8.7. Die Logarithmusfunktion


Dieser Abschnitt beschäftigt sich allein mit der Kehrwertfunktion. Sie ist die einzige Potenzfunktion, über deren Stammfunktionen wir bislang keine Kenntnis haben. Allerdings ist die Kehrwertfunktion stetig, auf Intervallen muss sie daher nach [8.1.5] integrierbar sein!

Eine ihrer Stammfunktionen auf >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@394B@ zeichnen wir durch einen eigenen Namen aus. Wir verwenden dabei den Hauptsatz [8.2.13]:

Definition:  Die Funktion ln: >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGG6aGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaOGaeyOKH4QaeSyhHekaaa@3F54@ , gegeben durch

ln(x) 1 x 1 X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaamiEaiaacMcacqGH9aqpdaWdXbqaamaalaaabaGaaGymaaqaaiaadIfaaaaaleaacaaIXaaabaGaamiEaaqdcqGHRiI8aaaa@40F5@
[8.7.1]

heißt der (natürliche) Logarithmus oder auch die (natürliche) Logarithmusfunktion.

Beachte:

  • Der Funktionsname ln MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gaaaa@37D0@ ist die Abkürzung des lateinischen Namens logarithmus naturalis.

  • Wie bei den trigonometrischen Funktionen auch ist es üblich, lnx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4baaaa@38CD@ statt ln(x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaamiEaiaacMcaaaa@3A26@ zu schreiben.

  • Die untere Integrationsgrenze 1 in [8.7.1] legt den Wert ln1=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaaIXaGaeyypa0JaaGimaaaa@3A4B@ fest.

  • Als Stammfunktion zu 1 X | >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiwaaaacaGG8bGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@3BF3@ ist ln bereits differenzierbar mit ln= 1 X | >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGNaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamiwaaaacaGG8bGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@3F88@ . Damit ist ln sofort beliebig oft differenzierbar,  ln C ( >0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacqGHiiIZcaWGdbWaaWbaaSqabeaacqGHEisPaaGccaGGOaGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaOGaaiykaaaa@4086@ , und insbesondere auch stetig ([7.5.2]).

  • Als stetige Funktion ist ln MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gaaaa@37D0@ auf dem Intervall >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@394B@ integrierbar. Mit Hilfe der partiellen Integration [8.3.1] (und einem kleinen Trick) können wir eine ihrer Stammfunktionen über den Hauptsatz errechnen. Für x >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaeyOpa4JaaGimaaaaaaa@3BCC@ ist nämlich:

    1 x ln = 1 x 1ln =Xln | 1 x 1 x X 1 X =xlnx 1 x 1 =xlnxx+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@70B2@

    Also ist (auch) XlnX=X(ln1) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabgwSixlGacYgacaGGUbGaeyOeI0Iaamiwaiabg2da9iaadIfacqGHflY1caGGOaGaciiBaiaac6gacqGHsislcaaIXaGaaiykaaaa@45D3@ eine Stammfunktion zu ln MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gaaaa@37D0@ .

  • Mit der Kettenregel [7.7.8] und der Ableitung von |X| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIfacaGG8baaaa@38C9@ (siehe [7.4.3]) berechnen wir

    (ln|X| ) = 1 |X| |X| X = 1 X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiGacYgacaGGUbGaeSigI8MaaiiFaiaadIfacaGG8bGabiykayaafaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaiiFaiaadIfacaGG8baaaiabgwSixpaalaaabaGaaiiFaiaadIfacaGG8baabaGaamiwaaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGybaaaaaa@4ABC@ ,

    so dass wir ln|X| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacqWIyiYBcaGG8bGaamiwaiaacYhaaaa@3BE7@ als eine Stammfunktion zur vollständigen Kehrwertfunktion 1 X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiwaaaaaaa@3794@ bestätigen können.


     

Weitere Eigenschaften der Logarithmusfunktion lassen sich direkt aus der Integraldarstellung gewinnen.

