The incomplete gamma function

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调研方向

Start by reading the referenced issue #1501 and Rubi file 8.6 to understand the proposed order and test harness; the payload names no repository paths or test files. The work includes complex upper incomplete gamma evaluation, branch conventions, derivatives, LaTeX and SymPy forms, and integration rules, with further Psi and lnGamma work noted. Done means implementing and validating these pieces against the stated identities and working precision.

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描述

The upper incomplete gamma function, Gamma(a, x) = int_x^oo t^(a - 1) e^(-t) dt, is what the last open item of #180 needs: int t^(-1/2) e^(-t^(-2)) dt is 2 sqrt(t) e^(-1/t^2) - 2 Gamma(3/4, 1/t^2). It is also most of Rubi's file 8.6, which #1469 counts at 233 problems. The file uses the two-argument GAMMA 628 times, against 34 for the one-argument one. @Happypig375's note there was to implement family 8's functions before v3.

What it takes, in #1501's order:

  1. The harness. Read 8.6, mapping GAMMA(a, x) to gamma(a, x), Psi(n, x) to polygamma(n, x) (#2068), and lnGAMMA. Then the count is visible before anything lands.
  2. The node. gamma(a, x), upper, as Mathematica's Gamma[a, x] and mpmath's gammainc(a, x) are. It needs:
    • its branch convention written down;
    • evaluation to the working precision for complex a and x: the series for small |x|, the continued fraction for large |x|, and Gamma(0, x) = E_1(x) and the other non-positive whole a handled on their own;
    • the derivative in x, -x^(a - 1) e^(-x);
    • its LaTeX and SymPy forms (uppergamma).
  3. Integration rules, each naming its identity: int x^(a - 1) e^(-x) dx = -Gamma(a, x) for a non-whole a, and int Gamma(a, x) dx = x Gamma(a, x) - Gamma(a + 1, x).

8.6's lnGAMMA problems need Psi(n, x) at negative n, the repeated integrals of ln Gamma, which are a further step.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

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