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Latex数学公式符号大全（超详细）

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Latex数学公式符号大全（超详细）

2024-07-01 11:31| 来源: 网络整理| 查看: 265

基本符号 小写希腊字母

注：部分希腊字母在数学公式中常以变量形式出现，例如 ϵ \epsilon ϵ在数学中一般写法为 ε \varepsilon ε， ϕ \phi ϕ在数学中通常写作 φ \varphi φ

符号语法符号语法符号语法 α \alpha α\alpha β \beta β\beta γ \gamma γ\gamma θ \theta θ\theta ε \varepsilon ε\varepsilon δ \delta δ\delta μ \mu μ\mu ν \nu ν\nu η \eta η\eta ζ \zeta ζ\zeta λ \lambda λ\lambda ψ \psi ψ\psi σ \sigma σ\sigma ξ \xi ξ\xi τ \tau τ\tau ϕ \phi ϕ\phi φ \varphi φ\varphi ρ \rho ρ\rho χ \chi χ\chi ω \omega ω\omega π \pi π\pi 大写希腊字母

大写希腊字母通常是小写希腊字母的LATEX语法第一个字母改为大写，见下表

符号语法符号语法符号语法 Σ \Sigma Σ\Sigma Π \Pi Π\Pi Δ \Delta Δ\Delta Γ \Gamma Γ\Gamma Ψ \Psi Ψ\Psi Θ \Theta Θ\Theta Λ \Lambda Λ\Lambda Ω \Omega Ω\Omega Φ \Phi Φ\Phi Ξ \Xi Ξ\Xi 常用字体

默认的字体为 A B C d e f ABCdef ABCdef，也就是\mathnormal{ABCdef}（当然，打公式的时候不需要加上这个\mathnormal，直接打字母就是这个效果）

字体语法字体语法 A B C d e f \mathrm{ABCdef} ABCdef\mathrm{ABCdef} A B C d e f \mathbf{ABCdef} ABCdef\mathbf{ABCdef} A B C d e f \mathit{ABCdef} ABCdef\mathit{ABCdef} A B C d e f \pmb{ABCdef} ABCdef\pmb{ABCdef} A B C d e f \mathscr{ABCdef} ABCdef\mathscr{ABCdef} A B C d e f \mathcal{ABCdef} ABCdef\mathcal{ABCdef} A B C d e f \mathfrak{ABCdef} ABCdef\mathfrak{ABCdef} A B C d e f \mathbb{ABCdef} ABCdef\mathbb{ABCdef} 常见运算符 运算符语法运算符语法运算符语法 + + ++ − - −- × \times ×\times ± \pm ±\pm ⋅ \cdot ⋅\cdot ∗ \ast ∗\ast ∪ \cup ∪\cup ∩ \cap ∩\cap ∘ \circ ∘\circ ∨ \lor ∨\lor或\vee ∧ \land ∧\land或\wedge ¬ \lnot ¬\lnot ⊕ \oplus ⊕\oplus ⊖ \ominus ⊖\ominus ⊗ \otimes ⊗\otimes ⊙ \odot ⊙\odot ⊘ \oslash ⊘\oslash ∙ \bullet ∙\bullet x \sqrt{x} x ​\sqrt{x} x n \sqrt[n]{x} nx ​\sqrt[n]{x} 大尺寸运算符 运算符语法运算符语法运算符语法 ∑ \sum ∑\sum ∏ \prod ∏\prod ∫ \int ∫\int ⋃ \bigcup ⋃\bigcup ⋂ \bigcap ⋂\bigcap ∮ \oint ∮\oint ⋁ \bigvee ⋁\bigvee ⋀ \bigwedge ⋀\bigwedge ∬ \iint ∬\iint ∐ \coprod ∐\coprod ⨆ \bigsqcup ⨆\bigsqcup ∯ \oiint ∬ ​\oiint 常见关系符号 符号语法符号语法符号语法 < < >> = = == ≤ \leq ≤\leq ≥ \geq ≥\geq ≠ \neq =\neq ≪ \ll ≪\ll ≫ \gg ≫\gg ≡ \equiv ≡\equiv ⊂ \subset ⊂\subset ⊃ \supset ⊃\supset ≈ \approx ≈\approx ⊆ \subseteq ⊆\subseteq ⊇ \supseteq ⊇\supseteq ∼ \sim ∼\sim ∈ \in ∈\in ∋ \ni ∋\ni ∝ \propto ∝\propto ⊢ \vdash ⊢\vdash ⊣ \dashv ⊣\dashv ⊨ \models ⊨\models ∣ \mid ∣\mid ∥ \parallel ∥\parallel ⊥ \perp ⊥\perp ∉ \notin ∈/\notin ⋈ \Join ⋈\Join ≁ \nsim ≁\nsim ⊊ \subsetneq ⊊\subsetneq ⊋ \supsetneq ⊋\supsetneq 数学模式重音符 符号语法符号语法符号语法 a ^ \hat{a} a^\hat{a} a ˉ \bar{a} aˉ\bar{a} a ~ \tilde{a} a~\tilde{a} a ⃗ \vec{a} a \vec{a} a ˙ \dot{a} a˙\dot{a} a ¨ \ddot{a} a¨\ddot{a} a b c ^ \widehat{abc} abc \widehat{abc} a b c ~ \widetilde{abc} abc \widetilde{abc} a b c ‾ \overline{abc} abc\overline{abc} 箭头