Bemerkung:  

1.    ln1=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaaIXaGaeyypa0JaaGimaaaa@3A4B@

[8.7.2]

2.    ln MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gaaaa@37D0@ ist streng monoton wachsend

[8.7.3]

3.    lnx<0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4bGaeyipaWJaaGimaaaa@3A8B@   für alle 0<x<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaadIhacqGH8aapcaaIXaaaaa@3A66@

[8.7.4]

4.    lnx>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4bGaeyOpa4JaaGimaaaa@3A8F@   für alle x>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaigdaaaa@38AC@

[8.7.5]

Beweis:  

1.     ln1= 1 1 1 X =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaaIXaGaeyypa0Zaa8qCaeaadaWcaaqaaiaaigdaaeaacaWGybaaaaWcbaGaaGymaaqaaiaaigdaa0Gaey4kIipakiabg2da9iaaicdaaaa@40E2@

2.    Für alle x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@ ist ln(x)= 1 x >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGNaGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamiEaaaacqGH+aGpcaaIWaaaaa@3F61@ . Die Behauptung folgt also aus der strengen Variante des Monotoniesatzes [7.10.5].

3. und 4. ergeben sich direkt aus 1. und 2.

Bereits aus diesen wenigen Angaben läßt sich der charakteristische Graphenverlauf des Logarithmus ableiten. Die unterstellte Unbeschränktheit wird allerdings erst am Ende dieses Abschnitts durch den Nachweis der Umkehrbarkeit erhärtet.

Mit Hilfe der Substitutionsregel gelingt es, die typischen Rechenregeln für den Logarithmus, die Logarithmengesetze, zu beweisen.

Bemerkung (Rechenregeln für ln):  Für alle a,b >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@3D4C@ , n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablssiIcaa@39DB@ gilt:

1.    ln a n =nlna MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWGHbWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaamOBaiabgwSixlGacYgacaGGUbGaamyyaaaa@40ED@

[8.7.6]

2.    ln 1 b =lnb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaWcaaqaaiaaigdaaeaacaWGIbaaaiabg2da9iabgkHiTiGacYgacaGGUbGaamOyaaaa@3E40@

[8.7.7]

3.    lnab=lna+lnb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWGHbGaeyyXICTaamOyaiabg2da9iGacYgacaGGUbGaamyyaiabgUcaRiGacYgacaGGUbGaamOyaaaa@4364@

[8.7.8]

4.    ln a b =lnalnb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaWcaaqaaiaadggaaeaacaWGIbaaaiabg2da9iGacYgacaGGUbGaamyyaiabgkHiTiGacYgacaGGUbGaamOyaaaa@4135@

[8.7.9]

5.    ln a n = 1 n lna MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaGcbaqaaiaadggaaSqaaiaad6gaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbaaaiabgwSixlGacYgacaGGUbGaamyyaaaa@41A6@    für n>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg6da+iaaicdaaaa@38A1@

[8.7.10]

Beweis:  

1.    Wir setzen die Substitutionsregel [8.3.5] ein:

ln a n = 1 a n 1 X = X n (1) X n (a) 1 X = 1 a 1 X X n n X n1 =n 1 a X n1 X n =n 1 a 1 X =nlna MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8181@

2.    ergibt sich mit n=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iabgkHiTiaaigdaaaa@398D@ direkt aus 1.

3.    Wir arbeiten noch einmal mit der Substitutionsregel:

lnab= 1 ab 1 X = bX( 1 b ) bX(a) 1 X = 1 b a 1 X bXb= 1 b a 1 X = 1 a 1 X 1 1 b 1 X =lnaln 1 b = 2. lna+lnb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@897B@

4.    führen wir auf 2. und 3. zurück:  ln a b =lna 1 b =lna+ln 1 b =lnalnb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaWcaaqaaiaadggaaeaacaWGIbaaaiabg2da9iGacYgacaGGUbGaamyyaiabgwSixpaalaaabaGaaGymaaqaaiaadkgaaaGaeyypa0JaciiBaiaac6gacaWGHbGaey4kaSIaciiBaiaac6gadaWcaaqaaiaaigdaaeaacaWGIbaaaiabg2da9iGacYgacaGGUbGaamyyaiabgkHiTiGacYgacaGGUbGaamOyaaaa@5149@ .

5.    Mit 1. hat man:  lna=ln a n n =nln a n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWGHbGaeyypa0JaciiBaiaac6gadaGcbaqaaiaadggaaSqaaiaad6gaaaGcdaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaWGUbGaeyyXICTaciiBaiaac6gadaGcbaqaaiaadggaaSqaaiaad6gaaaaaaa@46E3@ , und damit die Behauptung.

Für die weitere Untersuchung der Logarithmusfunktion benötigen wir als technisches Hilfsmittel eine Abschätzung, die sog. zentrale Ungleichung für ln MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gaaaa@37D0@ .

Bemerkung:  Für jedes x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@ ist

1 1 x lnxx1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaadIhaaaGaeyizImQaciiBaiaac6gacaWG4bGaeyizImQaamiEaiabgkHiTiaaigdaaaa@424C@
[8.7.11]

Beweis:  Für x=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaaigdaaaa@38AA@ ist nichts zu zeigen. Sei nun zunächst x>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaigdaaaa@38AC@ . Da 1 t 2 1 t 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiDamaaCaaaleqabaGaaGOmaaaaaaGccqGHKjYOdaWcaaqaaiaaigdaaeaacaWG0baaaiabgsMiJkaaigdaaaa@3E8C@ für alle t[1,x] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgIGiolaacUfacaaIXaGaaiilaiaadIhacaGGDbaaaa@3C91@ , folgt hier die Behauptung aus dem Monotonieverhalten des Integrals ([8.2.10]):

1 x +1= 1 X | 1 x = 1 x 1 X 2 1 x 1 X =lnx 1 x 1 =x1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamiEaaaacqGHRaWkcaaIXaGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaamiwaaaacaGG8bWaa0baaSqaaiaaigdaaeaacaWG4baaaOGaeyypa0Zaa8qCaeaadaWcaaqaaiaaigdaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaaaaaeaacaaIXaaabaGaamiEaaqdcqGHRiI8aOGaeyizIm6aaGbaaeaadaWdXbqaamaalaaabaGaaGymaaqaaiaadIfaaaaaleaacaaIXaaabaGaamiEaaqdcqGHRiI8aaWcbaGaeyypa0JaciiBaiaac6gacaWG4baakiaawIJ=aiabgsMiJoaapehabaGaaGymaaWcbaGaaGymaaqaaiaadIhaa0Gaey4kIipakiabg2da9iaadIhacqGHsislcaaIXaaaaa@602A@

Ist 0<x<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaadIhacqGH8aapcaaIXaaaaa@3A66@ , so ist 1 x >1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEaaaacqGH+aGpcaaIXaaaaa@3977@ . Also erhält man mit dem gerade gewonnenen Ergebnis:

1 1 1 x ln 1 x 1 x 1 1xlnx 1 x 1 x1lnx1 1 x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaaqaaiaaigdacqGHsisldaWcaaqaaiaaigdaaeaadaWcaaqaaiaaigdaaeaacaWG4baaaaaacqGHKjYOciGGSbGaaiOBamaalaaabaGaaGymaaqaaiaadIhaaaGaeyizIm6aaSaaaeaacaaIXaaabaGaamiEaaaacqGHsislcaaIXaaabaGaeyO0H4TaaGzbVdqaaiaaigdacqGHsislcaWG4bGaeyizImQaeyOeI0IaciiBaiaac6gacaWG4bGaeyizIm6aaSaaaeaacaaIXaaabaGaamiEaaaacqGHsislcaaIXaaabaGaeyO0H4TaaGzbVdqaaiaadIhacqGHsislcaaIXaGaeyyzImRaciiBaiaac6gacaWG4bGaeyyzImRaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaadIhaaaaaaaaa@6664@

Mit ln1=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaaIXaGaeyypa0JaaGimaaaa@3A4B@ steht uns bisher erst ein Funktionswert des Logarithmus zur Verfügung. Über die zentrale Ungleichung können wir nun zusätzlich den Funktionswert der Eulerschen Zahl e=lim (1+ 1 n ) n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabg2da9iGacYgacaGGPbGaaiyBaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaaaa@4080@ (siehe [5.7.7]) ermitteln.

Bemerkung:  

lne=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWGLbGaeyypa0JaaGymaaaa@3A7B@
[8.7.12]

Beweis:  Wir betrachten zunächst die Folge (ln (1+ 1 n ) n )=(nln(1+ 1 n )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiGacYgacaGGUbGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGPaGaeyypa0Jaaiikaiaad6gacqGHflY1ciGGSbGaaiOBaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaGaaiykaaaa@4B3B@ . Mit [8.7.11] gelingt die folgende Abschätzung:

n n+1 =n(1 n n+1 )=n(1 1 1+ 1 n )nln(1+ 1 n )n 1 n =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGUbaabaGaamOBaiabgUcaRiaaigdaaaGaeyypa0JaamOBaiabgwSixlaacIcacaaIXaGaeyOeI0YaaSaaaeaacaWGUbaabaGaamOBaiabgUcaRiaaigdaaaGaaiykaiabg2da9iaad6gacqGHflY1caGGOaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaaaacaGGPaGaeyizImQaamOBaiabgwSixlGacYgacaGGUbGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcacqGHKjYOcaWGUbGaeyyXIC9aaSaaaeaacaaIXaaabaGaamOBaaaacqGH9aqpcaaIXaaaaa@63BB@

Da n n+1 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGUbaabaGaamOBaiabgUcaRiaaigdaaaGaeyOKH4QaaGymaaaa@3C27@ , erhält man mit dem Schachtelsatz [5.5.8]: ln (1+ 1 n ) n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaad6gaaaGaaiykamaaCaaaleqabaGaamOBaaaakiabgkziUkaaigdaaaa@4056@ . Schließlich garantiert die Stetigkeit von ln die Behauptung:

lne=ln(lim (1+ 1 n ) n )=limln (1+ 1 n ) n =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWGLbGaeyypa0JaciiBaiaac6gacaGGOaGaciiBaiaacMgacaGGTbGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGPaGaeyypa0JaciiBaiaacMgacaGGTbGaciiBaiaac6gacaGGOaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaad6gaaaGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9iaaigdaaaa@5304@

Interessanterweise ist es über die zentralen Ungleichung sogar möglich, Zugang zu allen Funktionswerten von ln zu finden. Dazu benötigen wir zunächst eine Erweiterung der zentralen Ungleichung: Wendet man [8.7.11] auf x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbaaaaaa@37F7@ an, so erhält man mit [8.7.10] für alle x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@ die Abschätzung

n(1 1 x n )lnxn( x n 1) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaWaaOqaaeaacaWG4baaleaacaWGUbaaaaaakiaacMcacqGHKjYOciGGSbGaaiOBaiaadIhacqGHKjYOcaWGUbGaaiikamaakeaabaGaamiEaaWcbaGaamOBaaaakiabgkHiTiaaigdacaGGPaaaaa@4914@ .
[8.7.13]

Die beiden einschachtelnden Folgen erweisen sich dabei als konvergent gegen einen gemeinsamen Limes:

Bemerkung:  Für alle x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@ ist

lnx=limn( x n 1) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4bGaeyypa0JaciiBaiaacMgacaGGTbGaamOBaiaacIcadaGcbaqaaiaadIhaaSqaaiaad6gaaaGccqGHsislcaaIXaGaaiykaaaa@42AC@

lnx=limn(1 1 x n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4bGaeyypa0JaciiBaiaacMgacaGGTbGaamOBaiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaWaaOqaaeaacaWG4baaleaacaWGUbaaaaaakiaacMcaaaa@4377@

[8.7.14]

Beweis:  Wir beweisen beide Aussagen gleichzeitig und zeigen dazu der Reihe nach:

  1. (n( x n 1)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gacaGGOaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiaacMcacaGGPaaaaa@3D4E@ ist konvergent.

  2. n( x n 1)n(1 1 x n )0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacIcadaGcbaqaaiaadIhaaSqaaiaad6gaaaGccqGHsislcaaIXaGaaiykaiabgkHiTiaad6gacaGGOaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaamaakeaabaGaamiEaaWcbaGaamOBaaaaaaGccaGGPaGaeyOKH4QaaGimaaaa@465D@ .

