only, as Zeta and HurwitzZeta are. Before, a complex-typed argument compiled and ran the real kernel on it.

[arnog]

Thanks, merged, with a review pass on your branch as cd010c21. The main corrections:

• The shift loops ran once per unit of $−Re(z)$ with no cap (PolyGamma(0, -1e12 + i).N() hung), and the cancellation guard declined 11% of a left-half-plane grid. Both are replaced by the reflection formula for $Re(z) < 1/2$ (DLMF 5.15.6); the m-th derivative of $\cot(\pi z)$ comes from whichever of three series loses the fewest digits, and $\cot(\pi z)$ is built from $e^{2\pi iz}$ so it does not cancel for a large $|Im z|$. Every declined grid point now answers within $1e−14$ and the cost no longer depends on $Re(z)$.
• $m!$ and $ζ$ were combined outside the double range: PolyGamma(171, 1+i) gave ComplexInfinity (the value is $−1.6e283 + 9e248i$) and PolyGamma(100, 10000+i) a false $0$. Values are carried as a mantissa times a power of two; an unrepresentable result stays symbolic.
• The digamma tail overflowed for $|z|$ above $1e154$; Digamma(z) and Trigamma(z) now use the same complex kernel; an order above $10,000$ stays symbolic.

The one region that still declines (orders above 100 within $1e−5$ of a half-integer on the real axis) are tracked in ROADMAP.