MD数学公式大杂烩
Markdown / LaTeX 数学公式压力测试
本文档用于测试 Markdown 阅读器对 LaTeX / KaTeX / MathJax 数学公式的支持情况。
1. 行内公式测试
最简单的行内公式：E = mc^2。
欧拉恒等式：
e^{i\pi}+1=0
二次方程求根公式：
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
复杂一点：
\displaystyle \lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=e^x
嵌套根号：
\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\cdots}}}}
2. 基本 Display Math
a^2+b^2=c^2
\sum_{k=1}^{n}k=\frac{n(n+1)}{2}
\sum_{k=1}^{n}k^2
=
\frac{n(n+1)(2n+1)}{6}
\sum_{k=1}^{n}k^3
=
\left(\frac{n(n+1)}{2}\right)^2
3. 极限压力测试
\lim_{x\to0}\frac{\sin x}{x}=1
\lim_{n\to\infty}
\left(
1+\frac{1}{n}
\right)^n=e
\lim_{x\to+\infty}
\left(
\sqrt{x^2+x}-x
\right)
=
\frac12
\lim_{n\to\infty}
n
\left[
\left(
1+\frac{x}{n}
\right)^n-e^x
\right]
=
-\frac{x^2e^x}{2}
一个长得比较离谱的：
\lim_{n\to\infty}
\left[
\prod_{k=1}^{n}
\left(
1+\frac{k^2}{n^3}
\right)
\right]^{n}
4. 微积分
导数
\frac{d}{dx}
\left(
x^x
\right)
=
x^x(\ln x+1)
\frac{d^n}{dx^n}e^{ax}
=
a^ne^{ax}
\frac{\partial^3 f}
{\partial x^2\partial y}
梯度：
\nabla f
=
\begin{pmatrix}
\frac{\partial f}{\partial x_1}\\
\frac{\partial f}{\partial x_2}\\
\vdots\\
\frac{\partial f}{\partial x_n}
\end{pmatrix}
Hessian：
H_f(x)
=
\nabla^2f(x)
=
\begin{pmatrix}
\frac{\partial^2f}{\partial x_1^2}
&
\frac{\partial^2f}{\partial x_1\partial x_2}
&
\cdots
&
\frac{\partial^2f}{\partial x_1\partial x_n}
\\
\frac{\partial^2f}{\partial x_2\partial x_1}
&
\frac{\partial^2f}{\partial x_2^2}
&
\cdots
&
\frac{\partial^2f}{\partial x_2\partial x_n}
\\
\vdots & \vdots & \ddots & \vdots\\
\frac{\partial^2f}{\partial x_n\partial x_1}
&
\frac{\partial^2f}{\partial x_n\partial x_2}
&
\cdots
&
\frac{\partial^2f}{\partial x_n^2}
\end{pmatrix}
5. 积分
\int x^2\,dx=\frac{x^3}{3}+C
\int_{-\infty}^{+\infty}e^{-x^2}\,dx
=
\sqrt{\pi}
\int_0^\infty
\frac{x^{s-1}}{e^x-1}\,dx
=
\Gamma(s)\zeta(s)
多重积分：
\iiint_{\Omega}
\left(
\frac{\partial^2u}{\partial x^2}
+
\frac{\partial^2u}{\partial y^2}
+
\frac{\partial^2u}{\partial z^2}
\right)
\,dx\,dy\,dz
曲面积分：
\iint_{\partial\Omega}
\mathbf F\cdot\mathbf n\,dS
=
\iiint_{\Omega}
\nabla\cdot\mathbf F\,dV
6. 分段函数
f(x)=
\begin{cases}
x^2, & x<0,\\[4pt]
\sin x, & 0\le x<\pi,\\[4pt]
e^{-x}, & x\ge\pi.
\end{cases}
更加复杂：
F(x,y)=
\begin{cases}
\dfrac{x^2y}{x^4+y^2},
&
(x,y)\ne(0,0),\\[10pt]
0,
&
(x,y)=(0,0).
