Teorema de Bohr–Mollerup

Teorema de Bohr–Mollerup Em análise matemática, o teorema de Bohr-Mollerup é um teorema provado pelos matemáticos dinamarqueses Harald Bohr e Johannes Mollerup. O teorema caracteriza a função gama, defined for x > 0 por {displaystyle Gama (x)=int_{0}^{infty }t^{x-1}e^{-t},dt} as the only positive function f , with domain on the interval x > 0, que tem simultaneamente as três propriedades seguintes: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex.

Um tratamento deste teorema está no livro de Artin The Gamma Function, que foi reimpresso pela AMS em uma coleção de escritos de Artin.

O teorema foi publicado pela primeira vez em um livro sobre análise complexa, como Bohr e Mollerup pensavam que já estava provado.

Conteúdo 1 Declaração 2 Prova 3 Veja também 4 Declaração de Referências Teorema de Bohr–Mollerup. C(x) is the only function that satisfies f (x + 1) = x f (x) com log( f (x)) convex and also with f (1) = 1. Proof Let Γ(x) ser uma função com as propriedades assumidas estabelecidas acima: C(x + 1) = xC(x) e log(C(x)) é convexo, e Γ(1) = 1. De Γ(x + 1) = xC(x) podemos estabelecer {displaystyle Gama (x+n)=(x+n-1)(x+n-2)(x+n-3)cdots (x+1)xGamma (x)} O objetivo da estipulação de que Γ(1) = 1 força o Γ(x + 1) = xC(x) propriedade para duplicar os fatoriais dos inteiros, então podemos concluir agora que Γ(n) = (n - 1)! se n ∈ N e se Γ(x) existe em tudo. Por causa de nossa relação para Γ(x + n), se pudermos entender completamente Γ(x) por 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 então o fato de que S está aumentando monotonicamente faria 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