如果需要长箭头，只需要在语法前面加上\long，例如\longleftarrow即为 ⟵ \longleftarrow ⟵，如果加上\Long则变为双线长箭头，例如\Longleftarrow即为 ⟸ \Longleftarrow ⟸

符号语法符号语法符号语法 ← \leftarrow ←\leftarrow → \rightarrow →\rightarrow ↔ \leftrightarrow ↔\leftrightarrow ⇐ \Leftarrow ⇐\Leftarrow ⇒ \Rightarrow ⇒\Rightarrow ⇔ \Leftrightarrow ⇔\Leftrightarrow ↑ \uparrow ↑\uparrow ↓ \downarrow ↓\downarrow ↕ \updownarrow ↕\updownarrow ⇑ \Uparrow ⇑\Uparrow ⇓ \Downarrow ⇓\Downarrow ⇕ \Updownarrow ⇕\Updownarrow ↼ \leftharpoonup ↼\leftharpoonup ↽ \leftharpoondown ↽\leftharpoondown ⇀ \rightharpoonup ⇀\rightharpoonup ⇁ \rightharpoondown ⇁\rightharpoondown ⇌ \rightleftharpoons ⇌\rightleftharpoons ⇋ \leftrightharpoons ⇋\leftrightharpoons    ⟺    \iff ⟺\iff ↦ \mapsto ↦\mapsto 括号 括号语法括号语法括号语法 ( ) () ()() [ ] [] [][] { } \{\} {}\{\} ⌊ ⌋ \lfloor\rfloor ⌊⌋\lfloor\rfloor ⌈ ⌉ \lceil\rceil ⌈⌉\lceil\rceil ⟨ ⟩ \langle\rangle ⟨⟩\langle\rangle 大尺寸括号 括号语法括号语法 ( ) \left(\right) ()\left( \right) [ ] \left[ \right] []\left[ \right] x 1 x 2 … x n ⏞ n \overbrace{x_1x_2\ldots x_n}^{n} x1​x2​…xn​ ​n​\overbrace{x_1x_2\ldots x_n}^{n} x 1 x 2 … x n ⏟ n \underbrace{x_1x_2\ldots x_n}_{n} n x1​x2​…xn​​​\underbrace{x_1x_2\ldots x_n}_{n}

