Math Problems Vault

Problem 1.8.1

Define

f(x) = \begin{cases} e^{-1 / x^{2}} & (x \neq 0) \\ 0 & (x = 0)\end{cases}

Prove that $f$ has derivatives of all orders at $x = 0$ , and that $f^{(n)}(0) = 0$ for $n = 1,2,3, \ldots$ .

?

Problem 1.8.2

Let $a_{i j}$ be the number in the $i$ th row and $j$ th column of the array so that

a_{i j} = \begin{cases} 0 & (i < j) \\ -1 & (i = j) \\ 2^{j-i} & (i > j)\end{cases}

Prove that

\sum _{i} \sum _{j} a_{i j} = -2, \quad \sum _{j} \sum _{i} a_{i j} = 0

?

Problem 1.8.3

Prove that

\sum _{i} \sum _{j} a_{i j} = \sum _{j} \sum _{i} a_{i j}

if $a_{i j} \geq 0$ for all $i$ and $j$ (the case $+\infty = +\infty$ may occur).

?

Problem 1.8.4

(a)(b)(c)(d)

Prove the following limit relations:

?

(a)

$\lim_{x \rightarrow 0} \frac{b^{x}-1}{x} = \log b \quad (b > 0)$ .

(b)

$\lim_{x \rightarrow 0} \frac{\log (1+x)}{x} = 1$ .

(c)

$\lim_{x \rightarrow 0}(1+x)^{1 / x} = e$ .

(d)

$\lim_{n \rightarrow \infty }\left(1+\frac{x}{n}\right)^{n} = e^{x}$ .

Problem 1.8.5

(a)(b)(c)(d)

Find the following limits

?

(a)

$\lim_{x \rightarrow 0} \frac{e-(1+x)^{1 / x}}{x}$ .

(b)

$\lim_{n \rightarrow \infty } \frac{n}{\log n}\left[n^{1 / n}-1\right]$ .

(c)

$\lim_{x \rightarrow 0} \frac{\tan x-x}{x(1-\cos x)}$ .

(d)

$\lim_{x \rightarrow 0} \frac{x-\sin x}{\tan x-x}$ .

Problem 1.8.6

(a)(b)

Suppose $f(x) f(y) = f(x+y)$ for all real $x$ and $y$ .

?

(a)

Assuming that $f$ is differentiable and not zero, prove that

f(x) = e^{c x}

where $c$ is a constant.

(b)

Prove the same thing, assuming only that $f$ is continuous.

Problem 1.8.7

If $0 < x < \frac{\pi }{2}$ , prove that

\frac{2}{\pi } < \frac{\sin x}{x} < 1

?

Problem 1.8.8

For $n = 0,1,2, \ldots$ , and $x$ real, prove that

\left|\sin n x\right| \leq n\left|\sin x\right|

Note that this inequality may be false for other values of $n$ . For instance,

\left|\sin \frac{1}{2} \pi \right| > \frac{1}{2}\left|\sin \pi \right|

?

Problem 1.8.9

(a)(b)

?

(a)

Put $s_{N} = 1+\left(\frac{1}{2}\right)+\cdots +(1 / N)$ . Prove that

\lim _{N \rightarrow \infty }\left(s_{N}-\log N\right)

exists. (The limit, often denoted by $\gamma$ , is called Euler's constant. Its numerical value is $0.5772 \ldots$ It is not known whether $\gamma$ is rational or not).

(b)

Roughly how large must $m$ be so that $N = 10^{m}$ satisfies $s_{N} > 100$ ?

Problem 1.8.10

Prove that $\sum 1 / p$ diverges; the sum extends over all primes. (This shows that the primes form a fairly substantial subset of the positive integers).