1.    Gemäß [8.7.13] ist (n( x n 1)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gacaGGOaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiaacMcacaGGPaaaaa@3D4E@ nach unten beschränkt (durch lnx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4baaaa@38CD@ ). Nach [5.7.1] genügt es daher zu zeigen: (n( x n 1)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gacaGGOaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiaacMcacaGGPaaaaa@3D4E@ ist monoton fallend, d.h.

n( x n 1)(n+1)( x n+1 1) n x n +1(n+1) x n+1 [+] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaaqaaaqaaiaad6gacaGGOaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiaacMcacqGHLjYScaGGOaGaamOBaiabgUcaRiaaigdacaGGPaGaaiikamaakeaabaGaamiEaaWcbaGaamOBaiabgUcaRiaaigdaaaGccqGHsislcaaIXaGaaiykaaqaaiabgsDiBlaaywW7aeaacaWGUbWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaey4kaSIaaGymaiabgwMiZkaacIcacaWGUbGaey4kaSIaaGymaiaacMcadaGcbaqaaiaadIhaaSqaaiaad6gacqGHRaWkcaaIXaaaaOGaai4waiabgUcaRiaac2faaaaaaa@5CF8@

für alle n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3AEB@ . Über die Bernoullische Ungleichung

 i

1+ x1 nx [1] x n [2] 1+ x1 n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaamiEaiabgkHiTiaaigdaaeaacaWGUbGaamiEaaaadaWfqaqaaiabgsMiJcWcbaGaai4waiaaigdacaGGDbaabeaakmaakeaabaGaamiEaaWcbaGaamOBaaaakmaaxababaGaeyizImkaleaacaGGBbGaaGOmaiaac2faaeqaaOGaaGymaiabgUcaRmaalaaabaGaamiEaiabgkHiTiaaigdaaeaacaWGUbaaaaaa@4C6F@   für alle x,n>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWGUbGaeyOpa4JaaGimaaaa@3A4E@

Beweis[2] ergibt sich direkt aus der gewöhnlichen Bernoullischen Ungleichung [5.2.6], denn da x n 11 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiabgwMiZkabgkHiTiaaigdaaaa@3D17@ , hat man:

x= (1+ x n 1) n 1+n( x n 1)=1+n x n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaacIcacaaIXaGaey4kaSYaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gaaaGccqGHLjYScaaIXaGaey4kaSIaamOBaiaacIcadaGcbaqaaiaadIhaaSqaaiaad6gaaaGccqGHsislcaaIXaGaaiykaiabg2da9iaaigdacqGHRaWkcaWGUbWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaamOBaaaa@50C3@ ,

und damit:

x n x+n1 n =1+ x1 n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyizIm6aaSaaaeaacaWG4bGaey4kaSIaamOBaiabgkHiTiaaigdaaeaacaWGUbaaaiabg2da9iaaigdacqGHRaWkdaWcaaqaaiaadIhacqGHsislcaaIXaaabaGaamOBaaaaaaa@457E@

Zum Nachweis von [1] setzen wir das gerade erzielte Ergebnis ein und erhalten:

x n = 1 1 x n 1 1+ 1 x 1 n = nx nx+1x =1+ x1 nx+1x [+] 1+ x1 nx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOqaaeaadaWcaaqaaiaaigdaaeaacaWG4baaaaWcbaGaamOBaaaaaaGccqGHLjYSdaWcaaqaaiaaigdaaeaacaaIXaGaey4kaSYaaSaaaeaadaWcaaqaaiaaigdaaeaacaWG4baaaiabgkHiTiaaigdaaeaacaWGUbaaaaaacqGH9aqpdaWcaaqaaiaad6gacaWG4baabaGaamOBaiaadIhacqGHRaWkcaaIXaGaeyOeI0IaamiEaaaacqGH9aqpcaaIXaGaey4kaSYaaGbaaeaadaWcaaqaaiaadIhacqGHsislcaaIXaaabaGaamOBaiaadIhacqGHRaWkcaaIXaGaeyOeI0IaamiEaaaaaSqaaiaacUfacqGHRaWkcaGGDbaakiaawIJ=aiabgwMiZkaaigdacqGHRaWkdaWcaaqaaiaadIhacqGHsislcaaIXaaabaGaamOBaiaadIhaaaaaaa@6549@

