Math Problems Vault

Problem VI.2.1

Let $f \in C\left(\left[0, \frac{\pi }{2}\right]\right)$ . Find the limit

\lim _{n \rightarrow +\infty } n \int _{0}^{\pi / 2} x f(x) \cos ^{n} x d x

?

Problem VI.2.2

(a)(b)

Let $c>0$ and $I_{c}=\int_{0}^{c} e^{-t^{2}} d t$ . Prove that as $A \rightarrow +\infty$

?

(a)

$\int_{0}^{c / \sqrt{A}}\left(1-x^{2}\right)^{A} d x \sim \frac{I_{c}}{\sqrt{A}}$ ;

(b)

$\int_{0}^{c / \sqrt{A}} \frac{d x}{\left(1+x^{2}\right)^{A}} \sim \frac{I_{c}}{\sqrt{A}}$ .

Problem VI.2.3

Let $c>0$ . Find the asymptotics as $A \rightarrow +\infty$ of the integral

\int _{0}^{c / \sqrt{A}} \cos ^{A} x d x

?

Problem VI.2.4

Verify that the asymptotics as $A \rightarrow +\infty$ of the following integrals do not depend on the parameter $c>0$ :

?

(a)

$\int_{0}^{c}\left(1-x^{2}\right)^{A} d x \quad (c \leqslant 1)$ ;

(b)

$\int_{0}^{c} \frac{d x}{\left(1+x^{2}\right)^{A}}$ ;

(c)

$\int_{0}^{c} \cos^{A} x d x \quad \left(c \leqslant \frac{\pi }{2}\right)$ ;

(d)

$\int_{0}^{c} \frac{d x}{\left(1+x+x^{2}\right)^{A}}$ ;

(e)

$\int_{0}^{c}\left(\frac{\ln (1+x)}{x}\right)^{A} d x$ ;

(f)

$\int_{0}^{c}\left(\frac{\sin x}{x}\right)^{A} d x \quad (c \leqslant \pi )$ ;

(g)

$\int_{1}^{1+c} x^{-A x} d x$ ;

(h)

$\int_{0}^{c} \ln^{A}(e-x) d x \quad (c \leqslant e-1)$ .

Remark.

?

A characteristic feature of Problems $2.1-2.4$ is the phenomenon of localization — the asymptotics of the integral depend on the behavior of the integrand only in a neighborhood of a single point. This effect manifests itself distinctly also in the more general case, in the study of the important integrals of the form

\Phi (A)=\int _{a}^{b} f(x) \varphi ^{A}(x) d x

where the functions $f$ and $\varphi$ are nonnegative and $\varphi$ is piecewise monotone. For large values of the parameter $A$ , the graph of the function $\varphi^{A}$ has sharply pronounced "humps" in neighborhoods of those points at which the function $\varphi$ has a strict local maximum. By splitting the interval $\left[a, b\right]$ into several intervals if necessary, we may assume that $\varphi$ is monotone on $\left[a, b\right]$ . In doing so it suffices to consider only the case where $\varphi$ is decreasing. Then the values of the function $\varphi^{A}$ at points far from the point $a$ are negligibly small compared with its values at points close to $a$ , which give the main contribution to the integral $\Phi (A)$ . To find the principal part of $\Phi (A)$ it remains to approximate the functions $f$ and $\varphi$ near the point $a$ by simpler ones and to compute the resulting integral. The implementation of the scheme just described constitutes the Laplace method for studying integrals of the form $\Phi (A)$ , as well as their modifications. The most frequently encountered case is one in which the difference $\varphi (a)-\varphi (x)$ is an infinitesimal of power type, i.e.,

\varphi (a)-\varphi (x) \underset {x \rightarrow a+0}{\sim } C(x-a)^{p} \quad (C, p>0) .

This is investigated in Problem 2.7. The result obtained in this way is called Laplace's asymptotic formula (in Problem 2.7, for brevity of statement, it is assumed that $\varphi (a)=1$ and $f \equiv 1$ ). The reader will have no difficulty formulating the analogous statement if the function $\varphi$ increases on $\left[a, b\right]$ and

\varphi (b)-\varphi (x) \underset {x \rightarrow b-0}{\sim } C(b-x)^{p} .