\end{cases}
7. 矩阵
A=
\begin{pmatrix}
1&2&3\\
4&5&6\\
7&8&9
\end{pmatrix}
B=
\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}
行列式：
\det(A-\lambda I)
=
\begin{vmatrix}
a_{11}-\lambda&a_{12}&\cdots&a_{1n}\\
a_{21}&a_{22}-\lambda&\cdots&a_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
a_{n1}&a_{n2}&\cdots&a_{nn}-\lambda
\end{vmatrix}
=0
8. 线性代数
特征值问题：
A\mathbf v=\lambda\mathbf v
奇异值分解：
A=U\Sigma V^\ast
谱分解：
A
=
Q\Lambda Q^{-1}
=
\sum_{i=1}^{n}
\lambda_i
\mathbf v_i
\mathbf w_i^\ast
Moore–Penrose 伪逆：
A^+
=
V\Sigma^+U^\ast
最小二乘：
\hat{\boldsymbol\beta}
=
(X^\top X)^{-1}X^\top\mathbf y
9. RREF / Gaussian Elimination
\left[
\begin{array}{ccc|c}
1&2&-1&3\\
2&4&1&7\\
-1&-2&2&-2
\end{array}
\right]
\xrightarrow{R_2\leftarrow R_2-2R_1}
\left[
\begin{array}{ccc|c}
1&2&-1&3\\
0&0&3&1\\
-1&-2&2&-2
\end{array}
\right]
继续：
\xrightarrow{
\substack{
R_3\leftarrow R_3+R_1\\
R_2\leftarrow\frac13R_2
}}
\left[
\begin{array}{ccc|c}
1&2&-1&3\\
0&0&1&\frac13\\
0&0&1&1
\end{array}
\right]
10. 集合论
A\cap(B\cup C)
=
(A\cap B)\cup(A\cap C)
\mathcal P(A)
=
\{B\mid B\subseteq A\}
f:A\to B,
\qquad
x\mapsto f(x)
\forall x\in\mathbb R,\;
\exists y\in\mathbb R:
\quad
y>x
\neg
\left(
\forall x\in X,\exists y\in Y:P(x,y)
\right)
\iff
\exists x\in X,\forall y\in Y:\neg P(x,y)
11. 概率论
Bayes 定理：
P(A\mid B)
=
\frac{P(B\mid A)P(A)}
{P(B)}
全概率公式：
P(B)
=
\sum_{i=1}^{n}
P(B\mid A_i)P(A_i)
期望：
\mathbb E[X]
=
\int_{-\infty}^{\infty}
x f_X(x)\,dx
方差：
\operatorname{Var}(X)
=
\mathbb E[X^2]
-
\left(\mathbb E[X]\right)^2
协方差矩阵：
\Sigma
=
\mathbb E
\left[
(\mathbf X-\boldsymbol\mu)
(\mathbf X-\boldsymbol\mu)^\top
\right]
12. 正态分布
f(x)
=
\frac{1}{\sigma\sqrt{2\pi}}
\exp
\left[
-\frac{(x-\mu)^2}{2\sigma^2}
\right]
多元正态分布：
f_{\mathbf X}(\mathbf x)
=
\frac{
\exp\left(
-\frac12
(\mathbf x-\boldsymbol\mu)^\top
\Sigma^{-1}
(\mathbf x-\boldsymbol\mu)
\right)
}{
(2\pi)^{k/2}
|\Sigma|^{1/2}
}
13. 条件概率的复杂版本
P(X_i=x_i\mid X_{-i}=x_{-i})
=
\frac{
\exp\left(
-\beta E(x_i,x_{-i})
\right)
}{
\displaystyle
\sum_{x_i'}
\exp\left(