Hint: Given $N$ , let $p_{1}, \ldots , p_{k}$ be those primes that divide at least one integer $\leq N$ . Then

\begin{aligned} \sum _{n = 1}^{N} \frac{1}{n} & \leq \prod _{j = 1}^{k}\left(1+\frac{1}{p_{j}}+\frac{1}{p_{j}^{2}}+\cdots \right) \\ & = \prod _{j = 1}^{k}\left(1-\frac{1}{p_{j}}\right)^{-1} \\ & \leq \exp \sum _{j = 1}^{k} \frac{2}{p_{j}} \end{aligned}

The last inequality holds bocause

(1-x)^{-1} \leq e^{2 x}

if $0 \leq x \leq \frac{1}{2}$ . (There are many proofs of this result. See, for instance, the article by I. Niven in Amer. Math. Monthly, vol. 78, 1971, pp. 272-273, and the one by R. Bellman in Amer. Math. Monthly, vol. 50, 1943, pp. 318-319).

?

Problem 1.8.11

Suppose $f \in \mathscr {R}$ on $[0, A]$ for all $A < \infty$ , and $f(x) \rightarrow 1$ as $x \rightarrow +\infty$ . Prove that

\lim _{t \rightarrow 0} t \int _{0}^{\infty } e^{-t x} f(x) d x = 1 \quad (t > 0)

?

Problem 1.8.12

Suppose $0 < \delta < \pi , f(x) = 1$ if $\left|x\right| \leq \delta , f(x) = 0$ if $\delta < \left|x\right| \leq \pi$ , and $f(x+2 \pi ) = f(x)$ for all $x$ .

?

(a)

Compute the Fourier coefficients of $f$ .

(b)

Conclude that

\sum _{n = 1}^{\infty } \frac{\sin (n \delta )}{n} = \frac{\pi -\delta }{2} \quad (0 < \delta < \pi )

(c)

Deduce from Parseval's theorem that

\sum _{n = 1}^{\infty } \frac{\sin ^{2}(n \delta )}{n^{2} \delta } = \frac{\pi -\delta }{2}

(d)

Let $\delta \rightarrow 0$ and prove that

\int _{0}^{\infty }\left(\frac{\sin x}{x}\right)^{2} d x = \frac{\pi }{2}

(e)

Put $\delta = \pi / 2$ in (c). What do you get?

Problem 1.8.13

Put $f(x) = x$ if $0 \leq x < 2 \pi$ , and apply Parseval's theorem to conclude that

\sum _{n = 1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}

?

Problem 1.8.14

If $f(x) = (\pi -\left|x\right|)^{2}$ on $[-\pi , \pi ]$ , prove that

f(x) = \frac{\pi ^{2}}{3}+\sum _{n = 1}^{\infty } \frac{4}{n^{2}} \cos n x

and deduce that

\sum _{n = 1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}, \quad \sum _{n = 1}^{\infty } \frac{1}{n^{4}} = \frac{\pi ^{4}}{90}

(A recent article by E. L. Stark contains many references to series of the form $\sum n^{-s}$ , where $s$ is a positive integer. See Math. Mag., vol. 47, 1974, pp. 197-202).

?

Problem 1.8.15

(a)(b)(c)

With $D_{n}$ as defined in (77), put

K_{N}(x) = \frac{1}{N+1} \sum _{n = 0}^{N} D_{n}(x)

Prove that

K_{N}(x) = \frac{1}{N+1} \cdot \frac{1-\cos (N+1) x}{1-\cos x}

and that

?

(a)

$K_{N} \geq 0$

(b)

$\frac{1}{2 \pi } \int_{-\pi }^{\pi } K_{N}(x) d x = 1$ ,

(c)

$K_{N}(x) \leq \frac{1}{N+1} \cdot \frac{2}{1-\cos \delta }$ if $0 < \delta \leq \left|x\right| \leq \pi$

If $s_{N} = s_{N}(f ; x)$ is the $N$ th partial sum of the Fourier series of $f$ , consider the arithmetic means

\sigma _{N} = \frac{s_{0}+s_{1}+\cdots +s_{N}}{N+1}

Prove that

\sigma _{N}(f ; x) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } f(x-t) K_{N}(t) d t

and hence prove Fejér's theorem: If $f$ is continuous, with period $2 \pi$ , then $\sigma_{N}(f ; x) \rightarrow f(x)$ uniformly on $[-\pi , \pi ]$ . Hint: Use properties $(a),(b),(c)$ to proceed as in Theorem 7.26.

