海豹日記
海豹日記
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Math について

  • GROWI では MathJax external_link を活用して文章内に数式を挿入することが可能です
  • 以下にて紹介する代表的な MathJax の記法をそのまま引用し活用することが可能です

Inline Formula

When a≠0a \ne 0, there are two solutions to ax2+bx+c=0ax^2 + bx + c = 0 and they are
x=−b±b2−4ac2a.x = {-b \pm \sqrt{b^2-4ac} \over 2a}.

The Lorenz Equations

x˙=σ(y−x)y˙=ρx−y−xzz˙=−βz+xy\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}

The Cauchy-Schwarz Inequality

(∑k=1nakbk) ⁣ ⁣2≤(∑k=1nak2)(∑k=1nbk2)\left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)

A Cross Product Formula

V1×V2=∣ijk∂X∂u∂Y∂u0∂X∂v∂Y∂v0∣\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix}

The probability of getting (k)\left(k\right) heads when flipping (n)\left(n\right) coins is:

P(E)=(nk)pk(1−p)n−kP(E) = {n \choose k} p^k (1-p)^{ n-k}

An Identity of Ramanujan

1(ϕ5−ϕ)e25π=1+e−2π1+e−4π1+e−6π1+e−8π1+…\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } }

A Rogers-Ramanujan Identity

1+q2(1−q)+q6(1−q)(1−q2)+⋯=∏j=0∞1(1−q5j+2)(1−q5j+3),for ∣q∣<1.1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for $|q|<1$}.

Maxwell's Equations

∇×B⃗− 1c ∂E⃗∂t=4πcj⃗∇⋅E⃗=4πρ∇×E⃗ + 1c ∂B⃗∂t=0⃗∇⋅B⃗=0\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}