Revision as of 11:45, 16 November 2010

1.1 Basic Mathematics

1.1.1 Mathematical notation

• ≈ : approximately equal

• ~ : CST ·

Supremum and infimum vs. maximum and minimum

The concept of supremum, or least upper bound, is as follows: Let $ S={a[n]} $, the sequence with terms $ a[0],a[1],\cdots $ over all the nonnegative integers. $ S $ has a supremum, called $ \sup S $ , if for every $ n , a[n]\leq\sup S $ (i.e. no a[n] exceeds $ \sup S $ ), and furthermore, $ \sup S $ is the least value with this property; that is, if $ a[n]\leq b $ for all $ n $, then $ \sup S\leq b $ for all such $ b $ . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum $ \inf S $ , or greatest lower bound.

• Consider the set $ \left\{ x:\;0<x<1\right\} $ . There is no maximum or minimum, however $ 0 $ is the infimum and $ 1 $ is the supremum.

• Consider the set $ S={a[n]},\; a[n]=1/n $ where $ n $ is a positive integer.

– $ \sup S=1 $ , since $ 1/n>1/(n+1) $ for all such $ n $ , and so the largest term is the first. The maximum is also $ 1 $.

– $ \inf S=0 $ . However, the minimum does not exist.

Well-known sets

• $ \mathbb{N} $ : the set of natural numbers. It is countably infinite.

– $ \mathbb{N}_{0}=\left\{ 0,1,\cdots\right\} $

– $ \mathbb{N}^{*}=\mathbb{N}_{1}=\left\{ 1,2,\cdots\right\} $

• $ \mathbb{Z}_{n} $ : the set of modulo $ n $

Logarithm

• $ \log $ : base 2

• $ \ln $ : base $ e $

Partition

• A set $ \mathcal{A}=\left\{ A_{1},A_{2},\cdots,A_{k}\right\} $ of disjoint subsets of a set $ A $ is a partition of $ A $ if $ A=\bigcup_{i=1}^{k}A_{i} $ and $ A_{i}\neq\varnothing $ for every $ i $ .

• Another partition $ \left\{ A'_{1},A'_{2},\cdots,A'_{l}\right\} $ of $ A $ refines the partition $ \mathcal{A} $ if $ A'_{i} $ is contained in some $ A_{j} $ .

$ k $ -sets and $ k $ -subsets

• $ \left[A\right]^{k} $ is the set of all $ k $ -element subsets of $ A $ .

• Sets with $ k $ elements will be called $ k $ -sets.

• Similarly, subsets with $ k $ elements are $ k $ -subsets.

1.1.2 Taylor series

$ e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!} $.

$ \frac{x}{\left(1-x\right)^{2}}=\sum_{n=1}^{\infty}nx^{n}=\sum_{n=0}^{\infty}nx^{n} $ for $ \left|x\right|<1 $.

$ \log\left(1-x\right)=-\sum_{n=1}^{\infty}\frac{x^{n}}{n} $ for $ -1\leq x<1 $.

1.1.3 Differentiation and Integration

$ \left(\frac{b}{a}\right)^{\prime}=\frac{b^{\prime}a-ba^{\prime}}{a^{2}} $

$ \left(f\left(g\left(x\right)\right)\right)^{\prime}=f^{\prime}\left(g\left(x\right)\right)\cdot g^{\prime}\left(x\right) $

$ \int uv^{\prime}=uv-\int u^{\prime}v $

$ \int x^{n}dx=\frac{x^{n+1}}{n+1}+C\longleftrightarrow\left(x^{n}\right)^{\prime}=n\cdot x^{n-1} $

$ \int e^{x}dx=e^{x}+C\longleftrightarrow\left(e^{x}\right)^{\prime}=e^{x} $

$ \int a^{x}dx=\frac{a^{x}}{\ln a}+C\longleftrightarrow\left(a^{x}\right)^{\prime}=a^{x}\cdot\ln a $

$ \int e^{kx}dx=\frac{e^{kx}}{k}+C\longleftrightarrow\left(e^{kx}\right)^{\prime}=k\cdot e^{kx} $

$ \int\cos xdx=\sin x+C\longleftrightarrow\left(\sin x\right)^{\prime}=\cos x $

$ \int\sin xdx=-\cos x+C\longleftrightarrow\left(\cos x\right)^{\prime}=-\sin x $

$ \int\tan xdx=-\ln\left|\left(\cos x\right)^{-1}\right|+C=\ln\left|\sec x\right|+C\longleftrightarrow\left(\tan x\right)'=\sec^{2}x $

$ \int\frac{\sin x}{x}dx=? $

Integration using substitution

1. The problem is $ \int\frac{2t}{1+t^{2}}dt $.

2. We substitute $ 1+t^{2} $ as r . We can get $ 1tdt=dr $ because $ 1+t^{2}=r $ .

3. Now, we can get the solution using sutitution as $ \int\frac{1}{r}dr=\ln r=\ln\left(1+t^{2}\right) $.

1.1.4 Relations between cosine, sine and exponential functions

$ e^{\pm i\theta}=\cos\theta\pm i\sin\theta ${ (Euler's formula)}.

$ \sin\theta=\frac{1}{2i}\left(e^{i\theta}-e^{-i\theta}\right)=\frac{1}{2i}\left(\left(\cos\theta+i\sin\theta\right)-\left(\cos\theta-i\sin\theta\right)\right)=\frac{2i\sin\theta}{2i}=\sin\theta $.

$ \cos\theta=\frac{1}{2}\left(e^{i\theta}+e^{-i\theta}\right)=\frac{1}{2}\left(\left(\cos\theta+\sin\theta\right)+\left(\cos\theta-\sin\theta\right)\right)=\frac{2\cos\theta}{2}=\cos\theta $.

$ \cos\left(2x\right)=\cos^{2}\left(x\right)-\sin^{2}\left(x\right)=2\cos^{2}\left(x\right)-1=1-2\sin^{2}\left(x\right) $.

$ \cos^{2}\left(x\right)+\sin^{2}\left(x\right)=1 $.

$ \sec x=\frac{1}{\cos x} $.

$ \csc x=\frac{1}{\sin x} $.

$ \cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x} $.

1.1.5 Set operation

• $ A-B=A\cap\bar{B} $

• $ A\triangle B=A\cup B-A\cap B=\left(A\cup B\right)\cap\overline{\left(A\cap B\right)} $

• $ A=B\Longleftrightarrow A\subset B\textrm{ and }A\supset B $

• $ A=A\cap S=A\cap\left(B\cup\bar{B}\right)=\left(A\cap B\right)\cup\left(A\cap\bar{B}\right) $