By Zhaohui Luo
This e-book develops a sort idea, reports its houses, and explains its makes use of in desktop technological know-how. The booklet focuses specifically on how the learn of variety idea may well supply a strong and uniform language for programming, software specification and improvement, and logical reasoning. the kind idea constructed the following displays a conceptual contrast among logical propositions and computational info forms. ranging from an advent of the fundamental techniques, the writer explains the which means and use of the type-theoretic language with proof-theoretic justifications, and discusses a variety of concerns within the examine of kind thought. the sensible use of the language is illustrated by way of constructing an method of specification and information refinement in variety conception, which helps modular improvement of specification, courses, and proofs. scholars and researchers in desktop technology and common sense will welcome this interesting new ebook.
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Additional info for Computation and Reasoning: A Type Theory for Computer Science
Vergleichen Sie auch die Originalfunktion mit den Taylorpolynomen grafisch. Für f(x) = ln(x) kann an der Stelle 0 die Reihe nicht entwickelt werden, weil an dieser Stelle ein Pol der Funktion vorliegt ! B. an der Stelle x 0 = 1: 1 p1 ( x) ln ( x) reihe x = 1 grad o 1 ( x 1) nº ª « ( 1) n1 ( x 1) » n ¬ ¼ f ln ( x) = ¦ n 1 ( 1) nof ( 1) 1 3 1 3 ( x 1) 4 o1 n2 p2 ( x) ln ( 1 x) reihe x = 0 grad o 1 x ª n1 xnº «( 1) » n¼ ¬ f ¦ 1 1 2 2 x 1 3 3 x 1 4 4 x 1 5 5 x Taylorreihe für ln(1+x) mit der Entwicklungsstelle x0 = 0.
N1 ln ( 1 x) = 4 ( x 1) n1 n lim 2 2 ( x 1) = ln ( 1 x) ln ( 1 x) Seite 35 5 Taylorreihen § 1 x · reihe x = 0 grad o 2 x 2 x3 2 x5 3 5 © 1 x¹ p4 ( x) ln ¨ §1 ©1 ln ¨ 2n 1 f x· = 2 x¹ x ¦ n Taylorreihe für ln((1-x)/(1-x)) mit der Entwicklungsstelle x0 = 0. 2 n 1 0 Diese Reihe konvergiert sicher im Intervall -1 < x <1. 1 x Setzen wir 1 x = z , dann folgt: x= z1 z1 Wir setzen nun x in die vorhergende Reihe ein und ersetzen hinterher z durch x: reihe x = 0 grad § 1 x· p5 ( x) ln ¨ © 1 x¹ 3 5 z 1o 2 x 1 2 ( x 1) 2 ( x 1) ersetzen x = 3 5 x 1 3 5 z1 ( x 1) ( x 1) ersetzen z = x f ln ( x) = 2 ¦ n 0 2n 1 º ª 1 ( x 1) « » « ( 2 n 1) ( 1 x) 2n1 » ¬ ¼ Taylorreihe für ln(x).
Seite 25 Entwicklung der Funktion an der Stelle x 0 = 0 Taylorreihen fx ( x n) konstantes Glid a 0 f ( 0) = sin ( 0) = 0 n d n a0 = 0 Ableitungen f ( x) dx 1. Ableitung an der Stelle 0 1 a1 = 1 0 2. Ableitung an der Stelle 0 a2 = 0 1 3. Ableitung an der Stelle 0 1 a3 = 3 fx ( x 1) o cos ( x) fx ( 0 1) o cos ( 0) 1 fx ( x 2) o sin ( x) fx ( 0 2) o sin ( 0) fx ( x 3) o cos ( x) fx ( 0 3) o cos ( 0) fx ( x 4) o sin ( x) fx ( 0 4) o sin ( 0) 0 4. Ableitung an der Stelle 0 fx ( x 5) o cos ( x) fx ( 0 5) o cos ( 0) 1 5.