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The incomplete gamma function #2075
Description
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:
- The harness. Read 8.6, mapping
GAMMA(a, x)togamma(a, x),Psi(n, x)topolygamma(n, x)(The polygamma functions, and the derivative of the factorial #2068), andlnGAMMA. Then the count is visible before anything lands. - The node.
gamma(a, x), upper, as Mathematica'sGamma[a, x]and mpmath'sgammainc(a, x)are. It needs:- its branch convention written down;
- evaluation to the working precision for complex
aandx: the series for small|x|, the continued fraction for large|x|, andGamma(0, x) = E_1(x)and the other non-positive wholeahandled on their own; - the derivative in
x,-x^(a - 1) e^(-x); - its LaTeX and SymPy forms (
uppergamma).
- Integration rules, each naming its identity:
int x^(a - 1) e^(-x) dx = -Gamma(a, x)for a non-wholea, andint 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.
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Activity
The upper incomplete gamma function (#2077)
Integrate the upper incomplete gamma function beside a power or a pol…
Evaluate the incomplete gamma function's series faster (#2079)
Take the power's branch in the incomplete gamma function (#2080)
Integrate the incomplete gamma function of a whole order by parts aga…
Rafael-SOWNet commentedon Oct 23, 2026
Where this stands on master
e62a56d6:gamma(a, z), evaluated anywhere in the complex plane. Its branch is the power's (Take the power's branch in the incomplete gamma function #2080): principal, and the real root on the negative axis for an exact order of odd denominator below 10, so the derivative is the derivative of the value everywhere. Evaluating it is about ten times cheaper since Evaluate the incomplete gamma function's series faster #2079.xbut1/x.n <= 0over a power of another linear, by parts.Rubi's 8.6, read with a harness copy that maps
GAMMA(a, x): 156 of 177, 0 wrong. The node alone solved 63.Left:
x^m e^(b x^n)written in the function. That rule is written and measured, and it is not opened yet. On theGAMMAharness it takes family 2 from 690 to 761 of 793, and families 0, 3 and 8 from 3,760 to 3,816 of 3,900, with 0 wrong. A definite integral through 0 of one of its answers,integral(x^2*e^(x^4), x, 0, 1), runs the limit engine past 100 s, and it waits on that and on A definite integral through a special function's limit is NaN where it has a value #2081.Gamma(p, d (a + b log(c x^n))), which Rubi answers in closed form.GAMMA, after the rule above, so that the corpus figures stay one comparison.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Rafael-SOWNet commentedon Oct 23, 2026
Item 1 is done. The corpus harness now reads Maxima's
GAMMA(a, x)asgamma(a, x)andPsi(n, x)aspolygamma(n, x), andlnGAMMAstays excluded by its head. On mastere62a56d6, Rubi's suite goes from 48,305/48,305 to 48,502/49,052:GAMMAorPsiin the integrand or the answer0 wrong. Most of the 545 new unsolved rows are in 2.3, the 3.x logarithm files and 4.1.10/4.1.12, as
x^m e^(b x^n)shapes after a substitution. That is what the rule in my comment above answers.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Integrate a power beside an exponential of a power as an incomplete g…
Integrate a power beside an exponential of a power as an incomplete g…
Read a linear exponent written as a negation in the gamma rule (#2084)
Integrate a power beside a sine or a cosine as incomplete gamma funct…
7 remaining items
Integrate a power of x beside a power of a logarithm of a binomial in…
Integrate a power of a logarithm beside a linear in another as two po…
Integrate a power of a logarithm beside a linear in another as two po…
Integrate a polynomial beside a sine of a power of a shifted linear i…
Read the gamma rule's integrand when it is written otherwise (#2096)
Write an exponent given expanded as the power of a shifted linear it …
Take a power of a multiple of the shifted linear below the bar in its…
Take the real root in the precise gamma function where the power take…
Undo an inverse sine or cosine to a power that is not whole (#2100)
Integrate a power beside a Laurent polynomial in an exponential term …
Read a power of a power of a linear beside an exponential of a linear (…
Read a sum as the derivative of a lower power of a linear beside a fu…
Rafael-SOWNet commentedon Oct 24, 2026
With #2106 in, everything this issue listed is done, in #2077–#2106:
GAMMAandPsi;gamma(a, z)is the upper incomplete gamma function, with its convention written down and its evaluation;On the full fair Rubi corpus, measured on #2106 rebased onto master (
191510f3):What is left in 8.6
(c + d x)^k polygamma(n, a + b x)at a symbolic ordern. Their answers usepolygamma(n - 1, …)down topolygamma(n - 4, …).polygammahere is defined forn >= 0, so at smallnthose terms are the repeated integrals ofln Gamma, which the body calls a further step.-polygamma(n, a + b x)/x, but cannot be checked at the harness's non-integer pinned order.I'm closing this; the negative orders would be an issue of their own.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura