[Rafael-SOWNet]

Part of #2075.

The node. With a second argument, gamma(a, z) is the upper incomplete gamma function Gamma(a, z) = int_z^oo t^(a - 1) e^(-t) dt, as Mathematica's Gamma[a, z] and mpmath's gammainc(a, z) are. With one argument, gamma is still Gamma(x). A second argument used to be a FunctionArgumentCountException.

• Branch. z^a takes its principal branch, so the cut is the negative real axis, which keeps the value from above.
• Printing. In LaTeX it is \Gamma\left(a, z\right), and in SymPy uppergamma.
• Derivative. In z it is -z^(a - 1) e^(-z). If the parameter varies, it declines.
• Exact forms.
  • A whole a >= 1 is (n - 1)! e^(-z) times a polynomial (DLMF 8.4.8).
  • a = 1/2 is sqrt(pi) erfc(sqrt(z)) (8.4.6).
  • Gamma(a, 0) is Gamma(a) for a positive a.

Evaluation is to the working precision for any complex a and z. Which method runs:

• Legendre's continued fraction (8.9.2), by the modified Lentz algorithm, where |z| > 2|a| + 10, either in the right half-plane or past (P + 5) ln 10. Near the negative real axis at a moderate |z| it stops short of the value: Gamma(1/2, -10) was wrong in its fourth digit. With |a| large against |z| it converges slowly: Gamma(100, 1/10 - 10 i) had the wrong imaginary part. Both cases were measured, and both go to the series.
• The series everywhere else: Gamma(a) less the lower function's series (8.7.1), with twice |z|/ln 10 guard digits, and more where the difference shows it cancelled more.
• A whole a <= 0, where Gamma(a) has a pole: E_1(z) by its series, and the recurrence (8.4.15).
• Real values. The value is exactly real where it is real: for a positive z, and for a whole a >= 1 at any real z.
• No floor at zero. Each part is accurate to its own size, so nothing is rounded onto zero by size. Gamma(1/2, 300) is 3e-132, and Gamma(100, -5 + i/10) has an imaginary part 1e-86 of its real part.

Against mpmath's gammainc:

Precision	Points	Agreement	Covers
100 digits	168	1e-90 (worst 3.8e-100)
300 digits	54	1e-290	orders within 1e-30 of a pole, both sides of the cut, `

Cost: about 5 ms per evaluation at a hundred digits. At three hundred digits, an order near a pole costs tens of seconds the first time, mostly in setting up Gamma(a) at each new precision.

Not here: the integration rules, and the corpus harness reading Rubi's file 8.6. Those are #2075's next steps.

IncompleteGammaTest covers:

• ten values against mpmath, including the poles, the continued fraction far below the floor at zero, the cut, and a complex parameter;
• the elementary forms;
• the derivative, and the case where it declines;
• parsing and printing.

Results:

• Unit suite: passes (16,856).
• MCP tests: pass (329). The MCP syntax resource names the new form.
• Parser: regenerated, with its classes internal.
• PublicApi.txt: gains the node, the kernel and MathS.IncompleteGamma.
• BREAKING-CHANGES: has the entry.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura