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December 31

Generalization of Trigonometric, Hyperbolic, Gamma, and Zeta Functions

It is becoming increasingly clear that the expression $\prod _{n\in \mathbf {M} }\left[1+{\frac {x^{s}}{(n+a)^{s}}}\right],$ with $-a\not \in \mathbf {M} ,$ and $\Re (s)>1,$ is a natural generalization of all four classes of functions mentioned in the title, since

for $n\in \mathbb {N} ,$ $n\in \mathbb {N} ,$ and $\color {blue}s\in \mathbb {N} _{>1},$ $\color {blue}s\in \mathbb {N} _{>1},$ it yields a (finite) product of Gamma functions, which, in certain cases (depending on the parity of s, and on whether a is either an integer or a half-integer), can be further simplified to a (finite) product of trigonometric and/or hyperbolic functions. Thus,
- for even values of s, and natural values of a, it yields a (finite) product of trigonometric and/or hyperbolic sine functions.
- for even values of s, and half-integer values of a, it yields a (finite) product of trigonometric and/or hyperbolic cosine functions.
- for odd values of s, the only fortuitous case is $s=3,$ , since the multiples of the real part of the cube root of unity are either integers of half-integers, which fact, in conjunction with the two reflection formulas for the Gamma function, ultimately yields a (finite) product of hyperbolic functions, assuming that both $x$ and $2a$ are integers.
for $n\in \mathbb {P} ,~a=0,$ and $\color {green}|x|=1,$ it yields a (finite) product of Zeta functions.

My actual question would be what exactly happens in the first case for $\color {blue}s\in \mathbb {R} \setminus \mathbb {N} ,$ and in the latter for $\color {green}|x|\neq 1.$ — 79.113.196.8 (talk) 02:06, 31 December 2016 (UTC)[reply]

History of topology terms

Who coined the terms "first-countable space", "second-countable space"/"completely separable space", and "separable space"? —Tea2min (talk) 14:58, 31 December 2016 (UTC)[reply]

The earliest use of "first countable" with respect to a topological space that I can easily corroborate is in 1950, here [1]. Problem is, it's introduced without definition or citation, indicating it was in common usage then. But most all the earlier hits I see on google scholar are spurious. Hope that helps, SemanticMantis (talk) 22:05, 2 January 2017 (UTC)[reply]

Thanks. As you wrote, the terms probably are much older as they apparently already were in common usage in 1950. After all, Felix Hausdorff coined the term "topological space" in 1914, and Kazimierz Kuratowski generalized Hausdorff's definition to the one we use today in 1922, and the Polish School of Mathematics then probably laid the foundations of point-set topology. And of course there's Urysohn's metrization theorem from 1925/1926. – Tea2min (talk) 07:37, 3 January 2017 (UTC)[reply]

I would assume the terms "first countable" and "second countable" are based on the two axioms of countability (Abzählbarkeitsaxiome) presented in Hausdorff's book Grundzüge der Mengenlehre (a copy is available on the Internet Archive). They are referred to as "erstes Abzählbarkeitsaxiom" and "zweites Abzählbarkeitsaxiom". de:Abzählbarkeitsaxiom agrees that the axioms are due to Hausdorff, but their reference is just a set-theoretic topology textbook. I do not know when the terminology changed from "space satisfying the first axiom of countability" to "first countable space" (if that is your question). —Kusma (t·c) 13:18, 3 January 2017 (UTC)[reply]

That's exactly what I was looking for. Thanks! – Tea2min (talk) 14:25, 3 January 2017 (UTC)[reply]