Bei der letzten Abschätzung beachte man, dass das Weglassen des Summanden 1x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTiaadIhaaaa@3891@ den Nenner von [+] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaai4waiabgUcaRiaac2faaaa@388F@ vergrößert, falls der Zähler x1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgkHiTiaaigdaaaa@3891@ positiv ist, bzw. verkleinert, wenn der Zähler negativ ist. In jedem Fall aber verkleinert sich dabei der gesamte Bruch.

für die n-te Wurzel erhalten wir zunächst:

n x n+1 n +1=nx x n +1nx(1+ x1 nx )+1=nx+x=(n+1)x=(n+1) x n+1 n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66E3@ .

Die Abschätzung [+] ergibt sich nun, wenn man x durch x n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbGaey4kaSIaaGymaaaaaaa@3994@ ersetzt.

2.    Da x n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOKH4QaaGymaaaa@3AA9@ (siehe [5.7.9]) folgt die Behauptung aus dem dritten Grenzwertsatz [5.6.3]:

n( x n 1)n(1 1 x n )=n( x n 1)n x n 1 x n = n( x n 1) konvergent (1 1 x n ) 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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aiabgwSixpaayaaabaGaaiikaiaaigdacqGHsisldaWcaaqaaiaaigdaaeaadaGcbaqaaiaadIhaaSqaaiaad6gaaaaaaOGaaiykaaWcbaGaeyOKH4QaaGimaaGccaGL44pacqGHsgIRcaaIWaaaaa@7484@

Das Ergebnis in 2. zeigt nun: Mit (n( x n 1)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gacaGGOaWaaOqaaeaacaWG4baaleaacaWGUbaaaOGaeyOeI0IaaGymaiaacMcacaGGPaaaaa@3D4E@ ist auch (n(1 1 x n )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gacaGGOaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaamaakeaabaGaamiEaaWcbaGaamOBaaaaaaGccaGGPaGaaiykaaaa@3E19@ konvergent, und zwar gegen einen gemeinsamen Limes, der wegen [8.7.13] die Zahl lnx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4baaaa@38CD@ sein muss. Das Argument ist dabei der Schachtelsatz [5.5.8].

Eine weitere Anwendung der zentralen Ungleichung belegt nun die Umkehrbarkeit des Logarithmus.

Bemerkung:  Jedes y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgIGiolabl2riHcaa@39DE@ besitzt genau ein Urbild x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@ , d.h.

ln: >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGG6aGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaOGaeyOKH4QaeSyhHekaaa@3F54@   ist bijektiv.
[8.7.15]

Beweis:  Da ln(x)= 1 x 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGNaGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamiEaaaacqGHGjsUcaaIWaaaaa@4020@ für alle x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@ ist ln MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gaaaa@37D0@ nach [7.9.6] zunächst injektiv, d.h. jedes y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgIGiolabl2riHcaa@39DE@ hat höchstens ein Urbild.

Ein solches y hat aber auch mindestens ein Urbild, denn mit der zentralen Ungleichung [8.7.11] hat man zunächst 1 2 =1 1 2 ln2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqGH9aqpcaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacqGHKjYOciGGSbGaaiOBaiaaikdaaaa@3FFD@ und damit für ein beliebiges n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLcaa@39CF@ :

ln 2 2n =2nln22n 1 2 =n ln 2 2n =ln 2 2n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabiqaaaqaaiGacYgacaGGUbGaaGOmamaaCaaaleqabaGaaGOmaiaad6gaaaGccqGH9aqpcaaIYaGaamOBaiabgwSixlGacYgacaGGUbGaaGOmaiabgwMiZkaaikdacaWGUbGaeyyXIC9aaSaaaeaacaaIXaaabaGaaGOmaaaacqGH9aqpcaWGUbaabaGaciiBaiaac6gacaaIYaWaaWbaaSqabeaacqGHsislcaaIYaGaamOBaaaakiabg2da9iabgkHiTiGacYgacaGGUbGaaGOmamaaCaaaleqabaGaaGOmaiaad6gaaaGccqGHKjYOcqGHsislcaWGUbaaaaaa@5ADE@

Da MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyfHukaaa@3758@ in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375C@ unbeschränkt ist, gibt es nun zu y ein n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLcaa@39CF@ , so dass ln 2 2n nynln 2 2n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaaIYaWaaWbaaSqabeaacqGHsislcaaIYaGaamOBaaaakiabgsMiJkabgkHiTiaad6gacqGHKjYOcaWG5bGaeyizImQaamOBaiabgsMiJkGacYgacaGGUbGaaGOmamaaCaaaleqabaGaaGOmaiaad6gaaaaaaa@4A80@ . Damit liegt y zwischen zwei ln-Werten und muss daher aufgrund des Zwischenwertsatzes [6.6.2] ein Funktionswert von ln sein. ln ist also auch surjektiv.

[8.7.15] zeigt insbesondere, dass ln in beide Richtungen unbeschränkt wächst, d.h. lim x lnx= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHEisPaeqaaOGaciiBaiaac6gacaWG4bGaeyypa0JaeyOhIukaaa@42B2@ und lim x 0 + lnx= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqabeaacqGHRaWkaaaaleqaaOGaciiBaiaac6gacaWG4bGaeyypa0JaeyOeI0IaeyOhIukaaa@4403@ . Interessant ist dabei die Art dieses Wachstums:

  • ln strebt für x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgkziUkabg6HiLcaa@3A47@ langsamer gegen MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIukaaa@375D@ als jede positive Potenz, d.h. lnx< x a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4bGaeyipaWJaamiEamaaCaaaleqabaGaamyyaaaaaaa@3BE1@ für hinreichend große x.

  • ln strebt für x 0 + MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgkziUkaaicdadaahaaWcbeqaaiabgUcaRaaaaaa@3A9F@ langsamer gegen MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaeyOhIukaaa@384A@ als jede negative Potenz, d.h. lnx> x a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWG4bGaeyOpa4JaeyOeI0IaamiEamaaCaaaleqabaGaeyOeI0Iaamyyaaaaaaa@3DBF@ für hinreichend kleine x.

Die folgende Bemerkung präzisiert diese Vorstellungen.

Bemerkung:  Für jedes a >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolabl2riHoaaCaaaleqabaGaeyOpa4JaaGimaaaaaaa@3BB5@  *) gilt:

1.    lim x lnx x a =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHEisPaeqaaOWaaSaaaeaaciGGSbGaaiOBaiaadIhaaeaacaWG4bWaaWbaaSqabeaacaWGHbaaaaaakiabg2da9iaaicdaaaa@4425@

[8.7.16]

2.    lim x 0 + ( x a lnx)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqabeaacqGHRaWkaaaaleqaaOGaaiikaiaadIhadaahaaWcbeqaaiaadggaaaGccqGHflY1ciGGSbGaaiOBaiaadIhacaGGPaGaeyypa0JaaGimaaaa@481C@

[8.7.17]

___________
*) Potenzen mit beliebigen Exponenten werden in 8.9 eingeführt.

Beweis:  Wir wählen ein b>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg6da+iaaicdaaaa@3895@ so dass b<a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgYda8iaadggaaaa@38BD@ , also ab>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgkHiTiaadkgacqGH+aGpcaaIWaaaaa@3A68@ , und setzen wieder die zentrale Ungleichung [8.7.11] ein:

1.   Für alle x1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgwMiZkaaigdaaaa@396A@ ist

0 lnx x a = 1 b ln x b x a 1 b x b 1 x a 1 b x b x a = 1 b 1 x ab MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@67D6@ .

Da nun lim x 1 x ab =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHEisPaeqaaOWaaSaaaeaacaaIXaaabaGaamiEamaaCaaaleqabaGaamyyaiabgkHiTiaadkgaaaaaaOGaeyypa0JaaGimaaaa@43D3@ , folgt daher die Behauptung.

2.   Hier hat man für alle 0<x1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaadIhacqGHKjYOcaaIXaaaaa@3B17@ :

0 x a lnx= 1 b x a ln x b 1 b x a (1 1 x b )= 1 b ( x a x ab ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6A73@ ,

so dass die Behauptung jetzt aus lim x 0 + ( x a x ab )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqabeaacqGHRaWkaaaaleqaaOGaaiikaiaadIhadaahaaWcbeqaaiaadggaaaGccqGHsislcaWG4bWaaWbaaSqabeaacaWGHbGaeyOeI0IaamOyaaaakiaacMcacqGH9aqpcaaIWaaaaa@47CC@ folgt.


