teorema de Hölder

Hölder's theorem In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.[1] The theorem also generalizes to the {estilo de exibição q} -gamma function.

Statement of the theorem For every {estilo de exibição nin mathbb {N} _{0},} there is no non-zero polynomial {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},ldots ,Y_{n}]} de tal modo que {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gama (z),Gamma '(z),ldots ,{Gamma ^{(n)}}(z)certo)=0,} Onde {displaystyle Gama } is the gamma function. {displaystyle quad blacksquare } Por exemplo, definir {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},Y_{2}]} por {displaystyle P~{pilha {texto{df}}{=}}~X^{2}Y_{2}+XY_{1}+(X^{2}-nu ^{2})Y_{0}.} Then the equation {displaystyle Pleft(z;f(z),f'(z),f''(z)certo)=z^{2}f''(z)+zf'(z)+deixei(z^{2}-nu ^{2}certo)f(z)equivalente 0} is called an algebraic differential equation, que, nesse caso, has the solutions {displaystyle f=J_{não }} e {displaystyle f=Y_{não }} — the Bessel functions of the first and second kind respectively. Por isso, nós dizemos isso {estilo de exibição J_{não }} e {estilo de exibição Y_{não }} are differentially algebraic (also algebraically transcendental). Most of the familiar special functions of mathematical physics are differentially algebraic. All algebraic combinations of differentially algebraic functions are differentially algebraic. Além disso, all compositions of differentially algebraic functions are differentially algebraic. Hölder’s Theorem simply states that the gamma function, {displaystyle Gama } , is not differentially algebraic and is therefore transcendentally transcendental.[2] Proof Let {estilo de exibição nin mathbb {N} _{0},} and assume that a non-zero polynomial {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},ldots ,Y_{n}]} exists such that {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gama (z),Gamma '(z),ldots ,{Gamma ^{(n)}}(z)certo)=0.} As a non-zero polynomial in {estilo de exibição mathbb {C} [X]} can never give rise to the zero function on any non-empty open domain of {estilo de exibição mathbb {C} } (by the Fundamental Theorem of Algebra), we may suppose, sem perda de generalidade, este {estilo de exibição P} contains a monomial term having a non-zero power of one of the indeterminates {estilo de exibição Y_{0},Y_{1},ldots ,Y_{n}} .

Assume also that {estilo de exibição P} has the lowest possible overall degree with respect to the lexicographic ordering {estilo de exibição Y_{0}displaystyle deg left(-3X^{10}Y_{0}^{2}Y_{1}^{4}+iX^{2}Y_{2}certo)

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