Chapter 22

Prove the theorem of Cauchy: If \(\lim _{x \rightarrow \infty} f(x+1)-f(x)=\infty\), then \(\lim _{x \rightarrow \infty} f(x) / x=\infty\).

Use the theorem of Exercise 1 and the theorem on p. 768 to show that $$ \lim _{x \rightarrow \infty} \frac{a^{x}}{x}=\infty \quad \text { and } \quad \lim _{x \rightarrow \infty} \frac{\log x}{x}=0. $$

Use the modern definition of continuity and Cauchy's trigonometric identity $$ \sin (x+\alpha)-\sin x=2 \sin \frac{1}{2} \alpha \cos \left(x+\frac{1}{2} \alpha\right) $$ to show that \(\sin x\) is continuous at any value of \(x\).

Prove the following theorem of Cauchy: If \(f(x)\) is positive for sufficiently large values of \(x\) and if the ratio \(f(x+\) 1) \(/ f(x)\) converges to \(k\) as \(x\) increases indefinitely, then \([f(x)]^{1 / x}\) also converges to \(k\) as \(x\) increases indefinitely.

Use the Cauchy criterion to show that the series $$ 1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots $$ converges.

Show that if the sequence \(\left\\{a_{i}\right\\}\) converges to \(a\) and if \(f\) is continuous, then the sequence \(\left\\{f\left(a_{i}\right)\right\\}\) converges to \(f(a)\).

Show that the series \(\left\\{u_{k}(x)\right\\}\), where \(u_{1}(x)=x\) and \(u_{k}(x)=\) \(x^{k}-x^{k-1}\) for \(k>1\), satisfies the hypotheses of Cauchy's theorem \(6-1-1\) in a neighborhood of \(x=1\) but not the conclusion. Analyze Cauchy's proof for this case to see where it fails.

By putting \(a^{i}=1+\beta\), use Cauchy's definition of derivative to show that the derivative of \(y=a^{x}\) is \(y^{\prime}=a^{x} / \log _{a}(e)\).

Show that if \(f(x)\) is continuous on \([a, b]\) and if \(a=x_{0}<\) \(x_{1}<\cdots

Let \(f(x)=x^{2}+3 x\) on \([1,3]\). Partition \([1,3]\) into eight subintervals and determine the \(\theta\) that satisfies the property of Exercise \(13 .\)

Suppose \(M\) is the property that \(x<0\) or that \(x^{3}<3\). Since this property does not belong to all \(x\) but does belong to all \(x\) less than 1 , this property satisfies the conditions of Bolzano's least upper bound theorem. Beginning with the quantity \(V=1+1\), for which \(M\) is not valid for all \(x\) smaller than it, use Bolzano's proof method to construct an approximation to \(\sqrt[3]{3}\) accurate to three decimal places.

Show that \(\phi(x)=\alpha e^{m x}, \psi(y)=\beta \cos n y\), with \(m^{2}=n^{2}=\) \(\frac{1}{A}\), are solutions to $$ \frac{\phi(x)}{\phi^{\prime \prime}(x)}=-\frac{\psi(y)}{\psi^{\prime \prime}(y)}=A $$ Conclude that \(v=a e^{-n x} \cos n y\) is a solution to $$ \frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=0. $$

Show that $$ \int_{0}^{\pi} \sin m x \sin n x d x= \begin{cases}0, & \text { if } m \neq n; \\\ \frac{\pi}{2}, & \text { if } m=n.\end{cases} $$

Use Fourier's method of integration to calculate the Fourier series for \(\phi(x)=\frac{1}{2} x\) quoted in the text. Check the correctness of this result for \(x=\frac{\pi}{2}\) by using other known series sums.

Prove Cauchy's theorem on the continuity of the sum of a series of continuous functions under the additional assumption that the series converges uniformly.

Let $$ u_{k}=\frac{1}{k(k+1)} $$ and $$ v_{k}(h)=u_{k}+\frac{2 h}{((k-1) h+1)(k h+1)} $$ where \(h\) is a positive real variable. Show that \(\lim _{h \rightarrow 0} v_{k}(h)\) \(=u_{k}\), that \(\sum u_{k}\) converges, and that \(\sum v_{k}(h)\) converges for sufficiently small \(h\). Show also that $$ \lim _{h \rightarrow 0} \sum v_{k}(h) \neq \sum u_{k}. $$

Let \(\mathcal{R}\) be the set of all real numbers as defined by Dedekind via his cuts. Show that this set possesses the basic attribute of continuity. Namely, show that if \(\mathcal{R}\) is split into two classes \(\mathcal{A}_{1}, \mathcal{A}_{2}\) such that every real number in \(\mathcal{A}_{1}\) is less than every real number in \(\mathcal{A}_{2}\), then there exists exactly one real number \(\alpha\) that is either the greatest number in \(\mathcal{A}_{1}\) or the smallest number in \(\mathcal{A}_{2}\).

