Ueber die Transcendenz der Zahlen e und π

Abstract

Man nehme an, die Zahl e genüge der Gleichung n ^ten Grades

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgU % caRiaadggadaWgaaWcbaGaaGymaaqabaGccaWGLbGaey4kaSIaamyy % amaaBaaaleaacaaIYaaabeaakiaadwgadaahaaWcbeqaaiaaikdaaa % GccqGHRaWkcqWIVlctcqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqa % aOGaamyzamaaCaaaleqabaGaamOBaaaakiabg2da9iaaicdaaaa!48A9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$a + {a_1}e + {a_2}{e^2} + \cdots + {a_n}{e^n} = 0$$

, deren Coefficienten a, a ₁, ..., a _n ganze rationale Zahlen sind. Wird die linke Seite dieser Gleichung mit dem Integral

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaacq % GH9aqpaSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipakmaapedabaGa % amOEamaaCaaaleqabaGaeqyWdihaaOWaamWaaeaadaqadaqaaiaadQ % hacqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqaHvpGzcqGH % sislcaaIYaaacaGLOaGaayzkaaGaaiOlaiaac6cacaGGUaWaaeWaae % aacaWG6bGaeyOeI0IaamOBaaGaayjkaiaawMcaaaGaay5waiaaw2fa % amaaCaaaleqabaGaeqyWdiNaey4kaSIaaGymaaaakiaadwgadaahaa % WcbeqaaiabgkHiTiaadQhaaaGccaWGKbGaamOEaaWcbaGaaGimaaqa % aiabg6HiLcqdcqGHRiI8aaaa!5CC5!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\int_0^\infty = \int_0^\infty {{z^\rho }{{\left[ {\left( {z - 1} \right)\left( {\phi - 2} \right)...\left( {z - n} \right)} \right]}^{\rho + 1}}{e^{ - z}}dz} $$

multiplicirt, wo ρ eine ganze positive Zahl bedeutet, so entsteht der Ausdruck

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaape % dabaGaey4kaScaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccaWG % HbWaaSbaaSqaaiaaigdaaeqaaOGaamyzamaapedabaGaey4kaScale % aacaaIWaaabaGaeyOhIukaniabgUIiYdGccaWGHbWaaSbaaSqaaiaa % ikdaaeqaaOGaamyzamaaCaaaleqabaGaaGOmaaaakmaapedabaGaey % 4kaScaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccqWIVlctcqGH % RaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaamyzamaaCaaaleqaba % GaamOBaaaakmaapedabaaaleaacaaIWaaabaGaeyOhIukaniabgUIi % Ydaaaa!5857!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$a\int_0^\infty + {a_1}e\int_0^\infty + {a_2}{e^2}\int_0^\infty + \cdots + {a_n}{e^n}\int_0^\infty $$

und dieser Ausdruck zerlegt sich in die Summe der beiden folgenden Ausdrücke:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGqb % WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamyyamaapedabaGaey4k % aSIaamyyamaaBaaaleaacaaIXaaabeaakiaadwgadaWdXaqaaiabgU % caRiaadggadaWgaaWcbaGaaGOmaaqabaaabaGaaGimaaqaaiabg6Hi % LcqdcqGHRiI8aaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aOGaam % yzamaaCaaaleqabaGaaGOmaaaakmaapedabaGaey4kaSIaeS47IWKa % ey4kaScaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccaWGHbWaaS % baaSqaaiaad6gaaeqaaOGaamyzamaaCaaaleqabaGaamOBaaaakmaa % pedabaGaaiilaaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aaGcba % GaamiuamaaBaaaleaacaaIYaaabeaakiabg2da9iaadggadaWgaaWc % baGaaGymaaqabaGccaWGLbWaa8qmaeaacqGHRaWkcaWGHbWaaSbaaS % qaaiaaikdaaeqaaOGaamyzamaaCaaaleqabaGaaGOmaaaakmaapeda % baGaey4kaSIaeS47IWKaey4kaSIaamyyamaaBaaaleaacaWGUbaabe % aakiaadwgadaahaaWcbeqaaiaad6gaaaGcdaWdXaqaaiaac6caaSqa % aiaaicdaaeaacaWGUbaaniabgUIiYdaaleaacaaIWaaabaGaaGOmaa % qdcqGHRiI8aaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaaaa!7984!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{gathered} {P_1} = a\int_0^\infty { + {a_1}e\int_0^\infty { + {a_2}} } {e^2}\int_0^\infty { + \cdots + } {a_n}{e^n}\int_0^\infty, \hfill \\ {P_2} = {a_1}e\int_0^1 { + {a_2}{e^2}\int_0^2 { + \cdots + {a_n}{e^n}\int_0^n . } } \hfill \\ \end{gathered} $$

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Abdruck aus Nr. 2 der Göttinger Nachrichten v. J. 1893.