From this Laplace's asymptotic formula for a piecewise monotone function follows immediately.

By a suitable change of variable in the integral $\Phi (A)$ , the case of a nonconstant nonnegative function $f$ can be reduced to the case $f \equiv 1$ (as the new variable one should take the antiderivative $\int_{a}^{x} f(t) d t$ ). The result of Problem 2.8 makes it possible to substantially reduce the technical difficulties that arise in doing so.

Problem VI.2.5

We assume that $\varphi$ is a positive nonincreasing function on the interval $[a, b)(-\infty <a<b \leqslant +\infty )$ , with $\varphi (a)=\lim_{x \rightarrow a+0} \varphi (x)=1$ a strict maximum of $\varphi$ on $[a, b)$ . We also assume that $f \in C((a, b)), f \geqslant 0$ on $(a, b)$ and $f \not\equiv 0$ near the point $a$ , and that the integral $\int_{a}^{b} f(x) \varphi^{A}(x) d x$ is finite for sufficiently large $A$ . Prove the following properties of the integral $\Phi (A)=\int_{a}^{b} f(x) \varphi^{A}(x) d x$ as $A \rightarrow +\infty$ :

?

(a)

$\boldsymbol {\Phi }(A)$ decreases to zero;

(b)

$\Phi (A)$ cannot decrease exponentially: $\ln \Phi (A)=o(A)$ ;

(c)

statement b) cannot, in general, be strengthened: for any function $\omega (A)$ decreasing to zero as $A \rightarrow +\infty$ and such that $\ln \omega (A)=o(A)$ , there exists a nonnegative function $\varphi$ , decreasing on $[0,1)$ , with $\lim_{x \rightarrow 0} \varphi (x)=1$ , such that $\int_{0}^{1} \varphi^{A}(x) d x=o(\omega (A))$ ;

(d)

$\Phi (A)$ can decrease arbitrarily slowly: for any function $\omega (A)$ decreasing to zero as $A \rightarrow +\infty$ , there exists a function $\varphi$ such that $\int_{a}^{b} \varphi^{A}(x) d x \geqslant \omega (A)$ for sufficiently large $A$ ;

(e)

the asymptotics of $\Phi (A)$ is determined by the behavior of the functions $\varphi$ and $f$ in an arbitrarily small neighborhood of the point $a$ :

\Phi (A) \sim \int _{a}^{\beta } f(x) \varphi ^{A}(x) d x \quad \text{ for any number } \beta \in (a, b) ;

(f)

$\Phi (A) \sim \Phi (A+H)$ for any number $H \in \mathbb {R}$ .

Remark.

?

Note that statement f) allows one, without loss of generality, to assume that $\int_{a}^{b} f(x) d x<\infty$ , since otherwise, without changing the asymptotics of the integral $\Phi (A)$ , the function $f(x)$ can be replaced by $f(x) \varphi^{H}(x)$ for sufficiently large $H$ .

Problem VI.2.6

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

We assume that $\varphi$ is a positive nonincreasing function on the interval $[a, b)(-\infty <a<b \leqslant +\infty )$ , with $\varphi (a)=\lim_{x \rightarrow a+0} \varphi (x)=1$ a strict maximum of $\varphi$ on $[a, b)$ . We also assume that $f \in C((a, b)), f \geqslant 0$ on $(a, b)$ and $f \not\equiv 0$ near the point $a$ , and that the integral $\int_{a}^{b} f(x) \varphi^{A}(x) d x$ is finite for sufficiently large $A$ . Let $\Phi (A)=\int_{a}^{b} \varphi^{A}(x) d x$ . Prove that

?