-\beta E(x_i',x_{-i})
\right)
}
14. 信息论
熵：
H(X)
=
-\sum_{x\in\mathcal X}
p(x)\log p(x)
条件熵：
H(X\mid Y)
=
-\sum_{x,y}
p(x,y)
\log
p(x\mid y)
互信息：
I(X;Y)
=
\sum_{x,y}
p(x,y)
\log
\frac{p(x,y)}
{p(x)p(y)}
KL 散度：
D_{\mathrm{KL}}(P\|Q)
=
\sum_x
P(x)
\log
\frac{P(x)}{Q(x)}
15. 傅里叶变换
\hat f(\xi)
=
\int_{-\infty}^{+\infty}
f(x)e^{-2\pi i x\xi}\,dx
逆变换：
f(x)
=
\int_{-\infty}^{+\infty}
\hat f(\xi)
e^{2\pi i x\xi}\,d\xi
卷积定理：
\mathcal F\{f*g\}
=
\mathcal F\{f\}
\mathcal F\{g\}
其中
(f*g)(x)
=
\int_{-\infty}^{+\infty}
f(\tau)g(x-\tau)\,d\tau
16. Laplace Transform
\mathcal L\{f(t)\}(s)
=
\int_0^\infty
e^{-st}f(t)\,dt
\mathcal L
\left\{
\frac{d^nf}{dt^n}
\right\}
=
s^nF(s)
-
\sum_{k=0}^{n-1}
s^{n-1-k}f^{(k)}(0)
17. 微分方程
简单：
\frac{dy}{dx}+P(x)y=Q(x)
二阶：
a(x)y''+b(x)y'+c(x)y=f(x)
非线性：
\frac{d^2x}{dt^2}
+
\mu(x^2-1)\frac{dx}{dt}
+x
=
0
Lorenz 系统：
\begin{aligned}
\frac{dx}{dt}&=\sigma(y-x),\\
\frac{dy}{dt}&=x(\rho-z)-y,\\
\frac{dz}{dt}&=xy-\beta z.
\end{aligned}
18. 偏微分方程
热方程：
\frac{\partial u}{\partial t}
=
\alpha\nabla^2u
波动方程：
\frac{\partial^2u}{\partial t^2}
=
c^2\nabla^2u
Laplace 方程：
\nabla^2u=0
Poisson 方程：
\nabla^2u=f
19. Navier–Stokes
\rho
\left(
\frac{\partial\mathbf u}{\partial t}
+
(\mathbf u\cdot\nabla)\mathbf u
\right)
=
-\nabla p
+
\mu\nabla^2\mathbf u
+
\rho\mathbf f
不可压缩条件：
\nabla\cdot\mathbf u=0
分量形式：
\rho
\left(
\frac{\partial u_i}{\partial t}
+
u_j
\frac{\partial u_i}{\partial x_j}
\right)
=
-
\frac{\partial p}{\partial x_i}
+
\mu
\frac{\partial^2u_i}
{\partial x_j\partial x_j}
+
\rho f_i
20. Maxwell 方程组
\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}
积分形式：
\oint_{\partial S}
\mathbf E\cdot d\boldsymbol\ell
=
-
\frac{d}{dt}
\int_S
\mathbf B\cdot d\mathbf S
21. Schrödinger 方程
i\hbar
\frac{\partial}{\partial t}
\Psi(\mathbf r,t)
=
\left[
-\frac{\hbar^2}{2m}\nabla^2
+
V(\mathbf r,t)
\right]
\Psi(\mathbf r,t)
定态形式：
\left[
-\frac{\hbar^2}{2m}\nabla^2
+
V(\mathbf r)
\right]
\psi(\mathbf r)
=
E\psi(\mathbf r)
22. Dirac 方程
\left(
i\gamma^\mu\partial_\mu-m
\right)\psi=0
