Costante di Gauss

Costante di Gauss
Simbolo	G
Valore	0,8346268416740731862814297327990468... (sequenza A014549 dell'OEIS)
Origine del nome	Carl Friedrich Gauss
Frazione continua	[0; 1, 5, 21, 3, 4, 14, ...] (sequenza A053002 dell'OEIS)
Insieme	numeri trascendenti

In matematica, la costante di Gauss, indicata con la lettera $G$ , è definita come il reciproco della media aritmetico-geometrica tra 1 e ${\sqrt {2}}$ :

G={\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}=0,8346268\dots

La costante prende il nome dal matematico tedesco Carl Friedrich Gauss, il quale il 30 maggio 1799 scoprì che:

G={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}

e quindi:

G={\frac {1}{2\pi }}\beta \left({\frac {1}{4}},{\frac {1}{2}}\right)

dove $\beta$ indica la funzione beta di Eulero.

La costante di Gauss non deve essere confusa con la costante gravitazionale di Gauss.

Relazioni con altre costanti

La costante di Gauss può essere usata per esprimere la funzione gamma per 1/4:

\Gamma ({\tfrac {1}{4}})={\sqrt {2G{\sqrt {2\pi ^{3}}}}}

e, dato che $\pi$ e $\Gamma ({\tfrac {1}{4}})$ sono algebricamente indipendenti, con $\Gamma ({\tfrac {1}{4}})$ irrazionale, la costante di Gauss è necessariamente un numero trascendente.

Costanti delle lemniscate

La costante di Gauss può essere ustata per definire le costanti delle lemniscate, la prima delle quali è:

L_{1}=\pi G

e la seconda:

L_{2}={\frac {1}{2G}}

che compaiono nella ricerca della lunghezza d'arco di una lemniscata.

Altre formule

La seguente è una formula che esprime $G$ in relazione alla funzione theta di Jacobi:

G=\theta _{01}^{2}(e^{-\pi })

così come la seguente serie, rapidamente convergente:

G={\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}\left(\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}\right)^{2}.

La costante può anche essere espressa come prodotto infinito:

G=\prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).

La costante di Gauss ha come frazione continua [0, 1, 5, 21, 3, 4, 14, ...].

Collegamenti esterni

(EN) Eric W. Weisstein, Costante di Gauss, su MathWorld, Wolfram Research.

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