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# 8.7.E: Problemas en la Integración de Funciones Complejas y Vector-Valoradas

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## Ejercicio$$\PageIndex{1}$$

Demostrar el corolario$$1(\text { iii })-$$ (vii) en §4 componentwise para mapas integrables$$f: S \rightarrow E^{n}\left(C^{n}\right) .$$

## Ejercicio$$\PageIndex{2}$$

Demostrar teoremas 2 y 3 componentwise para$$E=E^{n}\left(C^{n}\right)$$.

## Ejercicio$$\PageIndex{2'}$$

Hazlo por Corolario 3 en §6.

## Ejercicio$$\PageIndex{3}$$

Demostrar el teorema 1 con

\ [\ int_ {A} |f|<
\ infty\]

\ [\ int_ {A}\ izquierda|f_ {k}\ derecha|<\ infty,\ quad k=1,\ ldots, n.
\]

## Ejercicio$$\PageIndex{4}$$

Demostrar que si$$f: S \rightarrow E^{n}\left(C^{n}\right)$$ es integrable en$$A,$$ así es$$|f| .$$ Disprove the converse.

## Ejercicio$$\PageIndex{5}$$

Desmentir Lema 1 para$$m A=\infty$$.

## Ejercicio$$\PageIndex{*6}$$

Completa el comprobante de Lema 3.

## Ejercicio$$\PageIndex{*7}$$

Completar la prueba del Teorema 3.

## Ejercicio$$\PageIndex{*8}$$

Hacer Problema 1 y$$2^{\prime}$$ para$$f: S \rightarrow E$$.

## Ejercicio$$\PageIndex{*9}$$

Demostrar la fórmula (1) a partir de definiciones de la Parte II de esta sección.

## Ejercicio$$\PageIndex{10}$$

$$\Rightarrow 10$$. Mostrar que
\ [
\ izquierda|\ int_ {A} f\ derecha|\ leq\ int_ {A} |f|
\]
para mapas integrables$$f: S \rightarrow E .$$ Ver también Problema 14.
[Pista: Si$$m A<\infty,$$ usa Corolario$$1(\text { ii ) of } §4 \text { and Lemma } 1 . \text { If } m A=\infty,$$ nos imitamos” la prueba de Lemma$$3 .$$]

## Ejercicio$$\PageIndex{11}$$

Hacer el Problema 11 en §6 para$$f_{n}: S \rightarrow E .$$ Hazlo componentwise para$$E=$$$$E^{n}\left(C^{n}\right) .$$

## Ejercicio$$\PageIndex{12}$$

Mostrar que si$$f, g: S \rightarrow E^{1}(C)$$ son integrables en$$A,$$ entonces
\ [
\ izquierda|\ int_ {A} f g\ derecha|^ {2}\ leq\ int_ {A} |f|^ {2}\ cdot\ int_ {A} |g|^ {2}.
\] ¿
En qué caso se sostiene la igualdad? Deducir Teorema$$4\left(\mathrm{c}^{\prime}\right)$$ en el Capítulo$$3,$$ §§1-3, a partir de este resultado.
[Pista: Argumentar como en ese teorema. Considera el caso
\ [
\ izquierda. \ left (\ existe t\ en E^ {1}\ derecha)\ quad\ int_ {A} |f-t g|=0. \ derecho]
\]

## Ejercicio$$\PageIndex{13}$$

Mostrar que si$$f: S \rightarrow E^{1}(C)$$ es integrable on$$A$$ y
\ [
\ izquierda|\ int_ {A} f\ right|=\ int_ {A} |f|,
\]
entonces
\ [
(\ existe c\ in C)\ quad c f=|f|\ quad\ text {a.e. on} A.
\]
[Pista: Vamos $$a=\int_{A} f .$$El caso$$a=0$$ es trivial. Si se$$a \neq 0,$$ deja
\ [
c=\ frac {|a|} {a}; |c|=1; c a=|a|.
\]
Vamos$$r=(c f)_{\mathrm{re}} .$$ Mostrar eso$$r \leq|c f|=|f|$$,
\ [
\ begin {alineado}\ izquierda|\ int_ {A} f\ derecha| &=\ int_ {A} c f=\ int_ {A} r\ leq\ int_ {A} |f|=\ int_ {A} f\ derecha|,\\ &\ int_ {A} |f|=\ int_ {A} r=\ int_ {A} (c f) _ {\ mathrm {re,}}\ end {alineado}
\]
$$\left.(c f)_{\mathrm{re}}=|c f|(\mathrm{a.e.}), \text { and } c f=|c f|=|f| \text { a.e. on } A .\right]$$

## Ejercicio$$\PageIndex{14}$$

Hacer Problema 10 por$$E=C$$ usar el método de Problema$$13 .$$

## Ejercicio$$\PageIndex{15}$$

Mostrar que si$$f: S \rightarrow E$$ es integrable en$$A,$$ es integrable en cada$$\mathcal{M}$$ -set$$B \subseteq A .$$ Si, además,
\ [
\ int_ {B} f=0
\]
para todos esos$$B,$$ muestran que$$f=0$$ a.e. en$$A .$$ Probarlo por$$E=E^{n}$$ primera vez.
[Pista para$$\left.E=E^{*}: A=A(f>0) \cup A(f \leq 0) . \text { Use Theorems } 1(\mathrm{h}) \text { and } 2 \text { from } §5 .\right]$$

## Ejercicio$$\PageIndex{16}$$

En Problem$$15,$$ mostrar que
\ [
s=\ int f
\]
es una función de conjunto$$\sigma$$ -aditivo en

\ [\ mathcal {M} _ {A} =\ {X\ in\ mathcal {M} | X\ subseteq A\}.
\]
(Nota$$4 \text { in } §5) ; s$$ se llama la integral indefinida de$$f$$ in$$A .$$

8.7.E: Problemas en la Integración de Funciones Complejas y Vector-Valoradas is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.