Mathematical analysis is the systematic study of real- and complex-valued functions. A large part of mathematics can be considered as mathematical analysis, including differentiation, integration, dynamical system, series and functional analysis.

Notes:

Handouts:

## Real Analysis

Real function $f: \mathbb{R} \mapsto \mathbb{R}$ is a map between (subsets of) the real line. Real-valued function $f: X \mapsto \mathbb{R}$ is a map whose codomain is the real line. Function of a real variable $f: \mathbb{R} \mapsto X$ is a map whose domain is the real line.

Zero set $f^{-1}(0)$ of a real/vector-valued function is the preimage of zero. Sublevel set $f^{-1}(-\infty, c]$ of a real-valued function on a set is the preimage of an upper-bounded interval.

Extremal point of a continuous real function on an interval is a point that maximizes the absolute function: $f \in C([a, b])$, $x_0 \in \arg\max_{x \in [a, b]} |f(x)|$. Critical point and regular point of a differentiable real-valued function is a point where the differential is zero or nonzero, respectively.

Partial derivative $\frac{\partial f}{\partial x^j}(a)$ of a real-valued function on an open subset of a Euclidean space at a point is its ordinary derivative w.r.t. a coordinate while the other coordinates are fixed: $\frac{\partial f}{\partial x^j}(a) = \lim_{h \to 0} [f(a + h e_j) - f(a)] / h$. Jacobian matrix $(\partial F^i / \partial x^j)$ of a function between (open subsets of) Euclidean spaces is the matrix of the partial derivatives of its component functions.

Continuously differentiable function between (open subsets of) Euclidean spaces is one such that all the partial derivatives of its component functions exist and are continuous at each point in its domain: $f \in C^1(U, \mathbb{R}^m)$. Smooth function between (open subsets of) Euclidean spaces is one such that all the partial derivatives of all orders of its component functions exist and are continuous at each point in its domain: $\{\frac{\partial^k F^i}{\partial x^J} : k \in \mathbb{N}, J \in n^k, i \in m\} \subset C^0(U)$; denoted as $f \in C^\infty(U, \mathbb{R}^m)$. Smooth function between subsets of Euclidean spaces is one that admits a smooth extension in a neighborhood of each point in its domain; a global extension is unnecessary.

Total derivative $DF(a)$ of a map $F: U \mapsto W$ between (open subsets of) finite-dimensional normed vector spaces that is differentiable at an interior point $a$ of its domain is the linear map $DF(a) \in \mathcal{L}(V, W)$ that satisfies $F(a+v) = F(a) + DF(a) v + R(v)$, $\lim_{v \to 0} |R(v)|/|v| = 0$. The total derivative of a function between (open subsets of) Euclidean spaces that is differentiable at an interior point of its domain is the linear map whose standard matrix representation is the Jacobian of the map at that point: $DF(a) = (\partial F^i / \partial x^j)(a)$. Directional derivative $D_v f(a)$ of a smooth real-valued function on an open subset of a Euclidean space at a point $a$ in the direction $v$ is the ordinary derivative of the composite function $g(t) = f(a + t v)$ at zero: $D_v f(a) = d f(a + t v) / d t |_{t=0}$; by chain rule, we have $D_v f(a) = Df(a) v$.

Real analytic function (解析函数) is a real function such that at every point in the domain there is a power series that converges to the function on a neighborhood of the point: $\forall x_0 \in U$, $\exists (a_n)_{n \in \mathbb{N}}$, $\exists U(x_0)$: $\sum_{n=0}^\infty a_n (x - x_0)^n = f|_{U(x_0)}$. Real-valued analytic function in k variables is a real-valued function on an open subset of the Euclidean k-space such that at every point in the domain there is a series of homogeneous polynomials that converges to the function on a neighborhood of the point: $\forall x_0 \in U$, $(P_n)_{n \in \mathbb{N}} \subset \mathbb{R}[x_i]_{i=1}^k$, $P_n(x_i)_{i=1}^k = \sum_{I \in \mathbb{N}^k}^{\|I\|_1 = n} a_I \prod_{i=1}^k (x_i - x_{0,i})(x_i)^{I_i}$: $\sum_{n=0}^\infty P_n = f|_{U(x_0)}$. The set of all analytic functions on a set is denoted as $C^\omega(X)$. Every analytic function is smooth: $C^\omega(X) \subset C^\infty(X)$; however, smooth real functions may not be analytic. Flat function is a smooth real function all of whose derivatives vanish at a given point. Nonconstant flat functions are non-analytic; for example, $f(x) = e^{-1/x^2}$, $f(0) = 0$.

### Asymptotic notation

Asymptotic notation, order notation, or Bachmann–Landau notation is a collection of notations that describe the limiting behavior of a function w.r.t. another when the argument tends towards a particular value or infinity. These notations first appeared in analytic number theory [@Bachmann1894; @Landau1909], and are useful in approximation of functions, computational complexity of algorithms, and statistics.

