Analysis

Cauchy's integral formula and Taylor series

Cauchy’s integral formula defines analytic function evaluation with path integral with denominator translation at evaluation point (\(\frac{1}{z-a}\)). \(f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz\,\) \(f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz\) Cauchy’s integral formula is a limit of path. \(\lim_{r \rightarrow 0}\gamma : \lvert z-z_{0} \rvert = r\) Taylor series evaluated a analytic function by approximation at an open disc \(D(z_{0}, r)\). \(f(x) = \sum_{n=0}^\infty a_n(x-b)^n\) \(\frac{f^{(n)}(b)}{n!} = a_n\)

Primitive function

The presence of primitive function is a strong condition that makes a function is analytic in a disc \(D(a,R)\). The meaning is the presence of primitive function is confusing at first to me. If a function is integrable, then integration value and a primitive function can be determined. But in complex analysis this is not the case. In real analysis, the integral interval \([a, b]\) is unique, but in complex analysis the integral interval should be determined by line path \(\Gamma = g(x)\).

Linear transformation

A norm of linear transformation \(\Lambda : X \rightarrow Y\) is defined by \(\| \Lambda \| = \sup \{ \| \Lambda(x) \|:x \in X, \|x\| \le 1 \}\). We can give a norm to a space or a set. A norm determines the size of a vector in the function space. The way of measure the size of a vector gives important properties of the space like boundness, completeness or orthogonality.

Convergence

It is the main subject of analysis that finding conditions making sequential mathematical objects like a set, sequence, series to be convergent. Induction changes \(S = \mathbb{N}\) to \(s_{1} \in S\) and if $ s_{n} S$ then \(s_{n+1} \in S\). The natural number has a property of endless addable with one. But, induction can prove only natural number \(\mathbb {N}\) not infinity \(\infty\). \[ Induction \\ s_{1} \in S \\ if\ s_{n} \in S \ then \ s_{n+1} \in S \\ Then\ S = \mathbb{N} \\ \] The limit is the way \(\mathbb {N}\) goes to \(\infty\).

Differential equations and Fourier transformation

Differential equations describe the change of state. The change relates to the state. The solutions of the differential equations are the status equations. The initial conditions set the time \(t\) and status \(y\). The boundary conditions are the value of boundary \(y_0\) and \(y_1\). \(dy \over dt\) \(= ay + q(t)\) starting from \(y(0)\) at $t=0. inital conditions \(t = 0\) and \(y=1\) \(q(t)\) is a input and \(y(t)\) is a response.

Taylor series

\[ f(x) = \sum_{k=0}^\infty c_k x^k = c_0 + c_1 x + c_2 x^2 + \dotsb. \] This is an approximation that is a function of h and derivatives of \(f(x)\) are elements of parameters. \(f(x \pm h) = f(x) \pm hf'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6}f'''(x) + O(h^4)\) Let’s think about \(\sin(x)\). \[ f(x) = \sin(x) \ f(0) = 0, f'(x)=\cos(x)\ f'(0)=1, f''(x)=-\sin(x)\ f''(0)=0 \] Thus, \[\begin{align*} \sin(x) &= 0 + \frac{1}{1!

Limit of inequality of sequence and epsilon

Here I summarize some tools for proof of the Riesz representation theorem. They are the limit of inequality of sequence and \(\epsilon\). The Rudin’s proof of the Riesz representation theorem construct measure \(\mu\) and measurable set \(\mathfrak{M}\), then prove the \(\mu\) and \(\mathfrak{M}\) have properties. Countable additivity (not subadditivity) is an important property. The strategy of proving equality (additivity) is bidirectional inequality. Limit of inequality of sequence gives us a tool that finite inequality makes infinite inequality.

Positive Borel measures

This is a note of real and complex analysis chapter 2. Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set (\(C\)) (\(\Lambda f\)) represents the integration of the function (\(\int f du\)) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let \(\Lambda\) be a positive linear functional on \(C_c(X)\).

Abstract integration

This is a note for Rudin’s real and complex analysis chapter 1. The key concepts are \(\sigma\)-algebra, measure (\(\mu\)) zero, and linear combination. The three concepts bring me abstract integration. The \(\sigma\)-algebra makes that countable sum and measure of complement (subtract measure) can be possible. Measure zero completes the system. linear combination integrates a measurable function. After a measure space established, Lebesgue’s monotone convergence theorem, Fatou’s lemma, and Lebesgue’s dominant convergence theorem follow.

Compactness

The compact is a property of space. In a nutshell, if space is compact, we can treat the space be a finite because space has a finite subcover. A continuous function on a compact space is uniformly continuous. Heine-Borel theorem describes the condition of compactness of finite dimensional space. Closed and bounded But the Heine-Borel theorem does not hold in an infinite-dimensional space. We need another condition. Previously, the compact space can be finite by taking subcover.