9.3.E: Problems on Cauchy Integrals
Fill in all proof details in Theorems \(1-3 .\) Verify also at least some of the cases other than \(\int_{a}^{\infty} f .\) Check the validity for \(L S\)-integrals (footnote 6).
Prove Theorem 4 in detail.
Verify Notes 2 and 3 and examples (A)-(D).
Assuming \(a>0,\) verify the following:
(i) \(\int_{1}^{\infty} \frac{1}{t} e^{-t} d t \leq \int_{1}^{\infty} e^{-t} d t=\frac{1}{e}\).
[Hint: Use Corollary 2.]
(ii) \(\int_{1}^{\infty} e^{-a t} d t=\frac{e^{-a}}{a}\).
(iii) \(\int_{0}^{\infty} e^{-a t} d t=\frac{1}{a}\).
(iv) \(\int_{0}^{\infty} e^{-a t} \sin b t d t=\frac{b}{a^{2}+b^{2}}\).
Verify the following:
(i) \(\int_{1}^{\infty} \int_{1}^{\infty} e^{-x y} d y d x=\int_{1}^{\infty} \frac{1}{x} e^{-x} d x \leq \frac{1}{e}(\text { converges, by } 3(\mathrm{i}))\).
(ii) \(\int_{0}^{\infty} \int_{0}^{\infty} e^{-x y} d y d x \geq \int_{1}^{\infty} \int_{0}^{\infty} e^{-x y} d y d x=\int_{1}^{\infty} \frac{1}{x}\left(1-e^{-x}\right) d x \geq \int_{1}^{\infty}\left(\frac{1}{x}-e^{-x}\right) d x=\infty\).
Does this contradict formula (4) in the text, or Problem 5, which follows?
Let \(f(x, y)=e^{-x y}\) and
\[
g(x)=L \int_{0}^{1} e^{-x y} d y ;
\]
so \(g(0)=1 .\) (Why?)
(i) Is \(g\) R-integrable on \(A=[0,1] ?\) Is \(f\) so on \(A \times A ?\)
(ii) Find \(g(x)\) using Corollary 1 in §1.
(iii) Find the value of
\[
R \int_{0}^{1} \int_{0}^{1} e^{-x y} d y d x=R \int_{0}^{1} g
\]
to within \(1 / 10 .\)
[Hint: Reduce it to Problem \(6(\mathrm{b}) \text { in } §1.]\)
\(\Rightarrow 6\). Let \(f, g: E^{1} \rightarrow E^{*}\) be \(m\)-measurable on \(A=[a, b), b \leq \infty .\) Prove the following:
(i) If
\[
C \int_{a}^{b-} f^{+}<\infty \text { or } C \int_{a}^{b-} f^{-}<\infty ,
\]
then \(C \int_{a}^{b-} f\) exists and equals
\[
C \int_{a}^{b-} f^{+}-C \int_{a}^{b-} f^{-}=\int_{A} f d m(\text { proper }) .
\]
(ii) If \(\int_{a}^{b-} f\) converges conditionally only, then
\[
\int_{a}^{b-} f^{+}=\int_{a}^{b-} f^{-}=+\infty .
\]
(iii) In case \(C \int_{a}^{b-}|f|<\infty,\) we have
\[
C \int_{a}^{b-}|f \pm g|=\infty
\]
iff \(C \int_{a}^{b-}|g|=\infty ;\) also,
\[
C \int_{a}^{b-}(f \pm g)=C \int_{a}^{b-} f \pm C \int_{a}^{b-} g
\]
if \(C \int_{a}^{b-} g\) exists (finite or not).
\(\Rightarrow 7\). Suppose \(f: E^{1} \rightarrow E^{*}\) is \(m\)-integrable and sign-constant on each
\[
A_{n}=\left[a_{n}, a_{n+1}\right), \quad n=1,2, \ldots
\]
but changes sign from \(A_{n}\) to \(A_{n+1},\) with
\[
\bigcup_{n=1}^{\infty} A_{n}=[a, \infty)
\]
and \(\left\{a_{n}\right\} \uparrow\) fixed.
Prove that if
\[
\left|\int_{A_{n}} f d m\right| \searrow 0
\]
as \(n \rightarrow \infty,\) then
\[
c \int_{a}^{\infty} f
\]
converges.
[Hint: Use Problem 10 in Chapter 4, §13.]
\(\Rightarrow 8\). Let
\[
f(x)=\frac{\sin x}{x}, \quad f(0)=1 .
\]
Prove that
\[
C \int_{0}^{\infty} f(x) d x
\]
converges conditionally only.
[Hints: Use Problem 7. Show that
\[
\left.C \int_{0}^{\infty}|f|=L \int_{(0, \infty)}|f|=L \int_{0}^{\infty} f^{+}=L \int_{0}^{\infty} f^{-}=\infty .\right]
\]
\(\Rightarrow 9\). (Additivity.) Given \(f: E^{1} \rightarrow E(E \text { complete) and } a<b<c \leq \infty\), suppose that
\[
\int_{a}^{x} f d m \neq \pm \infty
\]
(proper) exists for each \(x \in[a, c) .\) Prove the following:
(a) \(C \int_{a}^{b-} f\) and \(C \int_{a+}^{b} f\) converge.
(b) If
\[
C \int_{b}^{c-} f
\]
converges, so does
\[
C \int_{a}^{c-} f=C \int_{a}^{b-} f+C \int_{b}^{c-} f .
\]
(c) Countable additivity does not necessarily hold for C-integrals.
[Hint: Use Problem 8 suitably splitting \([0,\infty)\).]