(a)

if for some $L, p>0$ the inequality $\varphi (x) \geqslant 1-L(x-a)^{p}$ holds near the point $a$ , then

A^{1 / p} \Phi (A) \geqslant \frac{\Gamma \left(1+\frac{1}{p}\right)}{L^{1 / p}}+o(1) \quad \text{ as } \quad A \rightarrow +\infty ;

(b)

if $\frac{1-\varphi (x)}{(x-a)^{p}} \rightarrow 0$ as $x \rightarrow a+0$ , then $A^{1 / p} \Phi (A) \rightarrow +\infty$ as $A \rightarrow +\infty$ ;

(c)

if for some $l, p>0$ the inequality $\varphi (x) \leqslant 1-l(x-a)^{p}$ holds near the point $a$ , then

A^{1 / p} \Phi (A) \leqslant \frac{\Gamma \left(1+\frac{1}{p}\right)}{l^{1 / p}}+o(1) \quad \text{ as } \quad A \rightarrow +\infty ;

(d)

if $\frac{1-\varphi (x)}{(x-a)^{p}} \rightarrow +\infty$ as $x \rightarrow a+0$ , then

A^{1 / p} \Phi (A) \rightarrow 0 \quad \text{ as } \quad A \rightarrow +\infty .

Problem VI.2.7

We assume that $\varphi$ is a positive nonincreasing function on the interval $[a, b)(-\infty <a<b \leqslant +\infty )$ , with $\varphi (a)=\lim_{x \rightarrow a+0} \varphi (x)=1$ a strict maximum of $\varphi$ on $[a, b)$ . We also assume that $f \in C((a, b)), f \geqslant 0$ on $(a, b)$ and $f \not\equiv 0$ near the point $a$ , and that the integral $\int_{a}^{b} f(x) \varphi^{A}(x) d x$ is finite for sufficiently large $A$ . Let $1-\varphi (x) \underset {x \rightarrow a+0}{\sim } C(x-a)^{p}(C, p>0)$ . Prove the Laplace asymptotic formula

\Phi (A)=\int _{a}^{b} \varphi ^{A}(x) d x \underset {A \rightarrow +\infty }{\sim } \frac{\Gamma \left(1+\frac{1}{p}\right)}{(A C)^{1 / p}}

In particular,

\Phi (A) \sim \frac{1}{A C} \quad \text{ for } p=1, \quad \Phi (A) \sim \frac{1}{2} \sqrt{\frac{\pi }{A C}} \quad \text{ for } p=2 .

Supplement the Laplace formula by proving that for any $q>-1$

\int _{a}^{b}(x-a)^{q} \varphi ^{A}(x) d x \underset {A \rightarrow +\infty }{\sim } \frac{1}{q+1} \frac{\Gamma \left(1+\frac{q+1}{p}\right)}{(A C)^{\frac{q+1}{p}}}

?

Problem VI.2.8

We assume that $\varphi$ is a positive nonincreasing function on the interval $[a, b)(-\infty <a<b \leqslant +\infty )$ , with $\varphi (a)=\lim_{x \rightarrow a+0} \varphi (x)=1$ a strict maximum of $\varphi$ on $[a, b)$ . We also assume that $f \in C((a, b)), f \geqslant 0$ on $(a, b)$ and $f \not\equiv 0$ near the point $a$ , and that the integral $\int_{a}^{b} f(x) \varphi^{A}(x) d x$ is finite for sufficiently large $A$ . Let $g \in C((a, b))$ and let the function $g(x) \varphi^{A}(x)$ be absolutely integrable on $(a, b)$ for $A$ sufficiently large. Prove that

\int _{a}^{b} f(x) \varphi ^{A}(x) d x \underset {A \rightarrow \infty }{\sim } \int _{a}^{b} g(x) \varphi ^{A}(x) d x, \quad \text{ if } f(x) \underset {x \rightarrow a+0}{\sim } g(x)

In particular, if the function $f$ is continuous at the point $a$ and $f(a) \neq 0$ , then for any $\beta \in (a, b)$

\Phi (A)=\int _{a}^{b} f(x) \varphi ^{A}(x) d x \underset {A \rightarrow +\infty }{\sim } f(a) \int _{a}^{\beta } \varphi ^{A}(x) d x

?

Problem VI.2.9

Find the asymptotics as $A \rightarrow +\infty$ of the following integrals:

?

