[Rafael-SOWNet]

Part of #171. Follows #2068.

• Integrals. Each order integrates to the one before, int psi^(n)(u) = psi^(n - 1)(u), and the digamma function to ln Gamma(u) = ln((u - 1)!), for an argument linear in the variable. The integrator reads them from the table it uses for the other special functions. Where Gamma is negative, ln Gamma is i pi more, which is a constant on each interval between the poles.
• Exact values.
  • When. An odd order at a whole or half-whole argument is a rational times pi^(n + 1) plus a rational.
  • How. psi^(n)(1) = n! zeta(n + 1) and psi^(n)(1/2) = (2^(n + 1) - 1) n! zeta(n + 1) (DLMF 5.15.2, 5.15.3). zeta(2m) comes from the exact Bernoulli numbers, and the recurrence moves the argument to 1 or 1/2.
  • Examples. polygamma(1, 1) is pi^2/6, polygamma(1, 3) is pi^2/6 - 5/4, and polygamma(3, 5/2) is pi^4 - 2624/27.
  • Even orders. They want zeta at an odd number, or Euler's constant, so they stay as written.

PolygammaTest gains three groups:

• six exact values, each checked against its numerical value;
• three values that stay as written;
• four integrals, each differentiated back and compared at x = 7/3.

Results:

• Unit suite: passes (16,805).
• MCP tests: pass (329).
• Build: the library builds for netstandard2.0.
• BREAKING-CHANGES: The polygamma functions, and the derivative of the factorial #2068's entry gains the two rows, measured on 2.6.0, where both names were a product or a parse error.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura