By Herbert Amann, Joachim Escher
The second one quantity of this creation into research bargains with the combination concept of services of 1 variable, the multidimensional differential calculus and the speculation of curves and line integrals. the fashionable and transparent improvement that begun in quantity I is sustained. during this method a sustainable foundation is created which permits the reader to house attention-grabbing functions that typically transcend fabric represented in conventional textbooks. this is applicable, for example, to the exploration of Nemytskii operators which allow a clear creation into the calculus of diversifications and the derivation of the Euler-Lagrange equations.
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Additional info for Analysis II (v. 2)
4 Proposition (of the additivity of integrals) For f ∈ S(I, E) and a, b, c ∈ I we have b c f= b f+ a a f . c Proof It suﬃces to check this for a ≤ b ≤ c. If (fn ) is a sequence of staircase functions that converge uniformly to f and J is a compact perfect subinterval of I, then fn |J ∈ T (J, E) and fn |J −→ uniformly f |J . 4) The deﬁnition of the integral of staircase functions gives at once that c b fn = a c fn + fn . 1, we pass the limit n → ∞ and ﬁnd c b f= c f+ a a f . b Then we have b c f= a c f− a c f= b b f+ a f .
It is also known that G is doubly periodic, that is, there are two R-linearly independent periods ω1 , ω2 ∈ C such that G(z + ω1 ) = G(z + ω2 ) = G(z) for z ∈ C\M . Because elementary functions are at most “simply periodic”, G cannot be elementary. Therefore F is also not elementary. Hence the elementary function f has no elementary antiderivative. In the following we will show that rational functions have elementary antiderivatives. We begin with simple examples, which can then be extended to the general case.
K=0 ∞ n n=0 k=0 n ∞ j=0 Bj j z = j! n=0 n k=0 xn−k Bk n z (n − k)! k! n = and, alternately, Fx (z) = ∞ n z Bk xn−k n! k Bn (x)z n /n! The statement (ii) follows immediately from (i). Likewise from (i) we get (iii) because n n+1 Bk X n−k (n + 1 − k) Bn+1 (X) = k k=0 n = (n + 1) k=0 n Bk X n−k = (n + 1)Bn (X) . k Finally, (iv) and (v) follow from Fx+1 (z) − Fx (z) = zexz and F1−x (z) = Fx (−z) by comparing coeﬃcients. 7 Corollary The ﬁrst four Bernoulli polynomials read B0 (X) = 1 , B1 (X) = X − 1/2 , B2 (X) = X − X + 1/6 , B3 (X) = X 3 − 3X 2 /2 + X/2 .
Analysis II (v. 2) by Herbert Amann, Joachim Escher