(a)

$\int_{0}^{1}\left(1-x^{p}\right)^{A} d x(p>0)$ ;

(b)

$\int_{0}^{+\infty } \frac{d x}{\left(1+x^{p}\right)^{A}}(p>0)$ ;

(c)

$\int_{0}^{\pi / 2} x^{p} \cos^{A} x d x(p>-1)$ ;

(d)

$\int_{0}^{\pi / 2} x^{p} \sin^{A} x d x(p \in \mathbb {R})$ ;

(e)

$\int_{0}^{+\infty } \frac{\sin x}{\left(1+x^{2}\right)^{A}} d x$ ;

(f)

$\int_{0}^{+\infty } \frac{\cos x}{\left(1+x^{2}\right)^{A}} d x$ ;

(g)

$\int_{0}^{A} e^{x}(A-x)^{A} d x ;$

(h)

$\int_{0}^{+\infty } e^{A\left(x-x^{p}\right)} d x(p>1)$ ;

(i)

$\int_{0}^{+\infty } \frac{\sin x}{x^{p}} \cdot \frac{d x}{e^{A x}} \cdot (p<2)$ ;

(j)

$\int_{0}^{\pi / 2} \frac{\cos x}{e^{A \sqrt{\sin 2 x}}} d x$ ;

(k)

$\int_{-\infty }^{+\infty } \frac{x \cos \left(\frac{\pi }{2} x\right)}{1-x^{5}} \cdot \frac{d x}{e^{A x^{2}}}$ ;

(l)

$\int_{-\infty }^{+\infty } \frac{x \sin (\pi x)}{1-x^{1999}} \cdot \frac{d x}{e^{A x^{2}}}$ ;

(m)

$\int_{0}^{1} x^{A x} d x$ ;

(n)

$\int_{0}^{1}(A x)^{p x} d x \quad (p \in \mathbb {R})$ .

Problem VI.2.10

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

Find the asymptotics as $n \rightarrow +\infty (n \in \mathbb {N})$ of the following integrals:

?

(a)

$\int_{0}^{\pi }(a+\cos x)^{n} d x(a>0)$ ;

(b)

$\int_{0}^{+\infty }\left(\frac{\sin x}{x}\right)^{n} d x$ ;

(c)

$\int_{0}^{2}\left(1-4 x+2 x^{2}\right)^{n} \frac{d x}{\sqrt{x}}$ ;

(d)

$\int_{0}^{+\infty } \cos^{n} x \frac{d x}{e^{x}}$ .

Problem VI.2.11

Let $p, q, C>0$ and $r \in \mathbb {R}$ . Prove that

I(A)=\int _{0}^{+\infty } x^{r} e^{-C / x^{q}} e^{-A x^{p}} d x \sim \frac{C_{0}}{A^{\sigma }} e^{-\gamma (A)} \quad \text{ as } A \rightarrow +\infty

where $\sigma =\frac{1+r+\frac{q}{2}}{p+q}, \gamma (A)=(p+q)\left(\frac{C}{p}\right)^{\frac{p}{p+q}}\left(\frac{A}{q}\right)^{\frac{q}{p+q}}, C_{0}=\left(\frac{C q}{p}\right)^{\sigma } \sqrt{\frac{2 \pi }{C q(p+q)}}$ . In particular, as $A \rightarrow +\infty$

\int _{0}^{\infty } x^{r} e^{-C / x^{p}} e^{-A x^{p}} d x \sim \frac{\sqrt{\pi }}{p}\left(\frac{C}{A}\right)^{\frac{1+r}{2 p}} \frac{e^{-2 \sqrt{A C}}}{\sqrt[4]{A C}}

Note that the exponential factor $e^{-\gamma (A)}$ does not depend on $r$ and, consequently, this parameter does not affect the asymptotics of $\ln I(A)$ .

?

Problem VI.2.12

Compute the integral

\int _{-\infty }^{+\infty } \frac{d x}{\left(x^{4}+4 x+4\right)^{100}}

with relative error no more than $10 \%$ .

?

Problem VI.2.13

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

Find the asymptotics as $A \rightarrow +\infty$ of the integrals

?

(a)

$\int_{0}^{\infty }\left(\frac{2 x}{1+x^{2}}\right)^{A} \frac{d x}{e^{x \sqrt{A}}}$ ;

(b)

$\int_{0}^{A} e^{x^{p}}(A-x)^{A} d x(0<p<1)$ ;

(c)

$\int_{0}^{1 / e} e^{A / \ln x} d x$ ;

(d)

$\int_{0}^{1 / e}\left(1+\frac{1}{\ln x}\right)^{A} d x$ .

