Analysis

• \( \limsup \) , \( \overline{\lim} \)
  • Supremum Limit, Limit Supremum, Limsup, Superior Limit, Upper Limit, Outer Limit.
• \( \liminf \) , \( \underline{\lim} \)
  • Infimum Limit, Limit Infimum, Liminf, Inferior Limit, Lower Limit, Inner Limit

A sequence \( \{x_i\}_{i=1}^\infty \) in a metric space \( (X, d) \) is called Cauchy sequence if: \[ \forall \varepsilon > 0, \exists N > 0 : \forall m,n > N, d(x_m, x_n) <\varepsilon. \]

Useful in finding a root of a function.

Convert the problem \( f(x) = 0 \) into the form \( x = g(x) \). Initiate the iterations with \( x_{n+1} = g(x_n) \). It may converge to root \( \alpha \), since the root is the fixed point of the iteration.

The iteration is gueranteed to converge around the fixed-point if the slope is confined within -1 and 1. To be more precise, the graph of the function should not poke through the tops and bottoms of any square centered at the origin. Although there are possiblities that the point outside the square can converge.

The slope \( m \) around the fixed-point can be used to approximate how fast the iteration converges: \( O(|x_n - \alpha|) = |m|^n \).

• Monotone surjective function is continuous, and admits an inverse function.
• Given a sequence of functions \((f_n\colon I\to \mathbb{R})_{n=1}^\infty\) that converges pointwise, that is: \[ \forall x\in I, \lim_{n\to \infty}f_n(x) =: f(x) \]
  • the function \(f(x)\) that is referred to as the pointwise limit of the sequence, need not be continuous, even if all functions \(f_n\) are continuous.
  • \[ f_n(x)\ \text{continuous} \mathrel{\hskip9pt\not\phantom{0}\hskip-16.5pt\implies} \lim_{n\to \infty}(f_n(x))\ \text{continuous} \]

• 임계점

• If \( f \) is continuous on the interval \( [a,b] \) , then \( f \) has maxima and minima on \( [a,b] \) .

For a continuous function \( f: [a,b] \to \mathbb{R} \), \[ \min(f(a), f(b)) < u < \max(f(a), f(b)) \implies \exists c \in (a,b), f(c) = u. \]

Consider the subset \( S := \{ x \in [a,b] \mid f(x) < u \} \). \( c = \sup(S) \) exists due to the completeness of the real number. \( f(c) \) can only be \( u \) due to the continuity of \( f \) and the totally orderedness of the real number.

\begin{align*} (Df)&(\mathbf{a},\mathbf{b})\\[.5em] =& \left[\begin{array}{ccc|ccc} \frac{\partial f_1}{\partial x_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_1}{\partial x_n}(\mathbf{a},\mathbf{b}) & \frac{\partial f_1}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_1}{\partial y_m}(\mathbf{a},\mathbf{b}) \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial x_n}(\mathbf{a},\mathbf{b}) & \frac{\partial f_m}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial y_m}(\mathbf{a},\mathbf{b}) \end{array}\right]\\[.5em] =& \left[\begin{array}{c|c} J_{f,\mathbf{x}} & J_{f,\mathbf{y}} \end{array}\right] \end{align*}

• For a continuously differentiable function \(f: \mathbb{R}^{n+m}\to\mathbb{R}^m\), and a point \((\mathbf{a}, \mathbf{b})\) with \(f(\mathbf{a}, \mathbf{b}) = \mathbf{0}\),
• If the right-hand panel of the Jacobian \(\mathbf{J}_{f,\mathbf{y}}\) is invertible, then:
  • there exists an open set \(U\subset \mathbb{R}^n\) containing \(\mathbf{a}\)
  • such that there exists a unique continuously differentiable function \(g: U\to \mathbb{R}^m\)
  • such that \(g(\mathbf{a})= \mathbf{b}\), and \(f(\mathbf{x}, g(\mathbf{x}))=\mathbf{0}\) for all \(\mathbf{x}\in U\).
  • And the Jacobian matrix of partial derivatives of \(g\) in \(U\) is: \[ \left[\frac{\partial g_i}{\partial x_j}(\mathbf{x})\right]_{m\times n} = -[J_{f,\mathbf{y}}(\mathbf{x}, g(\mathbf{x}))]_{m\times m}^{-1}[J_{f,\mathbf{x}}(\mathbf{x},g(\mathbf{x}))]_{m\times n}. \]