Problem 1.8.16

Prove a pointwise version of Fejér's theorem:

If $f \in \mathscr {R}$ and $f(x+), f(x-)$ exist for some $x$ , then

\lim _{N \rightarrow \infty } \sigma _{N}(f ; x) = \frac{1}{2}[f(x+)+f(x-)]

?

Problem 1.8.17

(a)(b)(c)

Assume $f$ is bounded and monotonic on $[-\pi , \pi )$ , with Fourier coefficients $c_{n}$ , as given by (62).

?

(a)

Use Exercise 17 of Chap. 6 to prove that $\left\{ n c_{n}\right\}$ is a bounded sequence.

(b)

Combine (a) with Exercise 16 and with Exercise 14(e) of Chap. 3, to conclude that

\lim _{N \rightarrow \infty } s_{N}(f ; x) = \frac{1}{2}[f(x+)+f(x-)]

for every $x$ .

(c)

Assume only that $f \in \mathscr {R}$ on $[-\pi , \pi ]$ and that $f$ is monotonic in some segment $(\alpha , \beta ) \subset [-\pi , \pi ]$ . Prove that the conclusion of $(b)$ holds for every $x \in (\alpha , \beta )$ . (This is an application of the localization theorem).

Problem 1.8.18

Define

\begin{aligned} & f(x) = x^{3}-\sin ^{2} x \tan x \\ & g(x) = 2 x^{2}-\sin ^{2} x-x \tan x \end{aligned}

Find out, for each of these two functions, whether it is positive or negative for all $x \in (0, \pi / 2)$ , or whether it changes sign. Prove your answer.

?

Problem 1.8.19

Suppose $f$ is a continuous function on $\mathbb {R}^{1}, f(x+2 \pi ) = f(x)$ , and $\alpha / \pi$ is irrational. Prove that

\lim _{N \rightarrow \infty } \frac{1}{N} \sum _{n = 1}^{N} f(x+n \alpha ) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } f(t) d t

for every $x$ . Hint: Do it first for $f(x) = e^{i k x}$ .

?

Problem 1.8.20

The following simple computation yields a good approximation to Stirling's formula.

For $m = 1,2,3, \ldots$ , define

f(x) = (m+1-x) \log m+(x-m) \log (m+1)

if $m \leq x \leq m+1$ , and define

g(x) = \frac{x}{m}-1+\log m

if $m-\frac{1}{2} \leq x < m+\frac{1}{2}$ . Draw the graphs of $f$ and $g$ . Note that $f(x) \leq \log x \leq g(x)$ if $x \geq 1$ and that

\int _{1}^{n} f(x) d x = \log (n!)-\frac{1}{2} \log n > -\frac{1}{8}+\int _{1}^{n} g(x) d x

Integrate $\log x$ over $[1, n]$ . Conclude that

\frac{7}{8} < \log (n!)-\left(n+\frac{1}{2}\right) \log n+n < 1

for $n = 2,3,4, \ldots$ (Note: $\log \sqrt{2 \pi } \sim 0.918 \ldots$ ) Thus

e^{7 / 8} < \frac{n!}{(n / e)^{n} \sqrt{n}} < e

?

Problem 1.8.21

Let

L_{n} = \frac{1}{2 \pi } \int _{-\pi }^{\pi }\left|D_{n}(t)\right| d t \quad (n = 1,2,3, \ldots )

Prove that there exists a constant $C > 0$ such that

L_{n} > C \log n \quad (n = 1,2,3, \ldots )

or, more precisely, that the sequence

\left\{ L_{n}-\frac{4}{\pi ^{2}} \log n\right\}

is bounded.

