Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Log-Gamma branch on the right half-plane
Definition
On , define
Here is the principal logarithm from Complex logarithms, the principal logarithm, and principal and multivalued complex powers, and denotes Euler's constant from The Euler-Mascheroni constant. Both and lie in the principal-logarithm domain. The local uniform convergence of this series and the identity are established by Weierstrass log-Gamma series converges to a Gamma branch ↗.
Depends on
Used by
- Finite certified evaluation of Gamma and a continued log-Gamma branch Algorithm
- Log-Gamma branch on the slit plane Definition
- Binet integral identity for the right-half-plane log-Gamma branch Lemma
- Explicit sectorial remainder bound for the right-half-plane log-Gamma branch Lemma
- Explicit tail bound for the log-Gamma product series Lemma
- Shifted log-Gamma branch is independent of the shift Lemma
- Weierstrass log-Gamma series converges to a Gamma branch Lemma
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- DLMF, §5.8(ii), Weierstrass product (standard reference, not scraped)
- N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §§2 and 3 (standard reference, not scraped)