Define a natural ordering \(<\) on Dedekind's set of real numbers \(\mathcal{R}\) defined by the notion of cuts. That is, given two cuts \(\alpha=\left(\mathcal{A}_{1}, \mathcal{A}_{2}\right)\), and \(\beta=\left(\mathcal{B}_{1}, \mathcal{B}_{2}\right)\), define \(\alpha<\beta\). Show that this ordering \(<\) satisfies the same basic properties on \(\mathcal{R}\) as it satisfies on the set of rational numbers.

Define an addition on Dedekind's cuts. Show that \(\alpha+\beta=\) \(\beta+\alpha\) for any two cuts \(\alpha\) and \(\beta\).

Prove the theorem that every bounded increasing sequence of real numbers has a limit number, using Dedekind's cuts and also using Cantor's fundamental sequences. Which proof is easier?

Show that if \(\left\\{a_{i}\right\\}\) and \(\left\\{b_{i}\right\\}\) are fundamental sequences with \(\left\\{b_{i}\right\\}\) not defining the limit 0 , then \(\left\\{a_{i} / b_{i}\right\\}\) is also a fundamental sequence.

Define the product \(A B\) of two fundamental sequences \(A=\) \(\left\\{a_{i}\right\\}\) and \(B=\left\\{b_{i}\right\\}\) as the sequence consisting of the products \(\left\\{a_{i} b_{i}\right\\}\). Show that this definition makes sense and that if \(A B=C\), then \(B=C / A\), where division is defined as in Exercise \(29 .\)

Determine explicitly a point set \(P\) whose first and second derived sets, \(P^{\prime}\), and \(P^{\prime \prime}\), are different from \(P\) and from each other.

Fill in all the details of the calculation using residues that shows that $$ \int_{-\infty}^{\infty} \frac{\cos x}{1+x^{2}} d x=\frac{\pi}{e}. $$

Show using residues that $$ \int_{0}^{\infty} \frac{d x}{1+x^{6}}=\frac{2 \pi}{3}. $$

Let the complex function \(w(z)\) be given as the \(\operatorname{sum} u(x, y)\) \(+i v(x, y)\). Suppose that the Cauchy-Riemann equations are satisfied, that is, that \(\partial u / \partial x=\partial v / \partial y\) and \(\partial v / \partial x=\) \(-\partial u / \partial y\). Show that the derivative \(d w / d z\) is equal to \(\partial u / \partial x\) \(+i \partial v / \partial x\).

Suppose a surface \(S\) is defined by three parametric equations \(x=x(p, q), y=y(p, q), z=z(p, q)\). Show geometrically that the element of surface \(d S\) can be written in the form $$ d S=\left[\left(\frac{\partial(y, z)}{\partial(p, q)}\right)^{2}+\left(\frac{\partial(z, y)}{\partial(p, q)}\right)^{2}+\left(\frac{\partial(x, y)}{\partial(p, q)}\right)^{2}\right]^{1 / 2} d p d q. $$

Show that if \(\sigma=A i+B j+C k\) is a vector field with \(\operatorname{div} \sigma=0\), then \(\sigma=\operatorname{curl} \tau\) for some vector field \(\tau\).

What does Cauchy mean by his statement that an irrational number is the limit of the various fractions that approach it? What does Cauchy understand by the term "irrational number" or, even, by the term "number"?

Explain the differences between Cauchy's definition of continuity on an interval and the usual modern definition of continuity at a point. Does a function that satisfies Cauchy's definition satisfy the modern one for every point in the interval? Does a function that satisfies the modern definition for every point in an interval satisfy Cauchy's definition?

Develop a lesson plan for teaching the concept of uniform convergence by beginning with Cauchy's incorrect theorem and proof.

Compare the Eulerian and the Riemannian derivations of the Cauchy-Riemann equations. Which makes a better introduction for a course in complex analysis?