Remark.

?

In this and the following problems we consider integrals $\Phi (A)=\int_{a}^{b} f_{A}(x) d x$ , where the function $f_{A}$ , while no longer of the form considered in problems $2.5$ -- $2.8$ , nevertheless retains their characteristic feature: as $A$ grows, the graph of the function $f_{A}$ has an increasingly sharply pronounced "hump" in a neighborhood of the point $x_{A}$ at which $f_{A}$ attains its maximum value. Although the result of problem 2.7 does not apply here, the idea of the solution is nevertheless preserved: writing the function $f_{A}$ in the form $e^{\ln f_{A}}$ and replacing, in some neighborhood of the point $x_{A}$ , the function $\ln f_{A}$ by its Taylor expansion, and the integral over the interval $(a, b)$ by the integral over a neighborhood of the point $x_{A}$ , we find the principal part of $\Phi (A)$ . The main difficulty here is the choice of the neighborhood. On the one hand, it should not be too large, since otherwise the error caused by applying Taylor's formula will make itself felt. On the other hand, to keep down the error arising from replacing the integral over the interval $(a, b)$ by the integral over the neighborhood, it cannot be taken too small. A good choice of neighborhood, allowing both of the errors mentioned to be well estimated, constitutes the main content of the solution. Let us also note that in the case when $x_{A} \rightarrow x_{0}$ as $A \rightarrow +\infty$ , it may turn out to be more convenient to consider the Taylor expansion of the function $\ln f_{A}$ in a neighborhood of the point $x_{0}$ .

Problem VI.2.14

(a)(b)

Prove that as $A \rightarrow +\infty$

?

(a)

$\int_{0}^{1 / 2} x^{A x} d x \sim \frac{1}{A \ln A}$ ;

(b)

$\int_{0}^{1}\left|\ln x\right|^{p}(1-x)^{A} d x \sim \frac{\ln^{p} A}{A} \quad (p \in \mathbb {R})$ .

Problem VI.2.15

Let the functions $\psi , \theta$ be strictly increasing and not exceed 1 on $\left[0,1\right], \lim_{x \rightarrow 0} \psi (x)=\lim_{x \rightarrow 0} \theta (x)=0$ . Prove that if $\psi^{-1}(y) \sim \theta^{-1}(y)$ as $y \rightarrow +0$ , then

\int _{0}^{1}(1-\psi (x))^{A} d x \sim \int _{0}^{1}(1-\theta (x))^{A} d x \quad \text{ as } \quad A \rightarrow +\infty

?

Problem VI.2.16

Let

\Psi (A)=\int _{a}^{b} f(x) e^{-A \psi (x)} d x \quad \text{ and } \quad \Theta (A)=\int _{a}^{b} f(x) e^{-A \theta (x)} d x

where the function $f$ is nonnegative and integrable, and the functions $\psi$ and $\theta$ are continuous and increasing on $[a, b), \psi (a)=\theta (a)=0$ . Suppose that

$\psi (x)-\theta (x)=O\left(\psi^{2}(x)\right)$ as $x \rightarrow a$ ;
the integral $\Psi (A)$ does not decrease to zero too quickly:
$\ln \Psi (A)=o(\sqrt{A}) \quad \text{ as } \quad A \rightarrow +\infty .$

Prove that $\Psi (A) \sim \Theta (A)$ as $A \rightarrow +\infty$ . In particular, if $\psi <1$ , then, taking $\theta =-\ln (1-\psi )$ , we obtain

\int _{a}^{b} f(x)(1-\psi (x))^{A} d x \sim \int _{a}^{b} f(x) e^{-A \psi (x)} d x \quad \text{ as } \quad A \rightarrow +\infty

Can condition 2) be weakened by replacing it with $\ln \Phi (A)=O(\sqrt{A})$ ?

?

