New PDF release: Applied and Computational Complex Analysis - Vol 1: Power

By Peter Henrici

ISBN-10: 0471372447

ISBN-13: 9780471372448

At a mathematical point obtainable to the non-specialist, the 3rd of a three-volume paintings exhibits the best way to use equipment of complicated research in utilized arithmetic and computation. The booklet examines two-dimensional power idea and the development of conformal maps for easily and multiply hooked up areas. furthermore, it presents an advent to the speculation of Cauchy integrals and their purposes in capability concept, and provides an ordinary and self-contained account of de Branges' lately came across facts of the Bieberbach conjecture within the concept of univalent features. The evidence deals a few fascinating functions of fabric that seemed in volumes 1 and a pair of of this paintings. It discusses subject matters by no means prior to released in a textual content, reminiscent of numerical review of Hilbert remodel, symbolic integration to unravel Poisson's equation, and osculation tools for numerical conformal mapping.

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Extra resources for Applied and Computational Complex Analysis - Vol 1: Power Series, Integration, Conformal Mapping, Location of Zeros

Sample text

36 2. 3. 4. 5. 18 Elementare Funktionen a) f1 (x) = 3 + 5(x − 1) − 4(x − 1)2 b) f2 (x) = 3 c) f3 (t) = t − t2 √ √ d) f4 (x) = x(x − 1) + 2 e) f5 (t) = t3 − t f) f6 (x) = x+1 2 √ g) f7 (x) = (x − 2)(x + 13 ) h) f8 (x) = 3x − x−2 Durch die Angabe wie vieler Punkte ist eine kubische Funktion eindeutig bestimmt? Richtig oder falsch: a) Eine quadratische Funktion hat zwei reelle Nullstellen. b) Eine kubische Funktion hat mindestens eine reelle Nullstelle. Richtig oder falsch: a) Um die Nullstellen von f (x) = g(x)h(x) zu suchen, braucht man nur die Nullstellen von g(x) und h(x) zu suchen.

Die n-te Potenz von z ist also eine Zahl mit Betrag rn und n-fachem Winkel. Denn z n = (reiϕ )n = rn einϕ = rn (cos(nϕ) + i sin(nϕ)). 42 angewendet und das Ergebnis mithilfe von Kosinus und Sinus geschrieben. Die Formel ist benannt nach dem franz¨ osischen Mathematiker Abraham de Moivre, 1667–1754. 45 Sei z = reiϕ , dann heißt √ √ ϕ n z = n r ei n , −π < ϕ ≤ π, die n-te Wurzel von z. √ √ Statt 2 z schreibt man wie im reellen Fall einfach z. Aber Achtung: Die aus dem Reellen gewohnten Rechenregeln gelten f¨ ur√komplexe Wurzeln nicht immer!

Die gew¨ Eingabe der St¨ utzpunkte zu ersparen, k¨onnen wir leicht mit dem Table-Befehl eine Liste gleichm¨aßig verteilter St¨ utzpunkte erzeugen. Zum Beispiel: 34 18 Elementare Funktionen In[6]:= f[x ] = 1/(1 + 25x2 ); j j Stuetzpunkte[n ] := Table[{ , f[ ]}, {j, −n, n}]; n n P[x ] = InterpolatingPolynomial[Stuetzpunkte[6], x]//Expand 367051586875 x4 107641853578125 x6 551599221900 x2 + − + Out[8]= 1 − 28167484501 1847048164 112669938004 8 10 12 62017871484375 x 65809335937500 x 25628906250000 x − + 28167484501 28167484501 28167484501 Stuetzpunkte[n] erzeugt eine Liste von 2n + 1 St¨ utzpunkten, die gleichm¨ aßig im Intervall [−1, 1] verteilt sind.

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Applied and Computational Complex Analysis - Vol 1: Power Series, Integration, Conformal Mapping, Location of Zeros by Peter Henrici


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