Download Advanced Calculus: An Introduction to Classical Analysis by Louis Brand PDF

By Louis Brand

A direction in research dealing primarily with services of a true variable, this article for upper-level undergraduate scholars introduces the fundamental innovations of their least difficult surroundings and proceeds with various examples, theorems said in a realistic demeanour, and coherently expressed proofs. 1955 edition.

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Extra resources for Advanced Calculus: An Introduction to Classical Analysis (Dover Books on Mathematics)

Example text

Insbesondere ist fUr jede Menge A eX der Durchschnitt A aller A umfassenden abgeschlossenen Mengen abgeschlossen. Man nennt diese kleinste A enthaltende, abgeschlossene Menge die abgesehlossene Bulle A von A in X, es gilt A = A. 0 Eine Menge We X heiSt Umgebung der Menge Me X, wenn es eine in X offene Menge V mit MeV c W gibt. Man beaehte, daj3 naeh dieser Definition Umgebungen nieht notwendig offen sind. Offene Mengen sind Umgebungen all ihrer Punkte. Zu verschiedenen Punkten e, e' EX gibt es stets punktfremde Umgebungen: B,(e) nB,(e')=0 fUr 8: =-td(e, e'»O.

2) Es sei f: D ..... D' eine stetige Abbildung zwischen Bereichen in CC. a) Ist W eine Komponente von D, so ist f(W) in genau einer Komponente von D' enthalten. b) Ist f ein Homoomorphismus, so ist f(W) eine Komponente von D'. 3) Eine Teilmenge M cCC heiBt konvex, wenn flir je zwei Punkte w, zEM die Strecke [w, z] in M liegt. Konvexe Mengen in CC sind also wegzusammenhangend. a) Durchschnitte konvexer Mengen in CC sind konvex. b) Ist G ein konvexes Gebiet in CC, so ist auch G konvex. c) Sei G ein konvexes Gebiet in CC und cEaG.

HEIR. i den Wert 1 bzw. -1 und also keinen Limes. 3) Die Funktionen Re z, 1m z, Izl sind nirgends in CC komplex differenzierbar. Das zeigt man analog wie eben im Fall der Funktion z. 2. Cauchy-Riemannsche Differentialgleichungen. Wir schreiben c = a + i b = (a, b), z=x+iy=(x,y). 1st J(z)=u(x,y)+iv(x,y) komplex differenzierbar in cED, so gilt f'(c) = lim J(c + h) - J(c) h-O h lim J(c + i h) - J(c) . ih h-O § 1. Komplex differenzierbare Funktionen 39 c+ih 1 c<---c+h Wahlt man h reell, so foIgt '1' v(a+h,b)-v(a,b) ( 1· u(a+h,b)-u(a,b) f 'c)=lm h +llm h h~O h~O · u(a,b+h)-u(a,b) '1' v(a,b+h)-v(a,b) +llm .

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