注：大尺寸的()和[]是可以根据公式的高度自动调节的，例如

\arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]

arg ⁡ min ⁡ θ [ − ∑ i = 1 n [ y ( i ) ln ⁡ ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) ] ] \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right] argθmin​[−i=1∑n​[y(i)ln(hθ​(x(i)))+(1−y(i))ln(1−hθ​(x(i)))]]

可以看出，括号高度可以框住整个公式

因此在这种大型的公式中，使用大尺寸括号视觉效果更美观

其他常见符号 符号语法符号语法符号语法 ∀ \forall ∀\forall ∃ \exist ∃\exist ∠ \angle ∠\angle ∅ \emptyset ∅\emptyset ∂ \partial ∂\partial ∞ \infty ∞\infty … \ldots …\ldots ⋯ \cdots ⋯\cdots … \dots …\dots ⋮ \vdots ⋮\vdots ⋱ \ddots ⋱\ddots ′ \prime ′\prime ∵ \because ∵\because ∴ \therefore ∴\therefore □ \Box □\Box △ \triangle △\triangle § \S §\S 数学公式写法 上下标 ^：上标_：下标

例如:

\sum_{i=1}^{n}X_n表示 ∑ i = 1 n X n \sum_{i=1}^{n}X_n ∑i=1n​Xn​\int_{0}^{\infty}x^2dx表示 ∫ 0 ∞ x 2 d x \int_{0}^{\infty}x^2dx ∫0∞​x2dx\prod_{i=1}^{n}X_n表示 ∏ i = 1 n X n \prod_{i=1}^{n}X_n ∏i=1n​Xn​ 分数

使用\frac{}{}即可，例如\frac{a}{b}表示 a b \frac{a}{b} ba​

插入文字

使用\text，例如\text{hello,world!}表示 hello,world! \text{hello,world!} hello,world!

常见函数 函数语法函数语法函数语法 log ⁡ ( ) \log() log()\log() ln ⁡ ( ) \ln() ln()\ln() lg ⁡ ( ) \lg() lg()\lg() max ⁡ \max max\max min ⁡ \min min\min lim ⁡ x → ∞ \lim_{x \to \infty} limx→∞​\lim_{x \to \infty} arg ⁡ max ⁡ c ∈ C \arg\max_{c \in C} argmaxc∈C​\arg\max_{c \in C} arg ⁡ min ⁡ c ∈ C \arg\min_{c \in C} argminc∈C​\arg\min_{c \in C} exp ⁡ \exp exp\exp 矩阵、行列式

&表示分隔元素，\\表示换行

A= \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}

A = ( a 11 a 12 a 21 a 22 ) A= \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} A=(a11​a21​​a12​a22​​)

A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

A = [ a 11 a 12 a 21 a 22 ] A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} A=[a11​a21​​a12​a22​​]

A= \begin{Bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{Bmatrix}

A = { a 11 a 12 a 21 a 22 } A= \begin{Bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{Bmatrix} A={a11​a21​​a12​a22​​}

A= \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}

A = ∣ a 11 a 12 a 21 a 22 ∣ A= \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} A= ​a11​a21​​a12​a22​​ ​

A= \begin{Vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{Vmatrix}

A = ∥ a 11 a 12 a 21 a 22 ∥ A= \begin{Vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{Vmatrix} A= ​a11​a21​​a12​a22​​ ​

A= \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}

A = a 11 a 12 a 21 a 22 A= \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} A=a11​a21​​a12​a22​​

多行公式对齐

使用\begin{split} \end{split}，在需要对齐的地方添加&符号，注意需要用\\来换行。

例如：

\begin{split} L(\theta) &= \arg\max_{\theta}\ln(P_{All})\\ &= \arg\max_{\theta}\ln\prod_{i=1}^{n} \left[ (h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot (1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}} \right]\\ &= \arg\max_{\theta}\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right]\\ &= \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]\\ &= \arg\min_{\theta}\mathscr{l}(\theta) \end{split}

