Working definition
DLMF 8.2.1 defines the lower incomplete gamma as the integral of t to the a minus one times e to the minus t from zero to z, for real part of a greater than zero. 8.2.2 defines the upper form as the same integrand taken from z to infinity. 8.2.3 states the identity that binds them: the lower plus the upper equals the complete gamma at a, for a not zero or a negative integer. The normalized pair at 8.2.4 divides each by the complete gamma, and 8.2.5 states that they sum to one. The two halves are not symmetric in the parameter: the chapter records that the upper form is entire in a when z is non-zero, while the lower form is meromorphic with simple poles at the non-positive integers.
Notation
lower case gamma of a, z for the lower form and capital Gamma of a, z for the upperP of a, z and Q of a, z for the normalized paira for the parameter and z for the split pointAssumptions
- The lower definition at 8.2.1 is stated for real part of a greater than zero.
- The identity at 8.2.3 excludes a equal to zero and the negative integers, which are exactly the poles of the complete gamma.
- The two halves differ in the parameter: the upper is entire in a for non-zero z while the lower has simple poles at the non-positive integers.
Invariants
- Lower and upper integral definitions, DLMF 8.2.1 and 8.2.2
- The two halves sum to the complete gamma, DLMF 8.2.3
- The normalized pair sums to one, DLMF 8.2.5
Reproducible procedure
- Choose the half whose value is the smaller of the two, and take the other by subtraction only when precision allows.
- Use the normalized pair from 8.2.4 when comparing across parameters, since the complete gamma has been divided out.
- Check the parameter against zero and the negative integers before relying on 8.2.3, which excludes them.
Error and boundary controls
- The identity at 8.2.5 says the normalized pair sums to one exactly, which is why computing the small member by subtracting the large one from one destroys it. The same device appears at 7.2.2 for the error function, and for the same reason.
- Near the poles of the lower form the value is unbounded, so an algorithm tuned for one half is not automatically usable for the other.
- The chapter gives definitions rather than algorithms and makes no accuracy claim for any evaluation method.
What this does not establish
The chapter defines the two halves and their exact relation. It does not establish the accuracy of any algorithm, and the exactness of the identity is a mathematical statement rather than a numerical guarantee.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 8: Incomplete Gamma and Related Functions · National Institute of Standards and Technology
Establishes: Integral definitions for the lower and upper incomplete gamma functions, the identity that splits the complete gamma between them, the normalized pair and their sum, and the differing analytic structure of the two halves in the parameter.
Boundary: The chapter defines the functions and their relation. It does not establish the accuracy of any algorithm, and the identity that the normalized pair sums to one is exact rather than a numerical guarantee.