• For three interdependent variables \(x,y,z\): \[ \frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x} = -1. \]

• The interdependence implies the existence of a implicit function \(f(x, y, z) = 0\).
• By the implicit function theorem : \[ \frac{\partial z}{\partial x} = -\frac{\hskip2mm\displaystyle\frac{\partial f}{\partial x}\hskip2mm}{\displaystyle\frac{\partial f}{\partial z}}. \]
• The same thing can be said for the others, yielding the formula.

• The result of the inverse function theorem for the derivative of an inverse.

• Generalized Chain Rule

• Generalized Product Rule

Also known as the Lagrange-Bürmann Formula. The formula for Taylor series expansion of the inverse function.

For a analytic function \( f \) at a point \( a \) and \( f'(a) \neq 0 \), \[ f^{-1}(z) = a + \sum_{n=1}^{\infty} g_n \frac{(z - f(a))^n}{n!} \] where \[ g_n := \lim_{w \to a} \frac{d^{n-1}}{dw^{n-1}} \left[ \left( \frac{w-a}{f(w) - f(a)} \right)^n \right]. \] Futher, \( f^{-1} \) is also an analytic function at \( f(a) \).

A function \( f : D \to \mathbb{R}\) is real analytic on an open set \( D \subset \mathbb{R}\) if for any \( x_0 \in D \) one can write \[ f(x) = \sum_{n=0}^{\infty}a_n(x-x_0)^n \] in which the coefficients \( (a_n)_n \) are real numbers and the series is convergent to \( f(x) \) in a neighborhood of \( x_0 \).

• \[ f(x) = \begin{cases} e^{-\frac{1}{x}} & x\ge 0\\ 0 & \text{otherwise}.\\ \end{cases} \]
  • Smooth but not analytic.
• Fabius function
  • Another example

• A holomorphic funciton is analytic.
• At all non-zero point, the tranfromation given by the function is conformal.

Holomorphic, except for a set of isolated points, which are called the poles of the function.

\(f\) is meromorphic in \(U\) if every point of \(U\) has a neighborhood such that at least one of \(f\) and \(1/f\) is holomorphic in it.

It can be calculated via the

\(f(z)\) is meromorphic at infinity if \(f(1/z)\) is meromorphic in the neighborhood of \(0\). In other words, \(f\) is meromorphic at the complex infinity \(\infty\), or at the top of the Riemann sphere.

If a function is meromorphic and meromorphic at infinity, it is meromorphic on a Riemann sphere, and it is always a rational function.

Many definitions of the integral does not define what \( dx \) is, although the intuition is there. Therefore, the formalism must come separately.

The \( n \)th coefficient \( a_n \) in the power series of \( x \) in \( y \) is given by \[ a_n = \frac{1}{2\pi i} \oint_C \frac{x}{y^{n+1}} \dd{y} \] by Cauchy's integral formula.

Substituting \[ y = \frac{x}{\phi(x)}, \] and a bit of simplication give the above formula.

Method for assigning values to certain improper integrals.

The value for divergent integrals.

For a differentiable function \( f : \mathbb{R}^n \to \mathbb{R} \), if the transformation \[ p_i = \frac{\partial f}{\partial x_i}(x_1,x_2,\dots, x_n) \] is invertible, that is at least for the active variables the function has to be convex, then the Legendre transformation of \( f \) exists, and is defined by \[ f^*(p_1,p_2,\dots, p_n) = \sum_{i=1}^n p_ix_i - f(x_1, x_2,\dots, x_n), \] where, \( x_i \)s are thought of as functions of \( p_i \)s, which then satisfies \[ \frac{\partial f^*}{\partial p_i} = x_i. \]

This can be done on subset of variables. The variables participating in the transformation are called active variables, and the variables that are not are called passive variables.