带电磁场：
\left[
i\gamma^\mu
\left(
\partial_\mu+ieA_\mu
\right)
-m
\right]
\psi
=
0
23. Einstein 场方程
G_{\mu\nu}
+
\Lambda g_{\mu\nu}
=
\frac{8\pi G}{c^4}
T_{\mu\nu}
展开：
R_{\mu\nu}
-
\frac12Rg_{\mu\nu}
+
\Lambda g_{\mu\nu}
=
\frac{8\pi G}{c^4}
T_{\mu\nu}
Christoffel 符号：
\Gamma^\rho_{\mu\nu}
=
\frac12
g^{\rho\sigma}
\left(
\partial_\mu g_{\sigma\nu}
+
\partial_\nu g_{\sigma\mu}
-
\partial_\sigma g_{\mu\nu}
\right)
Riemann 曲率张量：
R^\rho_{\ \sigma\mu\nu}
=
\partial_\mu
\Gamma^\rho_{\nu\sigma}
-
\partial_\nu
\Gamma^\rho_{\mu\sigma}
+
\Gamma^\rho_{\mu\lambda}
\Gamma^\lambda_{\nu\sigma}
-
\Gamma^\rho_{\nu\lambda}
\Gamma^\lambda_{\mu\sigma}
24. 广义相对论指标地狱测试
\nabla_\lambda
R^\rho_{\ \sigma\mu\nu}
=
\partial_\lambda
R^\rho_{\ \sigma\mu\nu}
+
\Gamma^\rho_{\lambda\kappa}
R^\kappa_{\ \sigma\mu\nu}
-
\Gamma^\kappa_{\lambda\sigma}
R^\rho_{\ \kappa\mu\nu}
-
\Gamma^\kappa_{\lambda\mu}
R^\rho_{\ \sigma\kappa\nu}
-
\Gamma^\kappa_{\lambda\nu}
R^\rho_{\ \sigma\mu\kappa}
25. 量子力学 Bra-Ket
|\psi\rangle
=
\sum_{n=0}^{\infty}
c_n|n\rangle
\langle\phi|\psi\rangle
=
\sum_n
\phi_n^\ast\psi_n
\hat A
=
\sum_n
a_n
|a_n\rangle
\langle a_n|
不确定性原理：
\sigma_A\sigma_B
\ge
\frac12
\left|
\langle[\hat A,\hat B]\rangle
\right|
26. 机器学习
线性回归：
\hat{\mathbf y}
=
X\boldsymbol\beta
MSE：
\mathcal L(\theta)
=
\frac1N
\sum_{i=1}^{N}
\left(
y_i-f_\theta(x_i)
\right)^2
Softmax：
P(y=k\mid\mathbf x)
=
\frac{
e^{z_k}
}{
\sum_{j=1}^{K}e^{z_j}
}
Cross Entropy：
\mathcal L
=
-
\frac1N
\sum_{i=1}^{N}
\sum_{k=1}^{K}
y_{ik}
\log\hat y_{ik}
27. 神经网络链式法则
\frac{\partial\mathcal L}{\partial W^{(l)}}
=
\frac{\partial\mathcal L}
{\partial a^{(L)}}
\prod_{k=l+1}^{L}
\frac{\partial a^{(k)}}
{\partial a^{(k-1)}}
\frac{\partial a^{(l)}}
{\partial W^{(l)}}
28. Transformer Attention
\operatorname{Attention}(Q,K,V)
=
\operatorname{softmax}
\left(
\frac{QK^\top}{\sqrt{d_k}}
\right)V
Multi-Head Attention：
\operatorname{MultiHead}(Q,K,V)
=
\operatorname{Concat}
(
\operatorname{head}_1,
\dots,
\operatorname{head}_h
)
W^O
其中
\operatorname{head}_i
=
