压测目的 #
该文档用于评估 MathJax 在当前主题配置下的加载耗时、首屏渲染表现与滚动帧率。内容覆盖了多种数学排版场景，包括大规模对齐环境、嵌套分式、矩阵、化简步骤以及公式表格。
高密度对齐环境 #
\[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \]长链推导段落 #
在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。大型矩阵与行列式 #
\[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \]表格批量渲染 #
| 序号 | 公式类型 | 示例 |
|---|---|---|
| 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) |
| 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) |
| 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) |
| 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) |
| 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) |
滚动段落连续负载 #
我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \]多级对齐与算子级联 #
为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \]
大型张量与特殊函数 #
\[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \]级数与生成函数扩展 #
\[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \] ## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]## 高密度对齐环境 \[ \begin{aligned} R_{xx}(\tau) &= \mathbb{E}\left[x(t)\, x(t+\tau)\right] \\[-2pt] S_{xx}(\omega) &= \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j\omega \tau} d\tau \\[-2pt] H(j\omega) &= \frac{Y(j\omega)}{X(j\omega)} = \frac{\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt}{1} \\[-2pt] \Phi_{yy}(\omega) &= |H(j\omega)|^2 \Phi_{xx}(\omega) + \Phi_{nn}(\omega) \end{aligned} \] ## 长链推导段落 在通信系统噪声分析中，经常需要连续引用多个长公式： \[ \begin{aligned} P_e &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{\sqrt{2\pi}} \int_{\sqrt{2E_b/N_0}}^{\infty} e^{-u^2/2} du, \\ G(s) &= \sum_{k=0}^{\infty} p_k e^{-skT_s}, \\ I &= \int_{0}^{1} \frac{\ln(1+x)}{x} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. \end{aligned} \] 为了形成高密度压力，本段落重复引用相同公式： \( Q(x) = \tfrac{1}{2}\operatorname{erfc}\left(\tfrac{x}{\sqrt{2}}\right),\ \nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz},\ E\left\{e^{sX}\right\} = \int_{-\infty}^{\infty} e^{sx} f_X(x) dx \)，用于检测内联渲染的基线性能。 ## 大型矩阵与行列式 \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ - a_0 & - a_1 & - a_2 & \dots & - a_{n-1} \end{bmatrix} \] \[ \det(\mathbf{A}-\lambda \mathbf{I}) = \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_1 \lambda + a_0 \] ## 表格批量渲染 | 序号 | 公式类型 | 示例 | | ---- | -------- | ---- | | 1 | 贝塞尔函数 | \( J_n(x) = \sum_{m=0}^{\infty} (-1)^m \frac{(x/2)^{2m+n}}{m!\,(m+n)!} \) | | 2 | 复频谱 | \( X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \) | | 3 | 路径积分 | \( \oint_{\gamma} f(z) dz = 2\pi i \sum \operatorname{Res}(f, a_k) \) | | 4 | 线性估计 | \( \hat{x} = \mathbf{w}^T \mathbf{y} = \sum_{k=1}^{N} w_k y_k \) | | 5 | 香农容量 | \( C = B \log_2\left(1+\frac{S}{N}\right) \) | ## 滚动段落连续负载 我们通过一个长段落复制常见的积分与和式来模拟真实论文的滚动压力。下式重复嵌套多层结构，方便观测滚动过程中 MathJax 的重新排版开销： \[ \sum_{k=1}^{K} \int_{0}^{T} \left| \frac{1}{T} \int_{0}^{T} \left( x_k(t) - \sum_{m=1}^{M} \alpha_{km} \phi_m(t) \right) e^{-j2\pi ft} dt \right|^2 dt \] \[ S_n = \sum_{k=1}^{n} \frac{1}{k} = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + O\!\left(\frac{1}{n^4}\right) \] ## 多级对齐与算子级联 为了进一步增加渲染压力，加入多级对齐环境、偏导数与算子链式法则： \[ \begin{aligned} \frac{\partial}{\partial t} \int_{\Omega(t)} f(x,t) \, dx &= \int_{\Omega(t)} \frac{\partial f}{\partial t}(x,t) \, dx + \int_{\partial \Omega(t)} f(x,t) \; \mathbf{v}(x,t) \cdot \mathbf{n}(x,t) \, dS, \\ \nabla_{\mathbf{x}} (\mathbf{A}(\mathbf{x}) \mathbf{B}(\mathbf{x})) &= (\nabla_{\mathbf{x}} \mathbf{A}) \mathbf{B} + \mathbf{A} (\nabla_{\mathbf{x}} \mathbf{B}), \\ \mathrm{div}\,(\mathrm{curl}\, \mathbf{F}) &= 0, \qquad \mathrm{curl}\,(\nabla \phi) = \mathbf{0}. \end{aligned} \] \[ \begin{aligned} \mathbf{J}(\theta) &= \mathbb{E}\bigg[ \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right) \left( \frac{\partial}{\partial \theta} \log p(X;\theta) \right)^{\!T} \bigg], \\ \mathcal{I}(\theta) &= - \mathbb{E}\left[ \frac{\partial^2}{\partial \theta^2} \log p(X;\theta) \right], \qquad \mathcal{I}(\theta) = \mathbf{J}(\theta). \end{aligned} \] ## 大型张量与特殊函数 \[ \mathbf{T}_{ijk} = \sum_{m=1}^{M} \int_{\mathbb{R}^3} \int_{0}^{2\pi} r^2 \sin \phi \; Y_{im}(\theta,\phi) \, Y_{mj}(\theta,\phi) \, Z_{mk}(r) \, dr \, d\theta \, d\phi \] \[ \begin{aligned} \mathcal{H}_{\nu}^{(1)}(z) &= J_{\nu}(z) + i Y_{\nu}(z), \\ E_{\alpha,\beta}(z) &= \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \\ {}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) &= \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}. \end{aligned} \] ## 级数与生成函数扩展 \[ \log \Gamma(z) = \left(z - \tfrac{1}{2}\right) \log z - z + \tfrac{1}{2} \log(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k (2k-1) z^{2k-1}} \] \[ G(t) = \prod_{n=1}^{\infty} (1 - t^n)^{-1} = \sum_{k=0}^{\infty} p(k) t^k, \qquad p(k) \text{ 为分拆数} \] \[ \sum_{n=0}^{\infty} \frac{H_n}{n!} x^n = e^x \int_{0}^{x} \frac{1 - e^{-u}}{u} du, \qquad H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \]结论 #
当页面包含上述规模的公式时，可以通过浏览器性能工具记录首次内容绘制、脚本执行时间与滚动 FPS，以量化 MathJax 在站点中的表现差异。
