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November 9

Euler's gamma function

Latest comment: 3 days ago7 comments4 people in discussion

I was always puzzled by the definition $γ (z) = \int_{0}^{\infty} t^{z - 1} \exp (- t) d t$ instead of $\int_{0}^{\infty} t^{z} \exp (- t) d t$ that seems much more natural. Someone knows how it arose? Thanks ~2026-56087-11 (talk) 11:24, 9 November 2026 (UTC)Reply

Perhaps I have understood ... as a particular solution of f(x)=x*f(x) ... sorry for the disturb ~2026-56087-11 (talk) 12:53, 9 November 2026 (UTC)Reply

This paper cited in our article on the gamma function attributes this integral form (as well as the

γ

notation) to Adrien-Marie Legendre. I suspect that it was chosen so that

γ (z)

is defined for all

z > 0

rather than all

z > - 1

, but without having seen the actual original paper by Legendre, that is pure speculation on my part. GalacticShoe (talk) 14:24, 9 November 2026 (UTC)Reply

Do you not mean Gamma function with a capital gamma? JRSpriggs (talk) 15:11, 9 November 2026 (UTC)Reply

Many sources use "gamma function". ‑‑Lambiam 20:26, 9 November 2026 (UTC)Reply

I thank the author that has revised my formulae ~2026-56087-11 (talk) 20:48, 9 November 2026 (UTC)Reply

The article

Philip J. Davis (1959), "Leonhard Euler's Integral: A Historical Profile of the Gamma Function", The American Mathematical Monthly, 66:10, 849-869, doi:10.1080/00029890.1959.11989422,

after introducing Legendre's definition and noting that, as defined, the function satisfies the identity

Γ (n + 1) = n!

whenever

n

is a positive integer, contains a footnote that reads as follows:

Legendre's notation shifts the argument. Gauss introduced a notation

π (x)

free of this defect. Legendre's notation won out, but continues to plague many people. The notations

Γ

,

π

, and ! can all be found today.

It does not present a known or suspected rationale for Legendre's shift. ‑‑Lambiam 20:55, 9 November 2026 (UTC)Reply

@Lambian: Thanks, always very useful as in the propagandist/plugger case for the Murdoch mysteries

November 13