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Official Test

1. Inline Formula Test

  • Pythagorean theorem: a2+b2=c2 a^2 + b^2 = c^2
  • Quadratic equation root: x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Euler’s identity: eiπ+1=0 e^{i\pi} + 1 = 0

2. Independent Formula Test

Quadratic equation solution:

x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Gaussian integral:

∫−∞∞e−x2dx=π \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}

Matrix determinant:

∣abcd∣=ad−bc \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Limit definition:

lim⁡x→0sin⁡xx=1 \lim_{x \to 0} \frac{\sin x}{x} = 1

Fourier transform:

f^(ξ)=∫−∞∞f(x)e−2πixξdx \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} dx

Maxwell’s equations (differential form):

∇⋅E=ρε0∇⋅B=0∇×E=−∂B∂t∇×B=μ0J+μ0ε0∂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}

3. Complex Formula Test

Riemann zeta function:

ζ(s)=∑n=1∞1ns=∏p prime11−p−s \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}

Schrödinger equation:

iℏ∂∂tΨ=H^Ψ i\hbar \frac{\partial}{\partial t} \Psi = \hat{H} \Psi

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Last updated on • Xiao Di