The Abstract Of Mathematical Analysis I

于2020年11月8日2020年11月8日由Sukuna发布

1. Limits

Definition: series limit

$\left(\lim {n \rightarrow \infty} x{n}=A\right):=\forall \varepsilon>0\quad \exists N \in \mathbb{N}\quad \forall n>N\quad \left(\left|x_{n}-A\right|<\varepsilon\right)$

We say that the sequence ${x_n}$ converges to $A$ or tends to $A$ and write $x_n \to A$ as $n \to \infty$ .

Definition: fundamental or Cauchy sequence

A sequence ${x_n}$ is called a fundamental or Cauchy sequence if for any $\varepsilon > 0$ there exists an index $N \in N$ such that $|x_m − x_n| < \varepsilon$ whenever $n > N$ and $m > N$ .

Theorem: Weierstrass

In order for a nondecreasing sequence to have a limit, it is necessary and sufficient that it be bounded above.

Two important limit

$\text { e }:=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}$

$\lim_{x\to 0}\frac{\sin x}{x}=1$

Definition 3. inferior limit and superior limit

$\varliminf_{k \rightarrow \infty} x_{k}:=\lim {n \rightarrow \infty} \inf {k \geq n} x_{k}$

$\varlimsup_{k \rightarrow \infty} x_{k}:=\lim {n \rightarrow \infty} \sup {k \geq n} x_{k}$

Theorem 2. Stolz

Let $(a{n}){n\geq 1}$ and $(b{n}){n\geq 1}$ be two sequences of real numbers. Assume that $(b{n}){n\geq 1}$ is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching $+\infty$ , or strictly decreasing and approaching $-\infty$ ) and the following limit exists:

$\lim_ {n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l$

Then, the limit

$\lim _{n\to \infty }{\frac {a{n}}{b{n}}}=l$

Theorem 3. Toeplitz limit theorem

Supports that $n,k\subseteq \mathbb N^{+}$ , $t_{nk}\geq0$ and

$\sum_{k=1}^{n}{t_{nk}} = 1,\quad \lim_{n \rightarrow \infty}{t_{nk}} = 0$

$\lim_{n \rightarrow \infty}{a_{n}} = a$

, let

$x_{n} = \sum_{k=1}^{n}{t_{nk}a_{k}}$

, s.t.

$\lim_{n \rightarrow \infty}{x_{n}} = a$

By using $t_{nk}=\frac{1}{n}$ , we can quickly infer The Cauchy proposition theorem.
By using $t_{n k}=\frac{b_{k+1}-b_{k}}{b_{n+1}-b_{1}}$ , we can quickly infer The Stolz theorem.

Stirling’s formula

Specifying the constant in the $\mathcal O(\ln n)$ error term gives $\frac12 \ln(2\pi n)$ , yielding the more precise formula:

$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$

2. Continuity

Definition 0

A function $f$ is continuous at the point $a$ , if for any neighbourhood $V (f (a))$ of its value $f (a)$ at a there is a neighbourhood $U(a)$ of a whose image under the mapping $f$ is contained in $V (f (a))$ .

3. Differential calculus

Definition 0

The number

$f^{\prime}(a)=\lim _{E \ni x \rightarrow a} \frac{f(x)-f(a)}{x-a}$

is called the derivative of the function $f$ at $a$ .

Definition 1

A function $f : E \to R$ defined on a set $E\subset R$ is differentiable at a point x ∈ E that is a limit point of E if $f (x + h) − f (x) = A(x)h + \alpha(x;h)$ , where $h \mapsto A(x) h$ is a linear function in $h$ and $\alpha(x;h) = o(h)$ as $h \to 0$ , $x +h \in E$ .

Definition 2

The function $h \mapsto A(x) h$ of Definition 1, which is linear in $h$ , is called the differential of the function $f : E \to \mathcal R$ at the point $x\in E$ and is denoted $\mathrm d f (x)$ or $\mathrm D f (x)$ . Thus, $\mathrm d f (x)(h) = A(x)h$ .