?

Problem 1.8.22

If $\alpha$ is real and $-1 < x < 1$ , prove Newton's binomial theorem

(1+x)^{\alpha } = 1+\sum _{n = 1}^{\infty } \frac{\alpha (\alpha -1) \cdots (\alpha -n+1)}{n!} x^{n}

Hint: Denote the right side by $f(x)$ . Prove that the series converges. Prove that

(1+x) f^{\prime }(x) = \alpha f(x)

and solve this differential equation. Show also that

(1-x)^{-\alpha } = \sum _{n = 0}^{\infty } \frac{\Gamma (n+\alpha )}{n!\Gamma (\alpha )} x^{n}

if $-1 < x < 1$ and $\alpha > 0$ .

?

Problem 1.8.23

Let $\gamma$ be a continuously differentiable closed curve in the complex plane, with parameter interval $[a, b]$ , and assume that $\gamma (t) \neq 0$ for every $t \in [a, b]$ . Define the index of $\gamma$ to be

\operatorname {Ind}(\gamma ) = \frac{1}{2 \pi i} \int _{a}^{b} \frac{\gamma ^{\prime }(t)}{\gamma (t)} d t

Prove that Ind $(\gamma )$ is always an integer. Hint: There exists $\varphi$ on $[a, b]$ with $\varphi^{\prime } = \gamma^{\prime } / \gamma , \varphi (a) = 0$ . Hence $\gamma \exp (-\varphi )$ is constant. Since $\gamma (a) = \gamma (b)$ it follows that $\exp \varphi (b) = \exp \varphi (a) = 1$ . Note that $\varphi (b) = 2 \pi i$ Ind $(\gamma )$ .

Compute Ind $(\gamma )$ when $\gamma (t) = e^{i n t}, a = 0, b = 2 \pi$ . Explain why Ind $(\gamma )$ is often called the winding number of $\gamma$ around 0.

?

Problem 1.8.24

Let $\gamma$ be as in Exercise 23, and assume in addition that the range of $\gamma$ does not intersect the negative real axis. Prove that Ind $(\gamma ) = 0$ . Hint: For $0 \leq c < \infty$ , Ind $(\gamma +c)$ is a continuous integer-valued function of $c$ . Also, Ind $(\gamma +c) \rightarrow 0$ as $c \rightarrow \infty$ .

?

Problem 1.8.25

Suppose $\gamma_{1}$ and $\gamma_{2}$ are curves as in Exercise 23, and

\left|\gamma _{1}(t)-\gamma _{2}(t)\right| < \left|\gamma _{1}(t)\right| \quad (a \leq t \leq b)

Prove that Ind $\left(\gamma_{1}\right) =$ Ind $\left(\gamma_{2}\right)$ . Hint: Put $\gamma = \gamma_{2} / \gamma_{1}$ . Then $\left|1-\gamma \right| < 1$ , hence Ind $(\gamma ) = 0$ , by Exercise 24. Also,

\frac{\gamma ^{\prime }}{\gamma } = \frac{\gamma _{2}^{\prime }}{\gamma _{2}}-\frac{\gamma _{1}^{\prime }}{\gamma _{1}}

?

Problem 1.8.26

Let $\gamma$ be a closed curve in the complex plane (not necessarily diffierentiable) with parameter interval $[0,2 \pi ]$ , such that $\gamma (t) \neq 0$ for every $t \in [0,2 \pi ]$ .

