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| export const MATH_FORMULAS_MD = String.raw` |
| # Mathematical Formulas and Expressions |
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| This document demonstrates various mathematical notation and formulas that can be rendered using LaTeX syntax in markdown. |
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| ## Basic Arithmetic |
| |
| ### Addition and Summation |
| $$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$ |
| |
| ## Algebra |
| |
| ### Quadratic Formula |
| The solutions to $ax^2 + bx + c = 0$ are: |
| $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ |
| |
| ### Binomial Theorem |
| $$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$ |
| |
| ## Calculus |
| |
| ### Derivatives |
| The derivative of $f(x) = x^n$ is: |
| $$f'(x) = nx^{n-1}$$ |
| |
| ### Integration |
| $$\int_a^b f(x) \, dx = F(b) - F(a)$$ |
| |
| ### Fundamental Theorem of Calculus |
| $$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$ |
| |
| ## Linear Algebra |
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| ### Matrix Multiplication |
| If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then: |
| $$C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$$ |
| |
| ### Eigenvalues and Eigenvectors |
| For a square matrix $A$, if $Av = \lambda v$ for some non-zero vector $v$, then: |
| - $\lambda$ is an eigenvalue |
| - $v$ is an eigenvector |
| |
| ## Statistics and Probability |
| |
| ### Normal Distribution |
| The probability density function is: |
| $$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$ |
| |
| ### Bayes' Theorem |
| $$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$ |
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| ### Central Limit Theorem |
| For large $n$, the sample mean $\bar{X}$ is approximately: |
| $$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$$ |
| |
| ## Trigonometry |
| |
| ### Pythagorean Identity |
| $$\sin^2\theta + \cos^2\theta = 1$$ |
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| ### Euler's Formula |
| $$e^{i\theta} = \cos\theta + i\sin\theta$$ |
| |
| ### Taylor Series for Sine |
| $$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$ |
| |
| ## Complex Analysis |
| |
| ### Complex Numbers |
| A complex number can be written as: |
| $$z = a + bi = r e^{i\theta}$$ |
| |
| where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$ |
| |
| ### Cauchy-Riemann Equations |
| For a function $f(z) = u(x,y) + iv(x,y)$ to be analytic: |
| $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$ |
| |
| ## Differential Equations |
| |
| ### First-order Linear ODE |
| $$\frac{dy}{dx} + P(x)y = Q(x)$$ |
| |
| Solution: $y = e^{-\int P(x)dx}\left[\int Q(x)e^{\int P(x)dx}dx + C\right]$ |
| |
| ### Heat Equation |
| $$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$ |
| |
| ## Number Theory |
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| ### Prime Number Theorem |
| $$\pi(x) \sim \frac{x}{\ln x}$$ |
| |
| where $\pi(x)$ is the number of primes less than or equal to $x$. |
| |
| ### Fermat's Last Theorem |
| For $n > 2$, there are no positive integers $a$, $b$, and $c$ such that: |
| $$a^n + b^n = c^n$$ |
| |
| ## Set Theory |
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| ### De Morgan's Laws |
| $$\overline{A \cup B} = \overline{A} \cap \overline{B}$$ |
| $$\overline{A \cap B} = \overline{A} \cup \overline{B}$$ |
| |
| ## Advanced Topics |
| |
| ### Riemann Zeta Function |
| $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$$ |
| |
| ### Maxwell's Equations |
| $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$ |
| $$\nabla \cdot \mathbf{B} = 0$$ |
| $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ |
| $$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$$ |
| |
| ### Schrödinger Equation |
| $$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$$ |
| |
| ## Inline Math Examples |
| |
| Here are some inline mathematical expressions: |
| |
| - The golden ratio: $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$ |
| - Euler's number: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$ |
| - Pi: $\pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$ |
| - Square root of 2: $\sqrt{2} = 1.41421356...$ |
| |
| ## Fractions and Radicals |
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| Complex fraction: $\frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f} - \frac{g}{h}}$ |
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| Nested radicals: $\sqrt{2 + \sqrt{3 + \sqrt{4 + \sqrt{5}}}}$ |
| |
| ## Summations and Products |
| |
| ### Geometric Series |
| $$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{for } |r| < 1$$ |
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| ### Product Notation |
| $$n! = \prod_{k=1}^{n} k$$ |
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| ### Double Summation |
| $$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}$$ |
| |
| ## Limits |
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| $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ |
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| $$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$$ |
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| ## Further Bracket Styles and Amounts |
| |
| - \( \mathrm{GL}_2(\mathbb{F}_7) \): Group of invertible matrices with entries in \(\mathbb{F}_7\). |
| - Some kernel of \(\mathrm{SL}_2(\mathbb{F}_7)\): |
| \[ |
| \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \right\} = \{\pm I\} |
| \] |
| - Algebra: |
| \[ |
| x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a} |
| \] |
| - $100 and $12.99 are amounts, not LaTeX. |
| - I have $10, $3.99 and $x + y$ and $100x$. The amount is $2,000. |
| - Emma buys 2 cupcakes for $3 each and 1 cookie for $1.50. How much money does she spend in total? |
| - Maria has $20. She buys a notebook for $4.75 and a pack of pencils for $3.25. How much change does she receive? |
| - 1 kg の質量は |
| \[ |
| E = (1\ \text{kg}) \times (3.0 \times 10^8\ \text{m/s})^2 \approx 9.0 \times 10^{16}\ \text{J} |
| \] |
| 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。 |
| - Algebra: \[ |
| x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a} |
| \] |
| - Algebraic topology, Homotopy Groups of $\mathbb{S}^3$: |
| $$\pi_n(\mathbb{S}^3) = \begin{cases} |
| \mathbb{Z} & n = 3 \\ |
| 0 & n > 3, n \neq 4 \\ |
| \mathbb{Z}_2 & n = 4 \\ |
| \end{cases}$$ |
| - Spacer preceded by backslash: |
| \[ |
| \boxed{ |
| \begin{aligned} |
| N_{\text{att}}^{\text{(MHA)}} &= |
| h \bigl[\, d_{\text{model}}\;d_{k} + d_{\text{model}}\;d_{v}\, \bigr] && (\text{Q,K,V の重み})\\ |
| &\quad+ h(d_{k}+d_{k}+d_{v}) && (\text{バイアス Q,K,V）}\\[4pt] |
| &\quad+ (h d_{v})\, d_{\text{model}} && (\text{出力射影 }W^{O})\\ |
| &\quad+ d_{\text{model}} && (\text{バイアス }b^{O}) |
| \end{aligned}} |
| \] |
| |
| ## Formulas in a Table |
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| | Area | Expression | Comment | |
| |------|------------|---------| |
| | **Algebra** | \[ |
| x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a} |
| \] | Quadratic formula | |
| | | \[ |
| (a+b)^{n} = \sum_{k=0}^{n}\binom{n}{k}\,a^{\,n-k}\,b^{\,k} |
| \] | Binomial theorem | |
| | | \(\displaystyle \prod_{k=1}^{n}k = n! \) | Factorial definition | |
| | **Geometry** | \( \mathbf{a}\cdot \mathbf{b} = \|\mathbf{a}\|\,\|\mathbf{b}\|\,\cos\theta \) | Dot product & angle | |
| |
| ## No math (but chemical) |
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| Balanced chemical reaction with states: |
| |
| \[ |
| \ce{2H2(g) + O2(g) -> 2H2O(l)} |
| \] |
| |
| The standard enthalpy change for the reaction is: $\Delta H^\circ = \pu{-572 kJ mol^{-1}}$. |
| |
| --- |
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| *This document showcases various mathematical notation and formulas that can be rendered in markdown using LaTeX syntax.* |
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