We obtain

$\frac{\mathrm{d} f(x)(h)}{\mathrm{d} x(h)}=f^{\prime}(x)$

We denote the set of all such vectors by $T\mathbb R(x_0)$ or $T\mathbb R_{x_0}$ . Similarly, we denote by $T\mathbb R(x_0)$ or $T\mathbb R_{y_0}$ the set of all displacement vectors from the point $y_0$ along the y-axis. It can then be seen from the definition of the differential that the mapping

$\mathrm{d} f\left(x_{0}\right): T \mathbb{R}\left(x_{0}\right) \rightarrow T \mathbb{R}\left(f\left(x_{0}\right)\right)$

The derivative of an inverse function

If a function $f$ is differentiable at a point x0 and its differential $\mathrm{d} f\left(x_{0}\right): T \mathbb{R}\left(x_{0}\right) \rightarrow T \mathbb{R}\left(y_0\right)a$ is invertible at that point, then the differential of the function $f^{ −1}$ inverse to $f$ exists at the point $y_0 = f (x_0)$ and is the mapping

$\mathrm{d} f^{-1}\left(y_{0}\right)=\left[\mathrm{d} f\left(x_{0}\right)\right]^{-1}: T \mathbb{R}\left(y_{0}\right) \rightarrow T \mathbb{R}\left(x_{0}\right)$

inverse to $\mathrm{d} f\left(x_{0}\right): T \mathbb{R}\left(x_{0}\right) \rightarrow T \mathbb{R}\left(y_0\right)a$ .

The derivative of some common function formula

$(C)^{\prime}=0$
$\left(x^{\mu}\right)^{\prime}=\mu x^{\mu-1}$
$(\sin x)^{\prime}=\cos x$
$(\cos x)^{\prime}=-\sin x$
$(\tan x)^{\prime}=\sec ^{2} x$
$(\cot x)^{\prime}=-\csc ^{2} x$
$(\sec x)^{\prime}=\sec x \tan x$
$(\csc x)^{\prime}=-\csc x \cot x$
$\left(a^{x}\right)^{\prime}=a^{x} \ln a \quad(a>0, a \neq 1)$
$\left(\mathrm{e}^{x}\right)^{\prime}=\mathrm{e}^{x}$
$\left(\log _{a} x\right)^{\prime}=\frac{1}{x \ln a}(a>0, a \neq 1)$
$(\ln x)^{\prime}=\frac{1}{x}$
$(\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}}$
$(\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}}$
$(\arctan x)^{\prime}=\frac{1}{1+x^{2}}$
$(\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}}$
$(\sinh x)'= \cosh x$
$(\cosh x)'=\sinh x$
$(\tanh x)' =\frac{1}{\cosh ^{2} x}$
$(\operatorname{coth} x)' =-\frac{1}{\sinh ^{2} x}$
$(\operatorname{arsinh} x)'=\left(\ln \left(x+\sqrt{1+x^{2}}\right)\right)' = \frac{1}{\sqrt{1+x^{2}}}$
$(\operatorname{arcosh} x)'=(\ln \left(x \pm \sqrt{x^{2}-1}\right))'= \pm \frac{1}{\sqrt{x^{2}-1}}$
$(\operatorname{artanh} x)'=(\frac{1}{2} \ln \frac{1+x}{1-x})' = \frac{1}{1-x^{2}}$
$(\operatorname{arcoth} x)'=(\frac{1}{2} \ln \frac{x+1}{x-1})' = \frac{1}{x^{2}-1}$

L’Hôpital’s rule

The theorem states that for functions $f$ and $g$ which are differentiable on an open interval $I$ except possibly at a point $c$ contained in $I$ , if

$\lim_{x\to c}{\frac {f(x)}{g(x)}}=\lim_{x\to c}{\frac {f'(x)}{g'(x)}}$

Taylor’s theorem

Let $k \geq 1$ be an integer and let the function $f :\mathbb R\to\mathbb R$ be $k$ times differentiable at the point $a \in\mathbb R$ . Then there exists a function $R_k : \mathbb R \to\mathbb R$ such that ,