\operatorname{Attention}
(
QW_i^Q,
KW_i^K,
VW_i^V
)
29. 位置编码
PE_{(pos,2i)}
=
\sin
\left(
\frac{pos}
{10000^{2i/d_{\text{model}}}}
\right)
PE_{(pos,2i+1)}
=
\cos
\left(
\frac{pos}
{10000^{2i/d_{\text{model}}}}
\right)
30. 数论
Euler φ 函数：
\varphi(n)
=
n
\prod_{p\mid n}
\left(
1-\frac1p
\right)
Euler 定理：
a^{\varphi(n)}
\equiv1\pmod n
\qquad
\text{if }\gcd(a,n)=1
中国剩余定理：
\begin{cases}
x\equiv a_1\pmod{n_1}\\
x\equiv a_2\pmod{n_2}\\
\vdots\\
x\equiv a_k\pmod{n_k}
\end{cases}
31. Riemann Zeta Function
\zeta(s)
=
\sum_{n=1}^{\infty}
\frac1{n^s}
=
\prod_{p\ \mathrm{prime}}
\frac1{1-p^{-s}}
函数方程：
\zeta(s)
=
2^s
\pi^{s-1}
\sin
\left(
\frac{\pi s}{2}
\right)
\Gamma(1-s)
\zeta(1-s)
32. Gamma / Beta 函数
\Gamma(z)
=
\int_0^\infty
t^{z-1}e^{-t}\,dt
B(x,y)
=
\int_0^1
t^{x-1}(1-t)^{y-1}\,dt
=
\frac{\Gamma(x)\Gamma(y)}
{\Gamma(x+y)}
33. 复分析
Cauchy 积分公式：
f^{(n)}(a)
=
\frac{n!}{2\pi i}
\oint_\gamma
\frac{f(z)}
{(z-a)^{n+1}}
\,dz
留数定理：
\oint_\gamma f(z)\,dz
=
2\pi i
\sum_k
\operatorname{Res}
(f,z_k)
34. 无穷连分数
x
=
a_0+
\cfrac{1}{
a_1+
\cfrac{1}{
a_2+
\cfrac{1}{
a_3+
\cfrac{1}{
\ddots
}
}
}
}
黄金比例：
\varphi
=
1+
\cfrac1{
1+
\cfrac1{
1+
\cfrac1{
1+\ddots
}
}
}
35. 巨型求和
\mathcal Z
=
\sum_{n=1}^{\infty}
\sum_{k=0}^{n}
\sum_{j=0}^{k}
(-1)^{n+k+j}
\binom nk
\binom kj
\frac{
\Gamma(n+\alpha)
\zeta(2k+\beta)
}{
(n!)^2
}
e^{-\lambda j}
36. 巨型积分
\mathcal I(\alpha,\beta,\gamma)
=
\int_{-\infty}^{+\infty}
\int_0^\infty
\int_0^{2\pi}
\frac{
r^{\alpha-1}
e^{-\beta r^2}
\cos(m\theta)
}{
\left[
1+\gamma
\left(
x^2+r^2-2xr\cos\theta
\right)
\right]^{3/2}
}
\,d\theta\,dr\,dx
37. Optimization / Lagrangian
\min_{\mathbf x\in\mathbb R^n}
f(\mathbf x)
\quad
\text{subject to}
\quad
g_i(\mathbf x)\le0,
\quad
h_j(\mathbf x)=0
Lagrangian：
\mathcal L
(
\mathbf x,
\boldsymbol\lambda,
\boldsymbol\nu
)
=
f(\mathbf x)
+
\sum_{i=1}^{m}
\lambda_i g_i(\mathbf x)
+
\sum_{j=1}^{p}
\nu_jh_j(\mathbf x)
KKT：
\begin{aligned}
\nabla_x\mathcal L&=0,\\
g_i(\mathbf x)&\le0,\\
h_j(\mathbf x)&=0,\\
\lambda_i&\ge0,\\
\lambda_i g_i(\mathbf x)&=0.
