Math Problems Vault

Problem 2.23.1

Prove the asymptotic equality. $\int_{2}^{x} \frac{d t}{t^{\alpha } \ln^{\beta } t} \sim -\frac{1}{\alpha x^{\alpha } \ln^{\beta } x}, \quad x \rightarrow +\infty , \quad \alpha < 0, \quad \beta \in R$ .

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Problem 2.23.2

Prove the asymptotic equality. $\int_{0}^{x} t^{\alpha } e^{-1 / t} d t \sim x^{\alpha +2} e^{-1 / x}, \quad x \rightarrow +0$ .

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Problem 2.23.3

Prove the asymptotic equality. $\int_{a}^{x} \frac{(\ln t)^{\alpha }}{\sqrt{t^{2}+1}} d t \sim \frac{(\ln x)^{\alpha +1}}{\alpha +1}, \quad x \rightarrow +\infty , \quad \alpha > -1, \quad a > 1$ .

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Problem 2.23.4

Prove the asymptotic equality. $\int_{x}^{+\infty } \frac{\sin t}{\sqrt{t+1}} e^{-t} d t \sim \frac{1}{2 \sqrt{x}} e^{-x}(\sin x+\cos x), \quad x \rightarrow +\infty$ .

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Problem 2.23.5

Prove the asymptotic equality. $\int_{x}^{+\infty } \sqrt{t^{2}+1} \sin e^{t} d t \sim x e^{-x} \cos e^{x}, \quad x \rightarrow +\infty$ .

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Problem 2.23.6

Prove the asymptotic equality. $\int_{0}^{x} \sqrt{t^{2}+1} e^{t} \sin t d t \sim \frac{1}{2} x e^{x}(\sin x-\cos x), \quad x \rightarrow +\infty$ .

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Problem 2.23.7

Find an elementary function asymptotically equal, as $x \rightarrow +\infty$ , to the integral

\int _{1}^{x} \frac{\sin ^{4} t}{t} d t

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Problem 2.23.8

Prove that $\int_{x}^{+\infty } \cos t^{2} d t = -\frac{\sin x^{2}}{2 x}+O\left(\frac{1}{x^{3}}\right), \quad x \rightarrow +\infty$ .

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Problem 2.23.9

Prove that $\int_{x}^{+\infty } \sin t^{2} d t = \frac{\cos x^{2}}{2 x}+O\left(\frac{1}{x^{3}}\right), \quad x \rightarrow +\infty$ .

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Problem 2.23.10

Prove that

\int _{x}^{+\infty } \sin t^{m} d t = \frac{\cos x^{2}}{m x^{m-1}}+O\left(\frac{1}{x^{2 m-1}}\right), \quad x \rightarrow +\infty , \quad m > 1

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Problem 2.23.11

Prove Stirling's formula

\Gamma (s+1) \sim \sqrt{2 \pi } e^{-s} s^{s+1 / 2}, \quad s \rightarrow +\infty ,

where

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

is the gamma function.

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Problem 2.23.12

Prove: if a nonnegative, continuous function $f$ , not identically zero, has period $T$ for $t > 1$ :

f(t+T) = f(t), \quad T > 0

then

\int _{1}^{x} \frac{f(t)}{t} d t \sim A \ln x, \quad x \rightarrow +\infty , \quad A = \text{ const. }

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Problem 2.23.13

Prove that

\int _{1}^{x} \frac{\ln \left(1+\cos ^{2} t\right)}{t} d t \sim B \ln x, \quad x \rightarrow +\infty , \quad B = \text{ const. }

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Problem 2.23.14

Find the asymptotic series for the function $f(x) = e^{-x}, x > 0$ .

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Problem 2.23.15

Establish the following asymptotic expansions as $x \rightarrow +\infty$ . $\int_{x}^{+\infty } \frac{e^{-x t}}{1+t^{2}} d t \sim \frac{1}{x}-\frac{2 !}{x^{3}}+\frac{4 !}{x^{5}}-\ldots$

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Problem 2.23.16

Establish the following asymptotic expansions as $x \rightarrow +\infty$ . $\int_{x}^{+\infty } e^{x^{2}-t^{2}} d t \sim \frac{1}{2 x}-\frac{1}{2^{2} x^{3}}+\frac{1 \cdot 3}{2^{3} x^{5}}-\ldots$

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Problem 2.23.17

Prove that if series (1) converges to some function, then it is also the asymptotic expansion of this function as $x \rightarrow +\infty$ .

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Problem 2.23.18

Prove that for series (1) to be an asymptotic series as $x \rightarrow +\infty$ for the function $f$ , it is necessary and sufficient that

f(x)-S_{n}(x) = O\left(\frac{1}{x^{n+1}}\right), \quad x \rightarrow +\infty , \quad n \in N

where $S_{n}(x)$ is the partial sum of series (1).

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Problem 2.23.19

Prove that if

f(x) \sim \sum _{n = 0}^{\infty } \frac{a_{n}}{x^{n}}, \quad g(x) \sim \sum _{n = 1}^{\infty } \frac{b_{n}}{x^{n}}, \quad x \rightarrow +\infty

then for any numbers $\lambda$ and $\mu$ we have

\lambda f(x)+\mu g(x) \sim \sum _{n = 0}^{\infty } \frac{\lambda a_{n}+\mu b_{n}}{x^{n}}, \quad x \rightarrow +\infty

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Problem 2.23.20

Prove that if the asymptotic expansions (9) hold, then

f(x) g(x) \sim \sum _{n = 0}^{\infty } \frac{c_{n}}{x^{n}}, \quad x \rightarrow +\infty , \quad \text{ where } \quad c_{n} = \sum _{k = 0}^{\infty } a_{k} b_{n-k}.