$f(a+x)=f(a)+f'(a)(x)+{\frac {f''(a)}{2!}}x^{2}+\cdots +{\frac {f^{(k)}(a)}{k!}}x^{k}+R_k(x;a)$

and,

$R_{k}(x;a)=\int_{a}^{a+x} \frac{f^{(k+1)}(t)}{k !}(a+x-t)^{k} \mathrm d t$

prove:

q.e.d

remainder term

using little $o$ notation, $R_{k}(x;a)=o\left(|x|^{k}\right), \quad x \rightarrow 0$ (The Peano remainder term)

The Lagrange form remainder term( Mean-value forms)

$R_{n}(x;a)=\frac{f^{(n+1)}(\theta)}{(n+1) !}x^{n+1}\quad (\theta\in(a,a+x))$

4. Integral

Antiderivative

Definition

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function

$f$

is a differentiable function $F$ whose derivative is equal to the original function $f$

Suppose $F'(x)=f(x)$ , the notation is

$\int f(x)\mathrm dx=F(x)$

So all the antiderivative of $f$ become a family set ${F(x)+C|C\in\mathbb R}$ .
also the equation below is obviously.

$\mathrm d \int f(x) \mathrm{d} x=f(x) \mathrm{d} x, \quad \int F^{\prime}(x) \mathrm{d} x=F(x)+c$

Theorem: Integration by parts

$\int u(x)v'(x)\,\mathrm dx=u(x)v(x)-\int u'(x)v(x)\,\mathrm dx$

Example: Wallis product

the Wallis product for

$\pi$

, published in 1656 by John Wallis states that

Prove:

so that:

so that:

Simplify the Polynomial and Integral

$Q(z)=\left(z-z{1}\right)^{k{1}} \cdots\left(z-z{p}\right)^{k{p}}$

and

$\frac{P(z)}{Q(z)}$

is a proper fraction, there exists a unique representation of the fraction

$\frac{P(z)}{Q(z)}$

in the form

$\frac{P(z)}{Q(z)}=\sum_{j=1}^{p}\left(\sum_{k=1}^{k_{j}} \frac{a_{j k}}{\left(z-z_{j}\right)^{k}}\right)$

and if

$P(x)$

and

$Q(x)$

are polynomials with real coefficients and

$Q(x)=\left(x-x_{1}\right)^{k_{1}} \cdots\left(x-x_{l}\right)^{k_{l}}\left(x^{2}+p_{1} x+q_{1}\right)^{m_{1}} \cdots\left(x^{2}+p_{n} x+q_{n}\right)^{m_{n}}$

there exists a unique representation of the proper fraction

$\frac{P(x)}{Q(x)}$

in the form

$\frac{P(x)}{Q(x)}=\sum_{j=1}^{l}\left(\sum_{k=1}^{k_{j}} \frac{a_{j k}}{\left(x-x_{j}\right)^{k}}\right)+\sum_{j=1}^{n}\left(\sum_{k=1}^{m_{j}} \frac{b_{j k} x+c_{j k}}{\left(x^{2}+p_{j} x+q_{j}\right)^{k}}\right)$

where $a_{jk}, b_{jk},$ and $c_{j k}$ are real numbers.

and with these formulas below:

And from that we get the recursion:

$\int \frac{\mathrm{d} x}{\left(x^{2}+a^{2}\right)^{m+1}}=\frac{1}{2 m a^{2}} \frac{x}{\left(x^{2}+a^{2}\right)^{m}}+\frac{2 m-1}{2 m a^{2}} \int \frac{\mathrm{d} x}{\left(x^{2}+a^{2}\right)^{m}}$

Primitives of the Form
$\int R(\cos x, \sin x)\mathrm dx$

We make the change of variable $t = \tan \frac{x} {2}$ . Since:

$\cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}, \qquad \sin x=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}$

so that

$\mathrm{d} t=\frac{\mathrm{d} x}{2 \cos ^{2} \frac{x}{2}} \quad \Rightarrow\quad \mathrm{d} x=\frac{2 \mathrm{d} t}{1+\tan ^{2} \frac{x}{2}}=\frac{2\,\mathrm{d} t}{1+t^{2}}$