The negative y-intercept in terms of the slope is the Legendre transformation.

• For integrable function \(f\) and \(g\): \[ (f*g) (t)=\int_P f(\tau)g(t - \tau)\, d\tau. \]
• Note that this is also the case for the complex functions.

• Commutativity
• Associativity
• Distributivity
• It is multiplication within L1 space \(L^1(\mathbb{R}^n)\), since: \[ \|f * g\|_1 \le \| g\|_1\cdot \|g\|_1, \] forming the algebra \((L^1(\mathbb{R}^n), +, *)\) over \(\mathbb{R}\).

• The kernel \(\omega\) is convolved with the image \(f\) to produce the filtered image \(g\): \[ \omega \ast f = g. \]
• The kernel and the image are often represented with matrices.

• For a complex number \(z\) the Möbius transformation \(f\) is: \[ f(z) = \frac{az + b}{cz + d}. \]
  • with a invertible matrix \[ A = \begin{pmatrix} a & b \\ c& d\end{pmatrix} \in \mathrm{GL}(2, \mathbb{C}) \]

• is one of the ingredients.
• Any Möbius map can be mapped to a rigid motion of the Riemann sphere.
• It is the projective transformations of the complex projective line.

  • \[

    \begin{bmatrix} a & b \\ c & d\end{bmatrix}\begin{bmatrix} z \\ 1 \end{bmatrix} \sim \begin{bmatrix} \dfrac{az + b}{cz + d} \\[1em] 1 \end{bmatrix}

    \]

  • and forms the projective linear group \(\mathrm{PGL}(2, \mathbb{C})\).
• The subgroup projective special linear group \(\mathrm{PSL}(2,\mathbb{C})\) is isomorphic to the ((66ade534-e14a-4d82-b5b9-e5a712e29779)) \(\mathrm{SO}^+(1,3)\).

• Replace the partials \(\partial_{x_i}\) by variables \(\xi_i\) in the differential operator.

The linear operator \(P\colon C^\infty(E) \to C^\infty(F)\) on a smooth section of the vector bundles \(E\) and \(F\) of a manifold \(X\), is a differential operator of order \(k\) if, in local coordinates of \(X\): \[ Pu(x) = \sum_{|\alpha| = k} P^\alpha(x)\frac{\partial^\alpha u}{\partial x^\alpha} + \text{lower-order terms} \] where \(P^\alpha(x) \colon E\to F\) is a bundle map symmetric on the multi-indices \(\alpha\).

The \(k\)th order coefficients of \(P\) transforms as a symmetric tensor: \[ \sigma_P\colon S^k(T^*X)\otimes E\to F \] where \(S^k(T^*X)\) is the \(k\)th symmetric power of the cotangent bundle of \(X\).

This symmetric tensor \(\sigma_P\) is known as the principal symbol of \(P\). The principal symbol is obtained by replacing the partials \(\partial_{x_i}\) by variables \(\xi_i\) which is the covectors: \[ \sigma_P(x,\xi) = \sum_{|\alpha| = k} P^\alpha(x)\xi^\alpha. \]

\( \sigma_P \) is the coefficient of the highest order directional derivative in the direction given by its arguments \( \{\xi_i\} \).

\[ \Vert fg\Vert_1 \leq \Vert f\Vert_p\Vert g\Vert_q \] where \(f, g\) are functions on a measure space and \(p,q\in [1,\infty], 1/p+1/q=1\) and \[ \Vert\cdot\Vert_p=\left(\int_S|\cdot|^p\,d\mu\right)^{\frac{1}{p}}. \]