\end{aligned}
38. Functional Analysis
\|f\|_{L^p(\Omega)}
=
\left(
\int_\Omega
|f(x)|^p\,dx
\right)^{1/p}
\langle f,g\rangle
=
\int_\Omega
f(x)\overline{g(x)}\,dx
\|T\|
=
\sup_{\|x\|\le1}
\|Tx\|
39. 泛函积分
Z
=
\int\mathcal D\phi\,
\exp
\left[
\frac{i}{\hbar}
S[\phi]
\right]
其中
S[\phi]
=
\int d^4x
\left[
\frac12
\partial_\mu\phi
\partial^\mu\phi
-
\frac12m^2\phi^2
-
\frac{\lambda}{4!}\phi^4
\right]
40. 一个故意写得很夸张的公式
\boxed{
\mathfrak Z(\alpha,\beta,\gamma)
=
\lim_{N\to\infty}
\left\{
\prod_{n=1}^{N}
\left[
1+
\frac{
\displaystyle
\sum_{k=0}^{n}
(-1)^k
\binom nk
\Gamma
\left(
k+\frac12
\right)
\zeta(2k+\alpha)
}{
\displaystyle
n^\beta
+
\int_0^\infty
x^{n+\gamma}
e^{-x^2}
\,dx
}
\right]^{1/N}
\right\}
}
41. 多层上下标压力测试
T^{
\alpha_1\alpha_2\cdots\alpha_p
}_{
\beta_1\beta_2\cdots\beta_q
}
A_{
i_1i_2\cdots i_n
}^{
j_1j_2\cdots j_m
}
=
\sum_{
k_1,\ldots,k_r
}
B_{
i_1\cdots i_n
}^{
k_1\cdots k_r
}
C_{
k_1\cdots k_r
}^{
j_1\cdots j_m
}
42. 嵌套括号压力测试
\left[
\left\{
\left(
\frac{
1+
\left[
x+
\left(
y+z
\right)^2
\right]^3
}{
1-
\left\{
x-
\left[
y-z
\right]^2
\right\}^4
}
\right)^{1/2}
\right\}^{3/2}
\right]^{5/7}
43. 箭头与映射
A
\xrightarrow{\quad f\quad}
B
\xrightarrow{\quad g\quad}
C
x
\overset{f}{\longmapsto}
f(x)
\overset{g}{\longmapsto}
g(f(x))
A
\hookrightarrow
B
\twoheadrightarrow
C
\longrightarrow
0
44. Commutative Diagram 的纯 LaTeX 替代测试
\begin{array}{ccc}
A & \xrightarrow{f} & B\\
\downarrow g && \downarrow h\\
C & \xrightarrow{k} & D
\end{array}
45. 超长 aligned 环境
\begin{aligned}
\mathcal F(x)
&=
\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}}{n!}
\left[
\int_0^\infty
t^{n+\alpha-1}
e^{-\beta t}
J_\nu(\gamma t)
\,dt
\right]\\
&\quad+
\prod_{p\ \mathrm{prime}}
\left(
1-\frac1{p^s}
\right)^{-1}\\
&\quad+
\det
\left(
I+\lambda A^\top A
\right)^{-\frac12}\\
&\quad+
\operatorname{Tr}
\left[
e^{-\tau H}
\right].
\end{aligned}
46. Unicode + LaTeX 混排
\forall\,\varepsilon>0,\;
\exists\,\delta>0
\quad\text{使得}\quad
|x-a|<\delta
\Rightarrow
|f(x)-f(a)|<\varepsilon
\text{若 }A\subseteq B
\text{ 且 }B\subseteq C,
\text{ 则 }A\subseteq C.
47. 超长公式压力测试
\boxed{
\begin{aligned}
\mathscr Q
&=
\lim_{\substack{
N\to\infty\\
M\to\infty
}}
\frac{
\displaystyle
\sum_{n=1}^{N}
\sum_{m=1}^{M}
(-1)^{n+m}
\binom{N}{n}
\binom{M}{m}
\Gamma
\left(
n+m+\frac12
\right)
}{
\displaystyle
\prod_{p\le N}
\left(
1-\frac1{p^{2}}
\right)
}\\[6pt]
&\qquad\times
\exp
\left\{
-\int_0^\infty
\frac{
x^{\alpha-1}
}{
e^{\beta x}-1
}
\left[
\sum_{k=0}^{\infty}
\frac{
(-\gamma x^2)^k
}{
(k!)^2
}
\right]
dx
\right\}\\[6pt]
&\qquad\times
\det
\left[
\delta_{ij}
+
\lambda
\frac{
\partial^2
}{
\partial x_i\partial x_j
}
\log
\left(
1+\sum_{r=1}^{d}x_r^2
\right)
\right]_{i,j=1}^{d}.