It follows that

$\int R(\cos x, \sin x) \mathrm{d} x=\int R\left(\frac{1-t^{2}}{1+t^{2}}, \frac{2 t}{1+t^{2}}\right) \frac{2}{1+t^{2}} \mathrm{d} t$

not only $\sin ,\cos$ can to do this, but here are a lot of formula:
$\tan a=\frac{2 \tan \frac{a}{2}}{1-\tan ^{2} \frac{a}{2}}$
,
$\cot \alpha=\frac{1-\tan ^{2} \frac{\alpha}{2}}{2 \tan \frac{\alpha}{2}}$
,
$\sec \alpha=\frac{1+\tan ^{2} \frac{\alpha}{2}}{1-\tan ^{2} \frac{\alpha}{2}}$
,
$\csc \alpha=\frac{1+\tan ^{2} \frac{\alpha}{2}}{2 \tan \frac{\alpha}{2}}$

Integration

Riemann Sums

partition

A partition P of a closed interval $[a,b]$ , $a < b$ , is a finite system of points $x_0,\cdots,x_n$ of the interval such that $a = x_0 < x_1 <\cdots < x_n = b$ .

If a function $f$ is defined on the closed interval $[a, b]$ and $(P, \xi)$ is a partition with distinguished points on this closed interval, the sum

$\sigma(f ; P, \xi):=\sum_{i=1}^{n} f\left(\xi_{i}\right) \Delta x_{i}$

where $\Delta x_i = x_i − x_{i−1}$ , is the Riemann sum of the function $f$ corresponding to the partition $(P, \xi)$ with distinguished points on $[a,b]$ .

The largest of the lengths of the intervals of the partition $P$ , denoted $\lambda(P)$ , is called the mesh of the partition.

we define:

$\int_{a}^{b} f(x) \mathrm{d} x:=\lim {\lambda(P) \rightarrow 0} \sum{i=1}^{n} f\left(\xi_{i}\right) \Delta x_{i}$

Integral mean value theorem

If $f$ is a continuous function on the closed, bounded interval $[a,b]$ , then there is at least one number $\xi$ in $(a , b )$ for which

$\int_{a}^{b} f(x) \mathrm{d} x=f(\xi)(b-a)$

The second Integral mean value theorem

If $f , g$ are continuous functions on the closed, bounded interval $[a,b]$ , $g$ is monotonous on $[a, b]$ , then there is at least one number $\xi$ in $(a , b )$ for which

$\int_{a}^{b}(f \cdot g)(x) \mathrm{d} x=g(a) \int_{a}^{\xi} f(x) \mathrm{d} x+g(b) \int_{\xi}^{b} f(x) \mathrm{d} x$

Newton-Leibniz formula

Let $f$ be a continuous real-valued function defined on a closed interval $[a, b]$ . Let $F$ be the function defined, s.t.

$\frac{d}{\mathrm{d} x} \int_{a}^{x} f(t) \mathrm{d} t=f(x), \quad \forall x \in[a, b]$

Substitution Rule For Definite Integrals

Suppose $f \in C[a, b], \varphi:[\alpha, \beta] \rightarrow[a, b]$ and $\varphi^{\prime} \in \mathcal{R}[\alpha, \beta],\varphi(\alpha)=a, \varphi(\beta)=b$ , s.t.

$\int_{a}^{b} f(x) \mathrm{d} x=\int_{\alpha}^{\beta} f(\varphi(t)) \varphi^{\prime}(t) \mathrm{d} t$

The Abstract Of Mathematical Analysis I

1. Limits

Definition: series limit

Definition: fundamental or Cauchy sequence

Theorem: Weierstrass

Two important limit

Definition 3. inferior limit and superior limit

Theorem 2. Stolz

Theorem 3. Toeplitz limit theorem

Stirling’s formula

2. Continuity

Definition 0

3. Differential calculus

Definition 0

Definition 1

Definition 2

The derivative of an inverse function

The derivative of some common function formula

L’Hôpital’s rule

Taylor’s theorem

prove:

remainder term

4. Integral

Antiderivative

Definition

Theorem: Integration by parts

Example: Wallis product

Prove:

Simplify the Polynomial and Integral

Primitives of the Form

Integration

Riemann Sums

partition

Integral mean value theorem

The second Integral mean value theorem

Newton-Leibniz formula

Substitution Rule For Definite Integrals

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