Problem VI.2.17

Let $f, \psi \in C((a, b)), \int_{a}^{b} f(x) d x<\infty , 0<\psi <1, \psi$ be increasing and $\psi (x)=O\left((x-a)^{p}\right)$ as $x \rightarrow a(p>0)$ . Prove that if the function $f$ is positive and not too small near the point $a$ , more precisely, $\ln f(x)=o\left((x-a)^{-p}\right)$ as $x \rightarrow a$ , then

\int _{a}^{b} f(x)(1-\psi (x))^{A} d x \sim \int _{a}^{b} f(x) e^{-A \psi (x)} d x \quad \text{ as } \quad A \rightarrow +\infty .

?

Problem VI.2.18

Prove Stirling's formula

\Gamma (1+x) \underset {x \rightarrow +\infty }{\sim } \sqrt{2 \pi x}\left(\frac{x}{e}\right)^{x}

Supplement this result by proving that for any $r \in \mathbb {N}$

\Gamma ^{(r)}(1+x)=\Gamma (1+x) \ln ^{r} x\left(1+O\left(\frac{1}{\sqrt{x} \ln x}\right)\right) \quad \text{ as } \quad x \rightarrow +\infty .

?

Remark.

?

Here $\Gamma$ is Euler's $\Gamma$ -function

\Gamma (x)=\int _{0}^{+\infty } t^{x-1} e^{-t} d t, \quad x>0 .

Problem VI.2.19

Prove that $\Gamma (x+c) \underset {x \rightarrow +\infty }{\sim } x^{c} \Gamma (x)(c \in \mathbb {R})$ .

?

Remark.

?

Here $\Gamma$ is Euler's $\Gamma$ -function

\Gamma (x)=\int _{0}^{+\infty } t^{x-1} e^{-t} d t, \quad x>0 .

Problem VI.2.20

Prove that $\int_{0}^{x} t^{x} e^{-t} d t \underset {x \rightarrow +\infty }{\sim } \frac{1}{2} \Gamma (1+x)$ .

?

Remark.

?

Here $\Gamma$ is Euler's $\Gamma$ -function

\Gamma (x)=\int _{0}^{+\infty } t^{x-1} e^{-t} d t, \quad x>0 .

Problem VI.2.21

(a)(b)(c)

Let $\varphi$ be a positive function defined on $(0,+\infty )$ . Prove that as $x \rightarrow +\infty$

?

(a)

$\frac{\varphi (x)-x}{\sqrt{x}} \rightarrow 0 \quad \Longleftrightarrow \quad \frac{1}{\Gamma (1+x)} \int_{0}^{\varphi (x)} t^{x} e^{-t} d t \rightarrow \frac{1}{2}$ ;

(b)

$\frac{\varphi (x)-x}{\sqrt{x}} \rightarrow +\infty \quad \Longleftrightarrow \quad \frac{1}{\Gamma (1+x)} \int_{0}^{\varphi (x)} t^{x} e^{-t} d t \rightarrow 1$ ;

(c)

$\frac{\varphi (x)-x}{\sqrt{x}} \rightarrow -\infty \quad \Longleftrightarrow \quad \frac{1}{\Gamma (1+x)} \int_{0}^{\varphi (x)} t^{x} e^{-t} d t \rightarrow 0$ .

Remark.

?

Here $\Gamma$ is Euler's $\Gamma$ -function

\Gamma (x)=\int _{0}^{+\infty } t^{x-1} e^{-t} d t, \quad x>0 .

Problem VI.2.22

Let $f$ be the Cantor function (see problem III.3.14 and the text before it). Prove that

\int _{0}^{1} f^{2^{n}}(t) d t \sim \frac{C}{3^{n}} \quad \text{ as } \quad n \rightarrow +\infty

?

Remark.

?

The Cantor function is the function $f$ defined for $x \in [0,1]$ by

f(x)=\sup \left\{ \sum \frac{\varepsilon _{k}}{2^{k}} \left\lvert \, \sum \frac{2 \varepsilon _{k}}{3^{k}} \leqslant x\right., \varepsilon _{k}=0 \text{ or } 1\right\} .

Problem VI.2.23

Let $f$ be the Cantor function. Prove that

\int _{0}^{1} t^{3^{n}} d f(t) \sim \frac{C}{2^{n}} \quad \text{ as } \quad n \rightarrow +\infty

?