L ( θ ) = arg ⁡ max ⁡ θ ln ⁡ ( P A l l ) = arg ⁡ max ⁡ θ ln ⁡ ∏ i = 1 n [ ( h θ ( x ( i ) ) ) y ( i ) ⋅ ( 1 − h θ ( x ( i ) ) ) 1 − y ( i ) ] = arg ⁡ max ⁡ θ ∑ i = 1 n [ y ( i ) ln ⁡ ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) ] = arg ⁡ min ⁡ θ [ − ∑ i = 1 n [ y ( i ) ln ⁡ ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) ] ] = arg ⁡ min ⁡ θ l ( θ ) \begin{split} L(\theta) &= \arg\max_{\theta}\ln(P_{All})\\ &= \arg\max_{\theta}\ln\prod_{i=1}^{n} \left[ (h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot (1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}} \right]\\ &= \arg\max_{\theta}\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right]\\ &= \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]\\ &= \arg\min_{\theta}\mathscr{l}(\theta) \end{split} L(θ)​=argθmax​ln(PAll​)=argθmax​lni=1∏n​[(hθ​(x(i)))y(i)⋅(1−hθ​(x(i)))1−y(i)]=argθmax​i=1∑n​[y(i)ln(hθ​(x(i)))+(1−y(i))ln(1−hθ​(x(i)))]=argθmin​[−i=1∑n​[y(i)ln(hθ​(x(i)))+(1−y(i))ln(1−hθ​(x(i)))]]=argθmin​l(θ)​

上例中，在=前添加了&，因此实现等号对齐；

\begin{split} \end{split}语法默认为右对齐，也就是说如果不在任何地方添加&符号，则公式默认右侧对齐，例如：

\begin{split} L(\theta) = \arg\max_{\theta}\ln(P_{All})\\ = \arg\max_{\theta}\ln\prod_{i=1}^{n} \left[ (h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot (1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}} \right]\\ = \arg\max_{\theta}\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right]\\ = \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]\\ = \arg\min_{\theta}\mathscr{l}(\theta) \end{split}

上述LATEX代码没有添加&符号，则公式右对齐： L ( θ ) = arg ⁡ max ⁡ θ ln ⁡ ( P A l l ) = arg ⁡ max ⁡ θ ln ⁡ ∏ i = 1 n [ ( h θ ( x ( i ) ) ) y ( i ) ⋅ ( 1 − h θ ( x ( i ) ) ) 1 − y ( i ) ] = arg ⁡ max ⁡ θ ∑ i = 1 n [ y ( i ) ln ⁡ ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) ] = arg ⁡ min ⁡ θ [ − ∑ i = 1 n [ y ( i ) ln ⁡ ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) ] ] = arg ⁡ min ⁡ θ l ( θ ) \begin{split} L(\theta) = \arg\max_{\theta}\ln(P_{All})\\ = \arg\max_{\theta}\ln\prod_{i=1}^{n} \left[ (h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot (1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}} \right]\\ = \arg\max_{\theta}\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right]\\ = \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]\\ = \arg\min_{\theta}\mathscr{l}(\theta) \end{split} L(θ)=argθmax​ln(PAll​)=argθmax​lni=1∏n​[(hθ​(x(i)))y(i)⋅(1−hθ​(x(i)))1−y(i)]=argθmax​i=1∑n​[y(i)ln(hθ​(x(i)))+(1−y(i))ln(1−hθ​(x(i)))]=argθmin​[−i=1∑n​[y(i)ln(hθ​(x(i)))+(1−y(i))ln(1−hθ​(x(i)))]]=argθmin​l(θ)​

