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By F. W. J. Olver

A vintage reference, meant for graduate scholars, mathematicians, physicists, and engineers, this publication can be utilized either because the foundation for educational classes and as a reference device.

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B) Man behandle a) im Fall der Ersetzung von arcsinx durch 3x{2 + (1 - X2)1/2} -I. · a) Man berechne/ex) = limx" (0 ~ x ~ 1) für n-+ N die Ungleichung Ix· - /(x)1 < e gilt. Man skizziere N = N(e, x) bei festem e als Funktion von x (0 ~ x ~ 1). Ist es möglich ein N = N(e) anzugeben, das von x unabhängig ist? b) Ist wegen a) die Funktionsfolge {x·}, (0 ~ x ~ 1) gleichmäßig konvergent? (x) und die Grenzfunktion g(x) der Funktionenreihe 1 + I (x k - X k - I ) (0 ~ X ~ 1; k = 1,2, ...

Man be- rechne r(I), r(2) und stelle r(n) mit Hilfe von r(n - 1) dar, n = 3, 4, ... Welcher Zusammenhang besteht zwischen F(n) und n? 4. Betrachtet wird das uneigentliche Integral In = f x 2n o + 1. e- x2 dx, n = 0,1,2, ... Be- rechnen Sie zuerst 10 , 11 , und geben Sie dann eine Rekursionsformel für In an. Welchen Wert erhält man daraus für In? 5. Ermitteln Sie, bei Existenz, IX) den Wert des uneigentlichen Integrals bzw. ß) den Cauchyschen Hauptwert des uneigentlichen Integrals! 3 a) 3 Jx dx- 2 ' J b) += += d) = g) e)* _= Jf(x) dx mit 0 dx c) o (x - 2)2 ' 0 J x2x+l+ 4 dx, 3 J 0 f(X)~{ f x2 f) für 0< x < 1, J~ 1/2 xlnx' += h)* x~ dx 2 dx x 2 - 3x+ 2 ' für J o ~Ix- 21 ' 1, J x 2+ 6xx +.

23 9. Ableitungen c) man bestimme zu jedem e > 0 ein ~ = ~ (e) derart, daß If(x) - AI< e für alle x mit 0< x < ~ gilt. Zahlenwerte 1= 2 m, e = erl mit er = 0,1 % (er: relative Genauigkeit). 3. Wie müssen die Konstanten A und B gewählt werden, damit die Funktion = -2 sinx für x ~ -rrl2, fex) = A sinx + B für lxi< rrl2, fex) = cosx für x ~ rrl2 überall stetig wird? 4. Gesucht ist eine überall stetige Funktion f(x), für die gilt: = 1 für -1< x < 0, f'(x) = -1 für 0< x < rr, f(O) = 2, f'(x) = 0 für -00 < x< -1, f'(x) f'(x) = 0 für rr< x< +00.

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