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Problem 2.23.21

Prove that if $f_{n}(x) \sim \sum_{n = 0}^{\infty } \frac{a_{n}}{x^{n}}, x \rightarrow +\infty$ , and $a_{0} \neq 0$ , then the function $1 / f(x)$ also has an asymptotic expansion

\frac{1}{f(x)} \sim \frac{1}{a_{0}}+\sum _{n = 1}^{\infty } \frac{d_{n}}{x^{n}}, \quad x \rightarrow +\infty

where the coefficient $d_{n}, n \in N$ , of this expansion is expressed in terms of the coefficients $a_{0}, a_{1}, \ldots , a_{n}$ .

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Problem 2.23.22

Prove: if the function $f$ is continuous for $x \geqslant a > 0$ and has an asymptotic expansion beginning with a term of order $1 / x^{2}$ :

f_{n}(x) \sim \sum _{n = 2}^{\infty } \frac{a_{n}}{x^{n}}, \quad x \rightarrow +\infty

then

\int _{x}^{+\infty } f(t) d t \sim \sum _{n = 2}^{\infty } \frac{a_{n}}{(n-1) x^{n-1}}, \quad x \rightarrow +\infty

i.e., in the case indicated, the asymptotic series can be integrated term by term.

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Problem 2.23.23

Prove that if the function $f$ expands into the asymptotic series

f_{n}(x) \sim \sum _{n = 0}^{\infty } \frac{a_{n}}{x^{n}}, \quad x \rightarrow \infty

and if it has, for $x \geqslant a$ , a continuous derivative which also expands as $x \rightarrow +\infty$ into an asymptotic series, then this series is obtained by formal term-by-term differentiation of series (11):

f^{\prime }(x) \sim -\sum _{n = 1}^{\infty } \frac{n a_{n}}{x^{n+1}}, \quad x \rightarrow +\infty

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Problem 2.23.24

Prove: if the function $f$ has an asymptotic expansion (1) with no constant term $(a_{0} = 0)$ , then it can be formally exponentiated, i.e., the asymptotic expansion of the function $e^{f(x)}$ as $x \rightarrow +\infty$ can be obtained if, in the series

e^{f(x)} = \sum _{n = 0}^{\infty } \frac{(f(x))^{n}}{n}

replace the function $f$ by its asymptotic expansion (1), formally carry out the exponentiation, and combine like terms.

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Problem 2.23.25

Find the asymptotic expansion (1) for the function $f(x) = e^{-x} \sin e^{x}, x \rightarrow +\infty$ , and prove that the derivative $f^{\prime }(x)$ does not expand into a power-type asymptotic series as $x \rightarrow +\infty$ .

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Problem 2.23.26

Find the asymptotic expansion (1) as $x \rightarrow +\infty$ for the function

F(x) = \sum _{k = 1}^{\infty } \frac{c^{k}}{x+k}, \quad 0 < c < 1

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Problem 2.23.27

Prove that the following sequences are asymptotic (the sequence $\left\{ \alpha_{n}\right\}$ is strictly increasing):

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(1)

$\left\{ \left(x-x_{0}\right)^{n}\right\} , x \rightarrow x_{0}$

(2)

$\left\{ 1 / x^{n}\right\} , x \rightarrow +\infty$ ;

(3)

$\left\{ 1 / x^{\alpha_{n}}\right\} , x \rightarrow +\infty$ ;

(4)

$\left\{ x^{\alpha_{n}}\right\} , x \rightarrow 0$

(5)

$\left\{ \ln^{\alpha_{n}} x\right\} , x \geqslant 1, x \rightarrow +\infty$ ;

(6)

$\left\{ x^{\alpha_{n}} e^{-x}\right\} , x \rightarrow +\infty$ .

Problem 2.23.28

Prove that if $\left\{ \varphi_{n}(x)\right\}$ is an asymptotic sequence as $x \rightarrow x_{0}$ , then for series (5) to be an asymptotic expansion of the function $f$ as $x \rightarrow x_{0}$ , it is necessary and sufficient that

f(x)-S_{n}(x) = O\left(\varphi _{n+1}(x)\right), \quad x \rightarrow x_{0}, \quad n = 0,1,2, \ldots

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Problem 2.23.29

Prove that if $\varphi_{n}(x) \neq 0$ for $x \neq x_{0}, \quad n = 0,1, \ldots$ , and $f$ expands as $x \rightarrow x_{0}$ into the asymptotic series (5), then its coefficients are determined successively by the formulas

a_{0} = \lim _{x \rightarrow x_{0}} f(x), \quad a_{n} = \lim _{x \rightarrow x_{0}} \frac{1}{\varphi _{n}(x)}\left(f(x)-\sum _{k = 0}^{n-1} a_{k} \varphi _{k}(x)\right)

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Problem 2.23.30

Expand, as $x \rightarrow +\infty$ , into an asymptotic series the function

F(x ; \alpha ) = \int _{x}^{+\infty } \frac{e^{i t}}{t^{\alpha }} d t, \quad x > 0, \quad \alpha > 0

Prove that the real and imaginary parts of the integral $\frac{1}{2} F\left(x^{2} ; \frac{1}{2}\right)$ are the incomplete Fresnel integrals

\int _{x}^{+\infty } \cos t^{2} d t, \quad \int _{x}^{+\infty } \sin t^{2} d t

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Problem 2.23.31

Find the asymptotic expansion of the incomplete gamma function

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

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