如果希望左对齐，例如

\begin{split} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{split}

ln ⁡ h θ ( x ( i ) ) = ln ⁡ 1 1 + e − θ T x ( i ) = − ln ⁡ ( 1 + e θ T x ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) = ln ⁡ ( 1 − 1 1 + e − θ T x ( i ) ) = − θ T x ( i ) − ln ⁡ ( 1 + e θ T x ( i ) ) \begin{split} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{split} ​lnhθ​(x(i))=ln1+e−θTx(i)1​=−ln(1+eθTx(i))ln(1−hθ​(x(i)))=ln(1−1+e−θTx(i)1​)=−θTx(i)−ln(1+eθTx(i))​

除了\begin{split} \end{split}，也可以用\begin{align} \end{align}，用法与split相同，对齐方式也相同；

只有一点不同：采用align环境会默认为每一条公式编号（如下例），split则不会编号。

\begin{align} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{align}

ln ⁡ h θ ( x ( i ) ) = ln ⁡ 1 1 + e − θ T x ( i ) = − ln ⁡ ( 1 + e θ T x ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) = ln ⁡ ( 1 − 1 1 + e − θ T x ( i ) ) = − θ T x ( i ) − ln ⁡ ( 1 + e θ T x ( i ) ) \begin{align} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{align} ​lnhθ​(x(i))=ln1+e−θTx(i)1​=−ln(1+eθTx(i))ln(1−hθ​(x(i)))=ln(1−1+e−θTx(i)1​)=−θTx(i)−ln(1+eθTx(i))​​

但可以在align后加一个*号，则align环境也可以取消公式自动编号，如下： (也就是说align*和split的用法完全相同)

\begin{align*} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{align*}

ln ⁡ h θ ( x ( i ) ) = ln ⁡ 1 1 + e − θ T x ( i ) = − ln ⁡ ( 1 + e θ T x ( i ) ) ln ⁡ ( 1 − h θ ( x ( i ) ) ) = ln ⁡ ( 1 − 1 1 + e − θ T x ( i ) ) = − θ T x ( i ) − ln ⁡ ( 1 + e θ T x ( i ) ) \begin{align*} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{align*} ​lnhθ​(x(i))=ln1+e−θTx(i)1​=−ln(1+eθTx(i))ln(1−hθ​(x(i)))=ln(1−1+e−θTx(i)1​)=−θTx(i)−ln(1+eθTx(i))​

方程组

使用\begin{cases} \end{cases}

例如：

\begin{cases} \begin{split} p &= P(y=1|\mathbf{x})= \frac{1}{1+e^{-\theta^T\mathbf{X}}}\\ 1-p &= P(y=0|\mathbf{x})=1-P(y=1|\mathbf{x})= \frac{1}{1+e^{\theta^T\mathbf{X}}} \end{split} \end{cases}

{ p = P ( y = 1 ∣ x ) = 1 1 + e − θ T X 1 − p = P ( y = 0 ∣ x ) = 1 − P ( y = 1 ∣ x ) = 1 1 + e θ T X \begin{cases} \begin{split} p &= P(y=1|\mathbf{x})= \frac{1}{1+e^{-\theta^T\mathbf{X}}}\\ 1-p &= P(y=0|\mathbf{x})=1-P(y=1|\mathbf{x})= \frac{1}{1+e^{\theta^T\mathbf{X}}} \end{split} \end{cases} ⎩ ⎨ ⎧​p1−p​=P(y=1∣x)=1+e−θTX1​=P(y=0∣x)=1−P(y=1∣x)=1+eθTX1​​​

注意LATEX语法可以嵌套使用，上例即为\begin{cases} \end{cases}下嵌套了begin{split} \end{split}。

也可以将公式和文字结合起来，例如：

\text{Decision Boundary}= \begin{cases} 1\quad \text{if }\ \hat{y}>0.5\\ 0\quad \text{otherwise} \end{cases}

Decision Boundary = { 1 if y ^ > 0.5 0 otherwise \text{Decision Boundary}= \begin{cases} 1\quad \text{if}\quad \hat{y}>0.5\\ 0\quad \text{otherwise} \end{cases} Decision Boundary={1ify^​>0.50otherwise​ 注：\quad表示空格。



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