1/* eslint-disable no-irregular-whitespace */
2// Math Formulas Content
3export const MATH_FORMULAS_MD = String.raw`
4# Mathematical Formulas and Expressions
5
6This document demonstrates various mathematical notation and formulas that can be rendered using LaTeX syntax in markdown.
7
8## Basic Arithmetic
9
10### Addition and Summation
11$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
12
13## Algebra
14
15### Quadratic Formula
16The solutions to $ax^2 + bx + c = 0$ are:
17$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
18
19### Binomial Theorem
20$$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$
21
22## Calculus
23
24### Derivatives
25The derivative of $f(x) = x^n$ is:
26$$f'(x) = nx^{n-1}$$
27
28### Integration
29$$\int_a^b f(x) \, dx = F(b) - F(a)$$
30
31### Fundamental Theorem of Calculus
32$$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$
33
34## Linear Algebra
35
36### Matrix Multiplication
37If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then:
38$$C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$$
39
40### Eigenvalues and Eigenvectors
41For a square matrix $A$, if $Av = \lambda v$ for some non-zero vector $v$, then:
42- $\lambda$ is an eigenvalue
43- $v$ is an eigenvector
44
45## Statistics and Probability
46
47### Normal Distribution
48The probability density function is:
49$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$
50
51### Bayes' Theorem
52$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
53
54### Central Limit Theorem
55For large $n$, the sample mean $\bar{X}$ is approximately:
56$$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$$
57
58## Trigonometry
59
60### Pythagorean Identity
61$$\sin^2\theta + \cos^2\theta = 1$$
62
63### Euler's Formula
64$$e^{i\theta} = \cos\theta + i\sin\theta$$
65
66### Taylor Series for Sine
67$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
68
69## Complex Analysis
70
71### Complex Numbers
72A complex number can be written as:
73$$z = a + bi = r e^{i\theta}$$
74
75where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$
76
77### Cauchy-Riemann Equations
78For a function $f(z) = u(x,y) + iv(x,y)$ to be analytic:
79$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
80
81## Differential Equations
82
83### First-order Linear ODE
84$$\frac{dy}{dx} + P(x)y = Q(x)$$
85
86Solution: $y = e^{-\int P(x)dx}\left[\int Q(x)e^{\int P(x)dx}dx + C\right]$
87
88### Heat Equation
89$$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$
90
91## Number Theory
92
93### Prime Number Theorem
94$$\pi(x) \sim \frac{x}{\ln x}$$
95
96where $\pi(x)$ is the number of primes less than or equal to $x$.
97
98### Fermat's Last Theorem
99For $n > 2$, there are no positive integers $a$, $b$, and $c$ such that:
100$$a^n + b^n = c^n$$
101
102## Set Theory
103
104### De Morgan's Laws
105$$\overline{A \cup B} = \overline{A} \cap \overline{B}$$
106$$\overline{A \cap B} = \overline{A} \cup \overline{B}$$
107
108## Advanced Topics
109
110### Riemann Zeta Function
111$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$$
112
113### Maxwell's Equations
114$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
115$$\nabla \cdot \mathbf{B} = 0$$
116$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
117$$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$$
118
119### Schrödinger Equation
120$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$$
121
122## Inline Math Examples
123
124Here are some inline mathematical expressions:
125
126- The golden ratio: $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$
127- Euler's number: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$
128- Pi: $\pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$
129- Square root of 2: $\sqrt{2} = 1.41421356...$
130
131## Fractions and Radicals
132
133Complex fraction: $\frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f} - \frac{g}{h}}$
134
135Nested radicals: $\sqrt{2 + \sqrt{3 + \sqrt{4 + \sqrt{5}}}}$
136
137## Summations and Products
138
139### Geometric Series
140$$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{for } |r| < 1$$
141
142### Product Notation
143$$n! = \prod_{k=1}^{n} k$$
144
145### Double Summation
146$$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}$$
147
148## Limits
149
150$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$
151
152$$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$$
153
154## Further Bracket Styles and Amounts
155
156- \( \mathrm{GL}_2(\mathbb{F}_7) \): Group of invertible matrices with entries in \(\mathbb{F}_7\).
157- Some kernel of \(\mathrm{SL}_2(\mathbb{F}_7)\):
158 \[
159 \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \right\} = \{\pm I\}
160 \]
161- Algebra:
162\[
163x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a}
164\]
165- $100 and $12.99 are amounts, not LaTeX.
166- I have $10, $3.99 and $x + y$ and $100x$. The amount is $2,000.
167- Emma buys 2 cupcakes for $3 each and 1 cookie for $1.50. How much money does she spend in total?
168- Maria has $20. She buys a notebook for $4.75 and a pack of pencils for $3.25. How much change does she receive?
169- 1 kg の質量は
170 \[
171 E = (1\ \text{kg}) \times (3.0 \times 10^8\ \text{m/s})^2 \approx 9.0 \times 10^{16}\ \text{J}
172 \]
173 というエネルギーに相当します。これは約 21 百万トンの TNT が爆発したときのエネルギーに匹敵します。
174- Algebra: \[
175x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a}
176\]
177- Algebraic topology, Homotopy Groups of $\mathbb{S}^3$:
178$$\pi_n(\mathbb{S}^3) = \begin{cases}
179\mathbb{Z} & n = 3 \\
1800 & n > 3, n \neq 4 \\
181\mathbb{Z}_2 & n = 4 \\
182\end{cases}$$
183- Spacer preceded by backslash:
184\[
185\boxed{
186\begin{aligned}
187N_{\text{att}}^{\text{(MHA)}} &=
188h \bigl[\, d_{\text{model}}\;d_{k} + d_{\text{model}}\;d_{v}\, \bigr] && (\text{Q,K,V の重み})\\
189&\quad+ h(d_{k}+d_{k}+d_{v}) && (\text{バイアス Q,K,V）}\\[4pt]
190&\quad+ (h d_{v})\, d_{\text{model}} && (\text{出力射影 }W^{O})\\
191&\quad+ d_{\text{model}} && (\text{バイアス }b^{O})
192\end{aligned}}
193\]
194
195## Formulas in a Table
196
197| Area | Expression | Comment |
198|------|------------|---------|
199| **Algebra** | \[
200x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a}
201\] | Quadratic formula |
202| | \[
203(a+b)^{n} = \sum_{k=0}^{n}\binom{n}{k}\,a^{\,n-k}\,b^{\,k}
204\] | Binomial theorem |
205| | \(\displaystyle \prod_{k=1}^{n}k = n! \) | Factorial definition |
206| **Geometry** | \( \mathbf{a}\cdot \mathbf{b} = \|\mathbf{a}\|\,\|\mathbf{b}\|\,\cos\theta \) | Dot product & angle |
207
208## No math (but chemical)
209
210Balanced chemical reaction with states:
211
212\[
213\ce{2H2(g) + O2(g) -> 2H2O(l)}
214\]
215
216The standard enthalpy change for the reaction is: $\Delta H^\circ = \pu{-572 kJ mol^{-1}}$.
217
218---
219
220*This document showcases various mathematical notation and formulas that can be rendered in markdown using LaTeX syntax.*
221`;