Big O notation $f = \mathcal{O}(g)$ or $f = O(g)$ for functions on $X$ means that the former is absolutely bounded by a constant multiple of the latter: $\exists c: \forall x \in X, |f| \le c |g|$. Little O notation $f = o(g)$ for functions with limit point $x_0$ mean that the former is dominated by the latter asymptotically: $\lim_{x \to x_0} f(x) / g(x) = 0$. Soft-O notation $f = \tilde{\mathcal{O}}(g)$ or $f = \tilde{O}(g)$ for functions on $X$ is similar to the big O notation, but ignoring poly-logarithmic factors: $\exists k: f = \mathcal{O}(g \log^k g)$. This is a reasonable simplification because: $\forall k$, $\forall ε > 0$, $\log^k n = o(n^ε)$.

Order equivalence $f \asymp g$ or big theta notation $f = \Theta(g)$ for functions on $X$ means that both are absolutely bounded by a constant multiple of the other: $f \asymp g$ iff $f = \mathcal{O}(g)$ and $g = \mathcal{O}(f)$. Asymptotic equivalence $f \sim g$ for functions with limit point $x_0$ means that their ratio converges to one: $\lim_{x \to x_0} f(x) / g(x) = 1$.

Big omega notation $f = \Omega(g)$ and little omega notation $f = \omega(g)$ are the inverse of the big O notation and the little O notation, respectively: $f = \Omega(g)$ iff $g = \mathcal{O}(f)$; $f = \omega(g)$ iff $g = o(f)$. Knuth big omega notation $f = \Omega(g)$ in computational complexity theory means the former is bounded below by the latter as the argument goes to infinity: $\liminf_{n \to \infty} f(n) / g(n) > 0$.

Big O in probability notation $X_n = \mathcal{O}_p(a_n)$ means that the sequence of random variable divided by numbers is stochastically bounded: $\forall \varepsilon > 0$, $\exists C, N > 0$: $\forall n > N$, $P(|X_n / a_n| > C) < \varepsilon$. Little O in probability notation $X_n = o_p(a_n)$ means that the sequence of random variable divided by numbers converges in probability: $\forall \varepsilon > 0$, $\lim_{n \to \infty} P(|X_n / a_n| > \varepsilon) = 0$.

## Complex Analysis

Holomorphic function (全纯函数) is a complex function on an open subset that is differentiable everywhere. A complex function is holomorphic if and only if it is analytic. Entire function is a holomorphic function on the complex plane.

An element of an analytic function is its power series at a point that has a non-zero radius of convergence: $W_0(z) = \sum_{n \in \mathbb{N}} a_n (z - z_0)^n$, with disc of convergence $K_0 \supset B_r(z_0)$ and $W_0 = f|K_0$. A direct analytic continuation of an element $W_0$ is an element $W_1$ at another point in $W_0$'s disc of convergence: $W_1(z) = \sum_{n \in \mathbb{N}} b_n (z - z_1)^n$, where $z_1 \in K_0$, $z_1 \ne z_0$. An analytic continuation of an element $W_0$ is an element $W_n$ obtained by a chain of direct analytic continuations.

The complete analytic function generated by an element $W_0$ is the union of all analytic continuations of $W_0$: $W = {W_n : n \in \mathbb{N}, \forall i \in n, z_{i+1} \in K_i}$. The domain of existence of a complete function is the union of the discs of convergence of its elements: $D = \bigcup{K_n}$. A complete analytic function is generally a multi-valued function on $D$. A branch of an analytic function, given a domain $D' \subset D$ and an element $W_0$ centered in $D'$, is the union of all elements that can be obtained by analytic continuation of $W_0$ by means of chains with centres belonging to $D'$: $W' = {W_n : n \in \mathbb{N}, \forall i \in n, z_{i+1} \in K_i, z_n \in D'}$. If $D'$ is simply connected, the branch is single-valued in $D'$. The n-th root $\sqrt[n]{z}$ has exactly $n$ single-valued branches; The logarithm $\ln(z)$ has infinitely many single-valued branches.

A singular point of an analytic function is an obstacle to the analytic continuation of an element along any curve of a specific class ${L}$ in the z-plane. A regular point is a point that is not singular. An isolated singular point is a singular point $a$ such that analytic continuation is possible to all points sufficiently close to $a$. A single-valued singular point is an isolated singular point such that analytic continuations along closed curves are identical. A pole is a single-valued singular point if an infinite limit exists: $\lim f(z) = \infty$. An essential singular point is a single-valued singular point that is not a pole. A branch point or a many-valued singular point is an isolated singular point that is not single-valued. An algebraic branch point of order $m$, $m \ge 1$, is a branch point $a$ such that analytic continuations along $m + 1$ single loops around $a$ in the same direction are identical. A transcendental branch point is a branch point that is not algebraic.

## Misc

Vector calculus:

Curvilinear coordinates: