本篇示例专用于展示与测试博客正文中的 LaTeX 数学公式（Math / KaTeX） 解析能力。
公式在 Astro 构建期即被编译为无运行时损耗的纯 HTML/MathML 结构，并自带防溢出的弹性水平滚动容器。
一、行内数学公式（Inline Math）
在文本中使用单个美元符号 $ ... $ 包裹公式表达式：
质能方程：E = m c 2 E = mc^2 E = m c 2
欧拉恒等式：e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
高斯正态分布概率密度函数：f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
调和级数收敛极限：lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6 \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} lim n → ∞ ∑ k = 1 n k 2 1 = 6 π 2
- 质能方程：$E = mc^2$ - 欧拉恒等式：$e^{i\pi} + 1 = 0$ - 高斯分布：$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ 二、块级多行数学公式（Display Math）
使用双美元符号 $$ ... $$ 独立成段展示：
L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) } = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) \mathcal{L}\{\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t)\} = X(s)(s^2 + 2\zeta\omega_n s + \omega_n^2) L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t )} = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) 麦克斯韦方程组（微分形式）
∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 J + μ 0 ε 0 ∂ E ∂ t \begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_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 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned} ∇ ⋅ E ∇ ⋅ B ∇ × E ∇ × B = ε 0 ρ = 0 = − ∂ t ∂ B = μ 0 J + μ 0 ε 0 ∂ t ∂ E 高斯积分与线性代数矩阵
∫ − ∞ ∞ e − x 2 d x = π , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}, \quad
\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} ∫ − ∞ ∞ e − x 2 d x = π , A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn Dieses Beispiel dient der Demonstration und dem Test der Parsing-Fähigkeiten von LaTeX-Mathematikformeln (Math / KaTeX) im Blogtext.
Die Formeln werden während der Astro-Build-Phase in eine reine HTML/MathML-Struktur ohne Laufzeitverluste kompiliert und verfügen über einen flexiblen horizontalen Scroll-Container, der Überläufe verhindert.
Verwenden Sie im Text ein einzelnes Dollarzeichen $ ... $, um Formelausdrücke zu umschließen:
Energie-Masse-Äquivalenz: E = m c 2 E = mc^2 E = m c 2
Eulers Identität: e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
Gaußsche Normalverteilung (Wahrscheinlichkeitsdichtefunktion): f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
Konvergenzgrenzwert der harmonischen Reihe: lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6 \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} lim n → ∞ ∑ k = 1 n k 2 1 = 6 π 2
- Energie-Masse-Äquivalenz: $E = mc^2$ - Eulers Identität: $e^{i\pi} + 1 = 0$ - Gauß-Verteilung: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ Verwenden Sie doppelte Dollarzeichen $$ ... $$, um Formeln als separate Blöcke darzustellen:
L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) } = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) \mathcal{L}\{\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t)\} = X(s)(s^2 + 2\zeta\omega_n s + \omega_n^2) L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t )} = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) ∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 J + μ 0 ε 0 ∂ E ∂ t \begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_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 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned} ∇ ⋅ E ∇ ⋅ B ∇ × E ∇ × B = ε 0 ρ = 0 = − ∂ t ∂ B = μ 0 J + μ 0 ε 0 ∂ t ∂ E Gauß-Integral und lineare Algebra-Matrizen
∫ − ∞ ∞ e − x 2 d x = π , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}, \quad
\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} ∫ − ∞ ∞ e − x 2 d x = π , A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn This example is dedicated to demonstrating and testing the LaTeX mathematical formula (Math / KaTeX) rendering capabilities within blog posts.
Formulas are compiled during Astro’s build phase into pure HTML/MathML structures with no runtime overhead, and they come with a flexible horizontal scroll container that prevents overflow.
1. Inline Math
Wrap an expression with a single dollar sign $ ... $ in the text:
Mass–energy equation: E = m c 2 E = mc^2 E = m c 2
Euler’s identity: e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
Gaussian normal distribution probability density function: f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
Harmonic series convergence limit: lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6 \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} lim n → ∞ ∑ k = 1 n k 2 1 = 6 π 2
- Mass–energy equation: $E = mc^2$ - Euler's identity: $e^{i\pi} + 1 = 0$ - Gaussian distribution: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ 2. Display Math
Use double dollar signs $$ ... $$ to display a block of equations:
L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) } = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) \mathcal{L}\{\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t)\} = X(s)(s^2 + 2\zeta\omega_n s + \omega_n^2) L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t )} = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) ∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 J + μ 0 ε 0 ∂ E ∂ t \begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_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 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned} ∇ ⋅ E ∇ ⋅ B ∇ × E ∇ × B = ε 0 ρ = 0 = − ∂ t ∂ B = μ 0 J + μ 0 ε 0 ∂ t ∂ E Gaussian Integral and Linear Algebra Matrix
∫ − ∞ ∞ e − x 2 d x = π , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}, \quad
\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} ∫ − ∞ ∞ e − x 2 d x = π , A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn Este ejemplo está dedicado a demostrar y probar la capacidad de análisis de fórmulas matemáticas LaTeX (Math / KaTeX) en el cuerpo del blog.
Las fórmulas se compilan durante la fase de construcción de Astro en una estructura HTML/MathML pura sin coste de tiempo de ejecución, y vienen con un contenedor de desplazamiento horizontal elástico que evita desbordamientos.
1. Fórmulas matemáticas en línea (Inline Math)
En el texto se usan signos de dólar simples $ ... $ para envolver la expresión de la fórmula:
Ecuación de equivalencia masa‑energía: E = m c 2 E = mc^2 E = m c 2
Identidad de Euler: e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
Función de densidad de probabilidad de la distribución normal gaussiana: f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
Límite de convergencia de la serie armónica: lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6 \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} lim n → ∞ ∑ k = 1 n k 2 1 = 6 π 2
- 质能方程：$E = mc^2$ - 欧拉恒等式：$e^{i\pi} + 1 = 0$ - 高斯分布：$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ 2. Fórmulas matemáticas en bloque (Display Math)
Se usan dobles signos de dólar $$ ... $$ para presentar la fórmula en un bloque independiente:
L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) } = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) \mathcal{L}\{\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t)\} = X(s)(s^2 + 2\zeta\omega_n s + \omega_n^2) L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t )} = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) ∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 J + μ 0 ε 0 ∂ E ∂ t \begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_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 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned} ∇ ⋅ E ∇ ⋅ B ∇ × E ∇ × B = ε 0 ρ = 0 = − ∂ t ∂ B = μ 0 J + μ 0 ε 0 ∂ t ∂ E Integral de Gauss y matriz de álgebra lineal
∫ − ∞ ∞ e − x 2 d x = π , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}, \quad
\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} ∫ − ∞ ∞ e − x 2 d x = π , A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn Cet exemple est dédié à la démonstration et au test de la capacité d’analyse des formules mathématiques LaTeX (Math / KaTeX) dans le corps des articles du blog.
Les formules sont compilées lors de la phase de build d’Astro en structures HTML/MathML pures, sans surcharge d’exécution, et sont accompagnées d’un conteneur de défilement horizontal élastique qui empêche le débordement.
Pour utiliser une formule dans le texte, encadrez l’expression entre deux symboles dollar $ ... $ :
Équation de l’équivalence masse-énergie : E = m c 2 E = mc^2 E = m c 2
Identité d’Euler : e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
Fonction de densité de probabilité de la distribution normale de Gauss : f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
Limite de convergence de la série harmonique : lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6 \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} lim n → ∞ ∑ k = 1 n k 2 1 = 6 π 2
- Équation de l'équivalence masse-énergie : $E = mc^2$ - Identité d'Euler : $e^{i\pi} + 1 = 0$ - Distribution de Gauss : $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ Utilisez deux symboles dollar $$ ... $$ pour afficher la formule en paragraphe indépendant :
L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) } = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) \mathcal{L}\{\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t)\} = X(s)(s^2 + 2\zeta\omega_n s + \omega_n^2) L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t )} = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) ∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 J + μ 0 ε 0 ∂ E ∂ t \begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_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 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned} ∇ ⋅ E ∇ ⋅ B ∇ × E ∇ × B = ε 0 ρ = 0 = − ∂ t ∂ B = μ 0 J + μ 0 ε 0 ∂ t ∂ E Intégrale de Gauss et matrice d’algèbre linéaire
∫ − ∞ ∞ e − x 2 d x = π , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}, \quad
\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} ∫ − ∞ ∞ e − x 2 d x = π , A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn 本篇範例專用於展示與測試部落格正文中的 LaTeX 數學公式（Math / KaTeX） 解析能力。
公式在 Astro 建構期即被編譯為無執行時損耗的純 HTML/MathML 結構，並自帶防溢出的彈性水平捲動容器。
一、行內數學公式（Inline Math）
在文字中使用單個美元符號 $ ... $ 包裹公式表達式：
質能方程：E = m c 2 E = mc^2 E = m c 2
歐拉恆等式：e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
高斯常態分佈機率密度函數：f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
調和級數收斂極限：lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6 \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} lim n → ∞ ∑ k = 1 n k 2 1 = 6 π 2
- 質能方程：$E = mc^2$ - 歐拉恆等式：$e^{i\pi} + 1 = 0$ - 高斯分佈：$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ 二、區塊多行數學公式（Display Math）
使用雙美元符號 $$ ... $$ 獨立成段呈現：
L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) } = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) \mathcal{L}\{\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t)\} = X(s)(s^2 + 2\zeta\omega_n s + \omega_n^2) L { x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t )} = X ( s ) ( s 2 + 2 ζ ω n s + ω n 2 ) 馬克士威方程組（微分形式）
∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 J + μ 0 ε 0 ∂ E ∂ t \begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_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 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned} ∇ ⋅ E ∇ ⋅ B ∇ × E ∇ × B = ε 0 ρ = 0 = − ∂ t ∂ B = μ 0 J + μ 0 ε 0 ∂ t ∂ E 高斯積分與線性代數矩陣
∫ − ∞ ∞ e − x 2 d x = π , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}, \quad
\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} ∫ − ∞ ∞ e − x 2 d x = π , A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn 
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