Functions of Large Numbers

Let $p_{1}, p_{2}, \ldots , p_{l}, a_{1}, a_{2}, \ldots , a_{l}$ be arbitrary positive numbers. Then

exists and it is equal to the largest among the numbers $a_{1}, a_{2}, \ldots , a_{l}$.

Under the same hypotheses as in Problem 195

Let $f(x)$ be an arbitrary polynomial whose zeros are all real and positive and for which

Show that

exists and that it is equal to the smallest zero of $f(x)$.

The two functions $\varphi (x)$ and $f(x)$ are continuous and positive on the interval $\left[a,b\right]$. Then

exists and is equal to the maximum of $f(x)$ on $\left[a,b\right]$.

Under the same hypothesis as in Problem 198

Let $k$ be a positive constant and $a < \xi < b$. Show that for $a, b, \xi , k$ fixed and $n \to \infty$

The functions $\varphi (x)$, $h(x)$ and $f(x)=e^{h(x)}$ are defined on the finite or infinite interval $\left[a,b\right]$ and satisfy the following conditions:

(1) $\varphi (x)\left[f(x)\right]^{n} = \varphi (x)e^{nh(x)}$ is absolutely integrable over $\left[a,b\right]$; $n=0,1,2,\ldots$

(2) The function $h(x)$ attains its maximum only at the point $\xi$ in $(a,b)$; moreover, the least upper bound of $h(x)$ is smaller than $h(\xi )$ on any closed interval that does not contain $\xi$; there is, furthermore, a neighbourhood of $\xi$ where $h''(x)$ exists and is continuous; finally $h''(\xi ) < 0$,

(3) $\varphi (x)$ is continuous at $x=\xi$, $\varphi (\xi ) \neq 0$. Then the following asymptotic formula holds as $n \to \infty$

Let $n$ be an integer, $n \to +\infty$. Using the fact that

prove that

We assume that $\lambda$ is real, $\lambda > 1$; $P_{n}(x)$ denotes the $n$-th Legendre polynomial. As $n \to \infty$

The positive value of the roots must be used.

The Bessel function $J_{v}(t)$ can be defined by Hansen's expansion

Derive the following asymptotic formula:

Show that for positive $n$, $n \to +\infty$,

and, more accurately,

Let $k$ and $l$ be real numbers, $k > 1$. Prove that for $n \to +\infty$

Assume that $\alpha$ is real and that $t$ is positive and increases to infinity. Then

Let $0 < \alpha < 1$. The following approximation is justified as $\tau \to +0$:

Let $\alpha > 0$. As $t \to \infty$ we obtain

Let $\alpha$ and $\beta$ be two real constants. Then the relation

holds where

We denote by $\lambda$ a positive proper fraction and by $x_{n}$ the only positive root of the transcendental equation

As $n \to \infty$ the root is given by

where $\alpha$ and $\beta$ satisfy the equations:

(Continuation of Problem 201). Let $\alpha$ denote a real constant. Then, for $n \to \infty$,

The functions $\varphi (x)$, $h(x)$ and $f(x)=e^{h(x)}$ are defined on the finite or infinite interval $\left[a,b\right]$ and satisfy the following conditions:

(1) $\varphi (x)\left[f(x)\right]^{n} = \varphi (x)e^{nh(x)}$ is absolutely integrable over $\left[a,b\right]$, $n=0,1,2,\ldots$

(2) The value of the function $h(x)$ at a point $\xi$ of $(a,b)$ is larger than its least upper bound in any closed interval to the left of $\xi$ which does not contain $\xi$. Moreover there is a neighbourhood of $\xi$ in which $h''(x)$ exists and is bounded. Finally $h'(\xi ) > 0$.

(3) $\varphi (x)$ is continuous at $x=\xi$, $\varphi (\xi ) \neq 0$.

Prove, for $n \to \infty$, the asymptotic formula

where $\alpha$ and $\beta$ stand for real constants.

Let $\xi$ denote the only real root of the transcendental equation $e^{1+\xi }\xi = 1$. Then we have for $n \to \infty$

where $\alpha$ and $\beta$ are real constants and

Suppose that $n$ is odd and let $-x_{n}$ denote the only real root of the equation

As $n \to \infty$, $x_{n}$ is asymptotically given by

where $\xi$ is the only real root of the equation $e^{1+\xi }\xi = 1$ and $\alpha$ and $\beta$ are given by

Assume that the function $g(x)$ is monotone increasing for positive $x$ and that

We define

If there is a positive number $\gamma$ such that

exists and is a continuous function of $\alpha$ for $1-\gamma \leq \alpha \leq 1+\gamma$ then

Analogy to Problem 201 suggests sufficient conditions under which for $n \to +\infty$

where the partial derivatives of second order $h_{xx}, h_{yy}$, and $h_{xy}$ are taken at the point $(\xi ,\eta )$. Give a full statement and a proof.

Analogy to Problem 201 suggests sufficient conditions under which for $n \to +\infty$

where the partial derivatives of second order $h_{xx}, h_{yy}$, and $h_{xy}$ are taken at the point $(\xi ,\eta )$. Give a full statement and a proof.

The function

where $a > 1$, has the maximum $M_{n}$ on the interval $(n, +\infty )$. It can be approximated by

The function

where $a > 1$, has the maximum $M_{n}$ on the interval $(n, +\infty )$. It can be approximated by

We define $\sqrt{x} = Q_{0}(x)$,

The sequence of functions

is uniformly bounded for $x>0$ if $a \geq 2$; it is not uniformly bounded if $0 < a < 2$.

We define $x = P_{0}(x)$,

The sequence of functions

is uniformly bounded for $x>0$ if $a \geq 3+\sqrt{8}$; it is not uniformly bounded if $0 < a < 3+\sqrt{8}$.

Assume that $a>0$, $0<\mu <1$ and that $M_{n}$ is the maximum of $e^{-(x+ax^{\mu })} x^{n}$ in the interval $(0,+\infty )$. We find

The function $f(x,y)$ is continuous in the rectangle

Then its maximum for a given $x$ and $b \leq y \leq b'$ (along a segment parallel to the $y$-axis)

is a continuous function of $x$.

We interpret the surface

in a rectangular coordinate system $x,y,z$ with vertical $z$-axis as a topographical surface in a mountainous region. Then the curve

(in the $x,z$-plane) is the skyline of the range (as it appears when seen from a faraway point of the $y$-axis) and the minimax

(the minimum of the maxima) refers to the lowest point in the skyline.

(continued). Show that

(continued). Which one of the two signs, $<$ and $=$, is valid in the example

$a=0$, $a'=2$, $b=0$, $b'=4$?

Add to the assumptions of Problem 223 that $f(x,y) > 0$. Then