\end{aligned}
}
48. 最终 Boss：纯粹用于把数学渲染器往死里打
\boxed{
\begin{aligned}
\mathfrak{M}
&=
\underset{
\substack{
N,M,K\to\infty\\
\varepsilon\to0^+
}
}{\operatorname{lim\,sup}}
\;
\left[
\prod_{n=1}^{N}
\left\{
1+
\frac{
\displaystyle
\sum_{m=0}^{M}
\sum_{k=0}^{K}
(-1)^{m+k}
\binom{n+m}{m}
\frac{
\Gamma
\left(
k+\frac{\alpha}{2}
\right)
}{
\Gamma(k+\beta)
}
\zeta
\left(
2k+m+\gamma
\right)
}{
\displaystyle
n^{\delta}
+
\left|
\int_{-\infty}^{+\infty}
\frac{
e^{-x^2}
H_n(x)
}{
(1+x^2)^{m+\frac12}
}
\,dx
\right|^2
+
\varepsilon
}
\right\}^{\frac1{n^2}}
\right]
\\[8pt]
&\qquad\times
\exp
\left[
-\frac12
\int_{\mathbb R^d}
\int_{\mathbb R^d}
\phi(x)
\left(
-\Delta+m^2
\right)^{-1}(x,y)
\phi(y)
\,d^dx\,d^dy
\right]
\\[8pt]
&\qquad\times
\operatorname{Tr}
\left\{
\mathcal T
\exp
\left[
-\frac{i}{\hbar}
\int_{t_0}^{t_1}
\left(
\hat H_0
+
\lambda
\sum_{j=1}^{r}
\hat A_j(t)
\otimes
\hat B_j(t)
\right)
dt
\right]
\right\}
\\[8pt]
&\qquad\times
\det
\left[
g_{\mu\nu}
+
\alpha'
R_{\mu\nu}
+
(\alpha')^2
R_{\mu\rho\nu\sigma}
R^{\rho\sigma}
+
\frac{
\partial_\mu\phi
\partial_\nu\phi
}{
1+\phi^2
}
\right]^{1/2}
\\[8pt]
&\qquad\times
\left\{
\sum_{\ell=0}^{\infty}
\frac{
(-1)^\ell
}{
(2\ell+1)!
}
\left[
\oint_{\partial\Omega}
\frac{
\nabla^\ell f(z)
}{
(z-z_0)^{\ell+1}
}
\,dz
\right]^{2\ell+1}
\right\}.
\end{aligned}
}
49. 额外测试：各种数学字体
ABC
\qquad
\mathbf{ABC}
\qquad
\mathit{ABC}
\qquad
\mathrm{ABC}
\qquad
\mathsf{ABC}
\qquad
\mathtt{ABC}
\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
50. 各种符号
\alpha,\beta,\gamma,\delta,\epsilon,\varepsilon,
\zeta,\eta,\theta,\vartheta,\iota,\kappa,\lambda,
\mu,\nu,\xi,\pi,\varpi,\rho,\varrho,\sigma,\varsigma,
\tau,\upsilon,\phi,\varphi,\chi,\psi,\omega
\Gamma,\Delta,\Theta,\Lambda,\Xi,\Pi,\Sigma,\Upsilon,\Phi,\Psi,\Omega
\in,\notin,\subset,\subseteq,\supset,\supseteq,
\cup,\cap,\setminus,\emptyset,
\forall,\exists,\nexists,
\therefore,\because
\le,\ge,\neq,\approx,\sim,\simeq,\cong,\equiv,
\propto,\ll,\gg
\leftarrow,\rightarrow,\leftrightarrow,
\Leftarrow,\Rightarrow,\Leftrightarrow,
\mapsto,\hookrightarrow,\twoheadrightarrow
51. 最后测试 Markdown 和数学公式混排
普通文本 粗体、斜体、inline code。
- Euler：e^{i\pi}+1=0
- Einstein：E=mc^2
- Pythagoras：a^2+b^2=c^2
- Gaussian integral：\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt\pi
代码块中的 LaTeX 不应该被渲染：
\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi}
反引号内也通常不应该渲染：
$E=mc^2$
而这里应该恢复正常：
\boxed{
e^{i\pi}+1=0
}
END
如果你看到这里所有矩阵、积分、上下标、希腊字母、分段函数、aligned 环境、\boxed{} 和巨型公式都正常显示，那么你的 Markdown 数学渲染支持已经覆盖了相当多的常见 LaTeX 数学语法。
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