Skip to main content

# 8.7.E: Problems on Integration of Complex and Vector-Valued Functions

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Exercise $$\PageIndex{1}$$

Prove Corollary $$1(\text { iii })-$$ (vii) in §4 componentwise for integrable maps $$f: S \rightarrow E^{n}\left(C^{n}\right) .$$

## Exercise $$\PageIndex{2}$$

Prove Theorems 2 and 3 componentwise for $$E=E^{n}\left(C^{n}\right)$$.

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

Do it for Corollary 3 in §6.

## Exercise $$\PageIndex{3}$$

Prove Theorem 1 with
$\int_{A}|f|<\infty$
replaced by
$\int_{A}\left|f_{k}\right|<\infty, \quad k=1, \ldots, n .$

## Exercise $$\PageIndex{4}$$

Prove that if $$f: S \rightarrow E^{n}\left(C^{n}\right)$$ is integrable on $$A,$$ so is $$|f| .$$ Disprove the converse.

## Exercise $$\PageIndex{5}$$

Disprove Lemma 1 for $$m A=\infty$$.

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

Complete the proof of Lemma 3.

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

Complete the proof of Theorem 3.

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

Do Problem 1 and $$2^{\prime}$$ for $$f: S \rightarrow E$$.

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

Prove formula (1) from definitions of Part II of this section.

## Exercise $$\PageIndex{10}$$

$$\Rightarrow 10$$. Show that
$\left|\int_{A} f\right| \leq \int_{A}|f|$
for integrable maps $$f: S \rightarrow E .$$ See also Problem 14.
[Hint: If $$m A<\infty,$$ use Corollary $$1(\text { ii ) of } §4 \text { and Lemma } 1 . \text { If } m A=\infty,$$ us imitate" the proof of Lemma $$3 .$$ ]

## Exercise $$\PageIndex{11}$$

Do Problem 11 in §6 for $$f_{n}: S \rightarrow E .$$ Do it componentwise for $$E=$$ $$E^{n}\left(C^{n}\right) .$$

## Exercise $$\PageIndex{12}$$

Show that if $$f, g: S \rightarrow E^{1}(C)$$ are integrable on $$A,$$ then
$\left|\int_{A} f g\right|^{2} \leq \int_{A}|f|^{2} \cdot \int_{A}|g|^{2} .$
In what case does equality hold? Deduce Theorem $$4\left(\mathrm{c}^{\prime}\right)$$ in Chapter $$3,$$ §§1-3, from this result.
[Hint: Argue as in that theorem. Consider the case
$\left.\left(\exists t \in E^{1}\right) \quad \int_{A}|f-t g|=0 .\right]$

## Exercise $$\PageIndex{13}$$

Show that if $$f: S \rightarrow E^{1}(C)$$ is integrable on $$A$$ and
$\left|\int_{A} f\right|=\int_{A}|f| ,$
then
$(\exists c \in C) \quad c f=|f| \quad \text { a.e. on } A.$
[Hint: Let $$a=\int_{A} f .$$ The case $$a=0$$ is trivial. If $$a \neq 0,$$ let
$c=\frac{|a|}{a} ;|c|=1 ; c a=|a| .$
Let $$r=(c f)_{\mathrm{re}} .$$ Show that $$r \leq|c f|=|f|$$,
\begin{aligned}\left|\int_{A} f\right| &=\int_{A} c f=\int_{A} r \leq \int_{A}|f|=\left|\int_{A} f\right| , \\ & \int_{A}|f|=\int_{A} r=\int_{A} (c f)_{\mathrm{re,}} \end{aligned}
$$\left.(c f)_{\mathrm{re}}=|c f|(\mathrm{a.e.}), \text { and } c f=|c f|=|f| \text { a.e. on } A .\right]$$

## Exercise $$\PageIndex{14}$$

Do Problem 10 for $$E=C$$ using the method of Problem $$13 .$$

## Exercise $$\PageIndex{15}$$

Show that if $$f: S \rightarrow E$$ is integrable on $$A,$$ it is integrable on each $$\mathcal{M}$$-set $$B \subseteq A .$$ If, in addition,
$\int_{B} f=0$
for all such $$B,$$ show that $$f=0$$ a.e. on $$A .$$ Prove it for $$E=E^{n}$$ first.
[Hint for $$\left.E=E^{*}: A=A(f>0) \cup A(f \leq 0) . \text { Use Theorems } 1(\mathrm{h}) \text { and } 2 \text { from } §5 .\right]$$

## Exercise $$\PageIndex{16}$$

In Problem $$15,$$ show that
$s=\int f$
is a $$\sigma$$-additive set function on
$\mathcal{M}_{A}=\{X \in \mathcal{M} | X \subseteq A\} .$
(Note $$4 \text { in } §5) ; s$$ is called the indefinite integral of $$f$$ in $$A .$$

8.7.E: Problems on Integration of Complex and Vector-Valued Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

• Was this article helpful?