Choose $\delta > 0$ so that $\left|\gamma (t)\right| > \delta$ for all $t \in [0,2 \pi ]$ . If $P_{1}$ and $P_{2}$ are trigonometric polynomials such that $\left|P_{\jmath }(t)-\gamma (t)\right| < \delta / 4$ for all $t \in [0,2 \pi ]$ (their existence is assured by Theorem 8.15), prove that

\operatorname {Ind}\left(P_{1}\right) = \operatorname {Ind}\left(P_{2}\right)

by applying Exercise 25. Define this common value to be Ind $(\gamma )$ . Prove that the statements of Exercises 24 and 25 hold without any differentiability assumption.

?

Problem 1.8.27

(a)(b)(c)

Let $f$ be a continuous complex function defined in the complex plane. Suppose there is a positive integer $n$ and a complex number $c \neq 0$ such that

\lim _{\left|z\right| \rightarrow \infty } z^{-n} f(z) = c

Prove that $f(z) = 0$ for at least one complex number $z$ . Note that this is a generalization of Theorem 8.8. Hint: Assume $f(z) \neq 0$ for all $z$ , define

\gamma _{r}(t) = f\left(r e^{t t}\right)

for $0 \leq r < \infty , 0 \leq t \leq 2 \pi$ , and prove the following statements about the curves $\gamma_{r}$ :

?

(a)

Ind $\left(\gamma_{0}\right) = 0$

(b)

Ind $\left(\gamma_{r}\right) = n$ for all sufficiently large $r$ .

(c)

Ind $\left(\gamma_{r}\right)$ is a continuous function of $r$ , on [0, $\infty$ ).

Show that $(a),(b)$ , and $(c)$ are contradictory, since $n > 0$ .

Problem 1.8.28

Let $D$ be the closed unit disc in the complex plane. (Thus $z \in D$ if and only if $\left|z\right| \leq 1$ ). Let $g$ be a continuous mapping of $D$ into the unit circle $T$ . (Thus, $\left|g(z)\right| = 1$ for every $z \in D$ ).

Prove that $g(z) = -z$ for at least one $z \in T$ . Hint: For $0 \leq r \leq 1,0 \leq t \leq 2 \pi$ , put

\gamma _{r}(t) = g\left(r e^{t t}\right)

and put $\psi (t) = e^{-t t} \gamma_{1}(t)$ . If $g(z) \neq -z$ for every $z \in T$ , then $\psi (t) \neq -1$ for every $t \in [0,2 \pi ]$ . Hence Ind $(\psi ) = 0$ , by Exercises 24 and 26. It follows that Ind $\left(\gamma_{1}\right) = 1$ . But Ind $\left(\gamma_{0}\right) = 0$ . Derive a contradiction, as in Exercise 27.

?

Problem 1.8.29

Prove that every continuous mapping $f$ of $D$ into $D$ has a fixed point in $D$ . (This is the 2-dimensional case of Brouwer's fixed-point theorem). Hint: Assume $f(z) \neq z$ for every $z \in D$ . Associate to each $z \in D$ the point $g(z) \in T$ which lies on the ray that starts at $f(z)$ and passes through $z$ . Then $g$ maps $D$ into $T, g(z) = z$ if $z \in T$ , and $g$ is continuous, because

g(z) = z-s(z)[f(z)-z]

where $s(z)$ is the unique nonnegative root of a certain quadratic equation whose coefficients are continuous functions of $f$ and $z$ . Apply Exercise 28.

?

Problem 1.8.30

Use Stirling's formula to prove that

\lim _{x \rightarrow \infty } \frac{\Gamma (x+c)}{x^{c} \Gamma (x)} = 1

for every real constant $c$ .

?

Problem 1.8.31

In the proof of Theorem 7.26 it was shown that

\int _{-1}^{1}\left(1-x^{2}\right)^{n} d x \geq \frac{4}{3 \sqrt{n}}

for $n = 1,2,3, \ldots$ . Use Theorem 8.20 and Exercise 30 to show the more precise result

\lim _{n \rightarrow \infty } \sqrt{n} \int _{-1}^{1}\left(1-x^{2}\right)^{n} d x = \sqrt{\pi }

?