(Refined comparison test.) Given \(f, g: E^{1} \rightarrow E (E \text { complete) and } b \leq \infty,\) prove the following:
(i) If for some \(a<b\) and \(k \in E^{1}\),
\[
|f| \leq|k g| \quad \text { on }[a, b)
\]
then
\[
\int_{a}^{b-}|g|<\infty \text { implies } \int_{a}^{b-}|f|<\infty .
\]
(ii) Such \(a, k \in E^{1}\) do exist if
\[
\lim _{t \rightarrow b-} \frac{|f(t)|}{|g(t)|}<\infty
\]
exists.
(iii) If this limit is not zero, then
\[
\int_{a}^{b-}|g|<\infty \text { iff } \int_{a}^{b-}|f|<\infty .
\]
(Similarly in the case of \(\left.\int_{a+}^{b} \text { with } a \geq-\infty .\right)\)
Prove that
(I) \(\int_{1}^{\infty} t^{p} d t<\infty\) iff \(p<-1\);
(ii) \(\int_{0+}^{1} t^{p} d t<\infty\) iff \(p>-1\);
(iii) \(\int_{0+}^{\infty} t^{p} d t=\infty\).
Use Problems 10 and 11 to test for convergence of the following:
(a) \(\int_{0}^{\infty} \frac{t^{3 / 2} d t}{1+t^{2}}\);
(b) \(\int_{1}^{\infty} \frac{d t}{t \sqrt{1+t^{2}}}\);
(c) \(\int_{a}^{\infty} \frac{P(t)}{Q(t)} d t\)
\((Q, P \text { polynomials of degree } s \text { and } r, s>r ; Q \neq 0 \text { for } t \geq a)\);
(d) \(\int_{0}^{1-} \frac{d t}{\sqrt{1-t^{4}}}\);
(e) \(\int_{0+}^{1} t^{p} \ln t d t\);
(f) \(\int_{0}^{1-} \frac{d t}{\ln t}\);
(g) \(\int_{0+}^{\frac{\pi}{2}-} \tan ^{p} t d t\).
\(\Rightarrow 13\). (The Abel-Dirichlet test.) Given \(f, g: E^{1} \rightarrow E^{1},\) suppose that
(a) \(f \downarrow,\) with \(\lim _{t \rightarrow \infty} f(t)=0\);
(b) \(g\) is L-measurable on \(A=[a, \infty) ; \) and;
(c) \(\left(\exists K \in E^{1}\right)(\forall x \in A) \quad\left|L \int_{a}^{x} g\right|<K\).
Then \(C \int_{a}^{\infty} f(x) g(x) d x\) converges.
[Outline: Set
\[
G(x)=\int_{a}^{x} g ;
\]
so \(|G|<K\) on \(A .\) By Lemma 2 of §1, \(f g\) is L-integrable on each \([u, v] \subset A,\) and \((\exists c \in[u, v])\) such that
\[
\left|L \int_{u}^{v} f g\right|=\left|f(u) \int_{u}^{c} g\right|=|f(u)[G(c)-G(u)]|<2 K f(u) .
\]
Now, by \((\mathrm{a})\),
\[
(\forall \varepsilon>0)(\exists k \in A)(\forall u \geq k) \quad|f(u)|<\frac{\varepsilon}{2 K} ;
\]
so
\[
(\forall v \geq u \geq k) \quad\left|L \int_{u}^{v} f g\right|<\varepsilon .
\]
Now use Theorem \(2 .\)
\(\quad\) Now extend this to \(g: E^{1} \rightarrow E^{n}\left(C^{n}\right) .\)]
\(\Rightarrow 14\). Do Problem \(13,\) replacing assumptions (a) and (c) by
(a') \(f\) is monotone and bounded on \([a, \infty)=A,\) and
(c') \(C \int_{a}^{\infty} g(x) d x\) converges.
[Hint: If \(f \uparrow,\) say, set \(q=\lim _{t \rightarrow \infty} f(t)\) and \(F=q-f ;\) so
\[
f g=q g-F g .
\]
Apply Problem 13 to
\[
\left.C \int_{a}^{\infty} F(x) g(x) d x .\right]
\]
Use Problems 13 and 14 to test the convergence of the following:
(a) \(\int_{0}^{\infty} t^{p} \sin t d t\).
[Hint: The integral converges iff \(p<0 .]\)
(b) \(\int_{0+}^{\infty} \frac{\cos t}{\sqrt{t}} d t\).
\(\left[\text { Hint: Integrate } \int_{u}^{v} \frac{\cos t}{\sqrt{t}} d t \text { by parts; then let } u \rightarrow 0 \text { and } v \rightarrow \infty .\right]\)
(c) \(\int_{1}^{\infty} \frac{\cos t}{t^{p}} d t\).
(d) \(\int_{0}^{\infty} \sin t^{2} d t\).
\(\left[\text { Hint: Substitute } t^{2}=u ; \text { then use }(\mathrm{a}) .\right]\)
The Cauchy principal value \((\mathrm{CPV})\) of \(C \int_{-\infty}^{\infty} f(t) d t\) is defined by
\[
(\mathrm{CPV}) \int_{-\infty}^{\infty} f=\lim _{x \rightarrow \infty} \int_{-x}^{x} f(t) d t
\]
(if it exists). Prove the following:
(i) If \(C \int f(t) d t\) exists, so does \((\mathrm{CPV}) \int f,\) and the two are equal.
Disprove the converse.
[Hint: Take \(f(t)=\operatorname{sign}(t) / \sqrt{|t|} .]\)
(ii) Do the same for
\[
(\mathrm{CPV}) \int_{a}^{b} f=\lim _{\delta \rightarrow 0+}\left(\int_{a}^{p-\delta} f+\int_{p+\delta}^{b} f\right) ,
\]
\(p\) being the only singularity in \((a, b)\).