Hölder's theorem

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 {Anzeigestil q} -gamma function.

Statement of the theorem For every {Anzeigestil nin mathbb {N} _{0},} there is no non-zero polynomial {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},Punkte ,Y_{n}]} so dass {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gamma (z),Gamma '(z),Punkte ,{Gamma ^{(n)}}(z)Rechts)=0,} wo {Anzeigestil Gamma } is the gamma function. {displaystyle quad blacksquare } Zum Beispiel, definieren {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},Y_{2}]} durch {displaystyle P~{Stapel {Text{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)Rechts)=z^{2}f''(z)+zf'(z)+links(z^{2}-nu ^{2}Rechts)f(z)Äquiv 0} is called an algebraic differential equation, die, in diesem Fall, has the solutions {displaystyle f=J_{nicht }} und {displaystyle f=Y_{nicht }} — the Bessel functions of the first and second kind respectively. Somit, das sagen wir {Anzeigestil J_{nicht }} und {Anzeigestil Y_{nicht }} 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. Außerdem, all compositions of differentially algebraic functions are differentially algebraic. Hölder’s Theorem simply states that the gamma function, {Anzeigestil Gamma } , is not differentially algebraic and is therefore transcendentally transcendental.[2] Proof Let {Anzeigestil nin mathbb {N} _{0},} and assume that a non-zero polynomial {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},Punkte ,Y_{n}]} exists such that {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gamma (z),Gamma '(z),Punkte ,{Gamma ^{(n)}}(z)Rechts)=0.} As a non-zero polynomial in {Anzeigestil mathbb {C} [X]} can never give rise to the zero function on any non-empty open domain of {Anzeigestil mathbb {C} } (by the Fundamental Theorem of Algebra), we may suppose, ohne Verlust der Allgemeinheit, das {Anzeigestil P} contains a monomial term having a non-zero power of one of the indeterminates {Anzeigestil Y_{0},Y_{1},Punkte ,Y_{n}} .

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

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