Analysis 2 at Universität Heidelberg

Flashcards and summaries for Analysis 2 at the Universität Heidelberg

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Def Fktfolge (fn)n konv. gegen f in x

Exemplary flashcards for Analysis 2 at the Universität Heidelberg on StudySmarter:

Def (fn)n konv. gleichmäßig gegen f in D

Exemplary flashcards for Analysis 2 at the Universität Heidelberg on StudySmarter:

Grenzfunktion gleichmäßiger Konvergenz, stetig?

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Def Maximumsnorm

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||f||∞ und gleichmäßige Konvergenz

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Def Normeigenschaften

Exemplary flashcards for Analysis 2 at the Universität Heidelberg on StudySmarter:

Def Funktionenraum

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Exemplary flashcards for Analysis 2 at the Universität Heidelberg on StudySmarter:

Vollständigkeit vom Funktionenraum C[a,b]

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Integral von Limes von Funktionenfolge

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Konvergenz von Reihen von Funktionen

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Integration von Potenzreihen

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Def Integration Komplexer Funktionen

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Exemplary flashcards for Analysis 2 at the Universität Heidelberg on StudySmarter:

Analysis 2

Def Fktfolge (fn)n konv. gegen f in x

fa ε>0 ex N>0 mit |fn(x)-f(x)| < ε fa n>=N

standard epsilonkriterium

Analysis 2

Def (fn)n konv. gleichmäßig gegen f in D

fa ε>0 ex N>0  mit |fn(x)-f(x)| < ε fa n>=N, x in D

Analysis 2

Grenzfunktion gleichmäßiger Konvergenz, stetig?

gleichmäßiger Limes stetiger Funktionen ist stetig

(fn)n mit fn stetig fa n und glm. konv. gg f => f stetig in D

Analysis 2

Def Maximumsnorm

D beschränkt, abg., Intervall [a,b]=D, f stetig

||f||∞= max{|f(x)| x in D} 

Analysis 2

||f||∞ und gleichmäßige Konvergenz

D=[a,b], f stetig:

(fn)n konv glm gg f <=> ||fn-f||∞ -> 0, n ->∞  

(fn)n konv glm auf D <=> Cauchy-Kriterium: fa ε>0 ex N mit fa n,m>N:  ||fn-fm||∞

Analysis 2

Def Normeigenschaften

Definitheit: ||x|| = 0 => x=0

Homogenität: ||ax|| = |a|*||x||

Dreiecksugl: ||x+y|| ≤ ||x||+||y||

Analysis 2

Def Funktionenraum

C[a,b] = {f: [a,b]->R, f stetig}

mit ||f||∞ normierter VR

Analysis 2

Vollständigkeit vom Funktionenraum C[a,b]

C[a,b] ist vollständig bzgl gleichmäßiger Konverg., dh jede cauchyfolge aus C[a,b] besitzt Limes in C[a,b]

Analysis 2

Integral von Limes von Funktionenfolge

fn: [a,b]->R stetig und fn -> f glm. => f stetig und lim integral fn = integral lim fn

Analysis 2

Konvergenz von Reihen von Funktionen

fn stetig, Reihe ∑fn glm konv., dann

∑ integral fn(x) = integral ∑fn(x)

Analysis 2

Integration von Potenzreihen

p=∑an(x-x0)^n mit rho>0 =>

p kov. glm. in [x0-r,x0+r] für 0<r<rho, für [a,b] sset ]x0-rho,x0+rho[:

integral(a bis b) p= ∑an/(n+1)(x-x0)^(n+1)  |(a bis b)

Analysis 2

Def Integration Komplexer Funktionen

integral f(x) = integral Re(f(x)) + i*integral Im(f(x))

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