Teorema di Bohr-Mollerup

Teorema di Bohr-Mollerup In analisi matematica, il teorema di Bohr-Mollerup è un teorema dimostrato dai matematici danesi Harald Bohr e Johannes Mollerup. Il teorema caratterizza la funzione gamma, defined for x > 0 di {stile di visualizzazione Gamma (X)=int _{0}^{infty }t^{x-1}e^{-t},dt} as the only positive function f , with domain on the interval x > 0, che ha contemporaneamente le tre proprietà seguenti: f (1) = 1, and f (X + 1) = x f (X) for x > 0 and f is logarithmically convex.

Una trattazione di questo teorema è nel libro di Artin The Gamma Function, che è stato ristampato dall'AMS in una raccolta di scritti di Artin.

Il teorema è stato pubblicato per la prima volta in un libro di testo sull'analisi complessa, poiché Bohr e Mollerup pensavano che fosse già stato dimostrato.

Contenuti 1 Dichiarazione 2 Prova 3 Guarda anche 4 Riferimenti Dichiarazione Teorema di Bohr-Mollerup. C(X) is the only function that satisfies f (X + 1) = x f (X) con ceppo( f (X)) convex and also with f (1) = 1. Proof Let Γ(X) essere una funzione con le proprietà assunte stabilite sopra: C(X + 1) = xC(X) e log(C(X)) è convesso, e Γ(1) = 1. Da Γ(X + 1) = xC(X) possiamo stabilire {stile di visualizzazione Gamma (x+n)=(x+n-1)(x+n-2)(x+n-3)cdot (x+1)xGamma (X)} Lo scopo della disposizione che Γ(1) = 1 forza la Γ(X + 1) = xC(X) proprietà di duplicare i fattoriali degli interi quindi possiamo concludere ora che Γ(n) = (n - 1)! se n ∈ N e se Γ(X) esiste affatto. A causa della nostra relazione per Γ(X + n), se possiamo comprendere appieno Γ(X) per 0 < x ≤ 1 then we understand Γ(x) for all values of x. For x1, x2, the slope S(x1, x2) of the line segment connecting the points (x1, log(Γ (x1))) and (x2, log(Γ (x2))) is monotonically increasing in each argument with x1 < x2 since we have stipulated that log(Γ(x)) is convex. Thus, we know that {displaystyle S(n-1,n)leq S(n,n+x)leq S(n,n+1)quad {text{for all }}xin (0,1].} After simplifying using the various properties of the logarithm, and then exponentiating (which preserves the inequalities since the exponential function is monotonically increasing) we obtain {displaystyle (n-1)^{x}(n-1)!leq Gamma (n+x)leq n^{x}(n-1)!.} From previous work this expands to {displaystyle (n-1)^{x}(n-1)!leq (x+n-1)(x+n-2)cdots (x+1)xGamma (x)leq n^{x}(n-1)!,} and so {displaystyle {frac {(n-1)^{x}(n-1)!}{(x+n-1)(x+n-2)cdots (x+1)x}}leq Gamma (x)leq {frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}left({frac {n+x}{n}}right).} The last line is a strong statement. In particular, it is true for all values of n. That is Γ(x) is not greater than the right hand side for any choice of n and likewise, Γ(x) is not less than the left hand side for any other choice of n. Each single inequality stands alone and may be interpreted as an independent statement. Because of this fact, we are free to choose different values of n for the RHS and the LHS. In particular, if we keep n for the RHS and choose n + 1 for the LHS we get: {displaystyle {begin{aligned}{frac {((n+1)-1)^{x}((n+1)-1)!}{(x+(n+1)-1)(x+(n+1)-2)cdots (x+1)x}}&leq Gamma (x)leq {frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}left({frac {n+x}{n}}right)\{frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}&leq Gamma (x)leq {frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}left({frac {n+x}{n}}right)end{aligned}}} It is evident from this last line that a function is being sandwiched between two expressions, a common analysis technique to prove various things such as the existence of a limit, or convergence. Let n → ∞: {displaystyle lim _{nto infty }{frac {n+x}{n}}=1} so the left side of the last inequality is driven to equal the right side in the limit and {displaystyle {frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}} is sandwiched in between. This can only mean that {displaystyle lim _{nto infty }{frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}=Gamma (x).} In the context of this proof this means that {displaystyle lim _{nto infty }{frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}} has the three specified properties belonging to Γ(x). Also, the proof provides a specific expression for Γ(x). And the final critical part of the proof is to remember that the limit of a sequence is unique. This means that for any choice of 0 < x ≤ 1 only one possible number Γ(x) can exist. Therefore, there is no other function with all the properties assigned to Γ(x). The remaining loose end is the question of proving that Γ(x) makes sense for all x where {displaystyle lim _{nto infty }{frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}} exists. The problem is that our first double inequality {displaystyle S(n-1,n)leq S(n+x,n)leq S(n+1,n)} was constructed with the constraint 0 < x ≤ 1. If, say, x > 1 allora il fatto che S sia monotonicamente crescente renderebbe S(n + 1, n) < S(n + x, n), contradicting the inequality upon which the entire proof is constructed. However, {displaystyle {begin{aligned}Gamma (x+1)&=lim _{nto infty }xcdot left({frac {n^{x}n!}{(x+n)(x+n-1)cdots (x+1)x}}right){frac {n}{n+x+1}}\Gamma (x)&=left({frac {1}{x}}right)Gamma (x+1)end{aligned}}} which demonstrates how to bootstrap Γ(x) to all values of x where the limit is defined. See also Wielandt theorem References "Bohr–Mollerup theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Bohr–Mollerup Theorem". MathWorld. "Proof of Bohr–Mollerup theorem". PlanetMath. "Alternative proof of Bohr–Mollerup theorem". PlanetMath. Artin, Emil (1964). The Gamma Function. Holt, Rinehart, Winston. Rosen, Michael (2006). Exposition by Emil Artin: A Selection. American Mathematical Society. Mollerup, J., Bohr, H. (1922). Lærebog i Kompleks Analyse vol. III, Copenhagen. (Textbook in Complex Analysis) Categories: Gamma and related functionsTheorems in complex analysis