 

Funktionen des Typs f f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgwSixlqadAgagaqbaaaa@3A18@ und f f 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaceWGMbGbauaaaeaacaWGMbWaaWbaaSqabeaacaaIYaaaaaaaaaa@38C7@ konnten wir in [8.1.11-12] leicht mit einer Stammfunktion versorgen. Mit Hilfe der Logarithmusfunktion finden wir auch für den Typ f f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaceWGMbGbauaaaeaacaWGMbaaaaaa@37DE@ ein passendes Verfahren.

Bemerkung (Logarithmische Integration):  Ist f:A >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGbbGaeyOKH4QaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@3DA7@ differenzierbar, so ist f f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaceWGMbGbauaaaeaacaWGMbaaaaaa@37DE@ integrierbar und

lnf MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacqWIyiYBcaWGMbaaaa@39F5@ ist eine Stammfunktion zu f f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaceWGMbGbauaaaeaacaWGMbaaaaaa@37DE@ .
[8.7.18]

Beweis:   lnf MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacqWIyiYBcaWGMbaaaa@39F5@ ist gemäß Kettenregel ([7.7.8]) differenzierbar, und zwar mit folgender Ableitung:

(lnf ) =lnf f = 1 X f f = f f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiGacYgacaGGUbGaeSigI8MaamOzaiqacMcagaqbaiabg2da9iGacYgacaGGUbGaai4jaiablIHiVjaadAgacqGHflY1ceWGMbGbauaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGybaaaiablIHiVjaadAgacqGHflY1ceWGMbGbauaacqGH9aqpdaWcaaqaaiqadAgagaqbaaqaaiaadAgaaaaaaa@4F61@

So hat man zum Beispiel:

  • 0 1 2X X 2 +1 =ln( X 2 +1) | 0 1 =ln2ln1=ln2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWcaaqaaiaaikdacaWGybaabaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaaaaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaeyypa0JaciiBaiaac6gacqWIyiYBcaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdacaGGPaGaaiiFamaaDaaaleaacaaIWaaabaGaaGymaaaakiabg2da9iGacYgacaGGUbGaaGOmaiabgkHiTiGacYgacaGGUbGaaGymaiabg2da9iGacYgacaGGUbGaaGOmaaaa@5558@

  • e e 2 1 Xln = e e 2 1 X ln =lnln | e e 2 =ln(2lne)ln(lne)=ln2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6978@

  • Für x] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@40DC@ ist 0 x tan = 0 x sin cos =lncos | 0 x =ln(cosx)+ln1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGG0bGaaiyyaiaac6gaaSqaaiaaicdaaeaacaWG4baaniabgUIiYdGccqGH9aqpcqGHsisldaWdXbqaamaalaaabaGaeyOeI0Iaci4CaiaacMgacaGGUbaabaGaci4yaiaac+gacaGGZbaaaaWcbaGaaGimaaqaaiaadIhaa0Gaey4kIipakiabg2da9iabgkHiTiGacYgacaGGUbGaeSigI8Maci4yaiaac+gacaGGZbGaaiiFamaaDaaaleaacaaIWaaabaGaamiEaaaakiabg2da9iabgkHiTiGacYgacaGGUbGaaiikaiGacogacaGGVbGaai4CaiaadIhacaGGPaGaey4kaSIaciiBaiaac6gacaaIXaaaaa@60FF@ , so dass wir mit lncos|] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaciiBaiaac6gacqWIyiYBciGGJbGaai4BaiaacohacaGG8bGaaiyxaiabgkHiTmaalaaabaGaeqiWdahabaGaaGOmaaaacaGGSaWaaSaaaeaacqaHapaCaeaacaaIYaaaaiaacUfaaaa@4639@ eine Stammfunktion zu tan|] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacggacaGGUbGaaiiFaiaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@422C@ ermittelt haben.


8.6. 8.8.