---

---

---

---

Latex

Katex¹

---

Repeating fractions

$\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} \equiv 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }$

$\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} \equiv 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }$(ϕ5 ​ ​−ϕ)e52​π1​≡1+1+1+1+1+⋯e−8π​e−6π​e−4π​e−2π​

---

Summation notation

$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$

$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$(∑k=1n​ak​bk​)2≤(∑k=1n​ak2​)(∑k=1n​bk2​)

---

Sum of a Series

$\displaystyle\sum_{i=1}^{k+1}i$

$\displaystyle\sum_{i=1}^{k+1}i$i=1∑k+1​i

$\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)$

$\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)$=(i=1∑k​i)+(k+1)

$\displaystyle= \frac{k(k+1)}{2}+k+1$

$\displaystyle= \frac{k(k+1)}{2}+k+1$=2k(k+1)​+k+1

$\displaystyle= \frac{k(k+1)+2(k+1)}{2}$

$\displaystyle= \frac{k(k+1)+2(k+1)}{2}$=2k(k+1)+2(k+1)​

$\displaystyle= \frac{(k+1)(k+2)}{2}$

$\displaystyle= \frac{(k+1)(k+2)}{2}$=2(k+1)(k+2)​

$\displaystyle= \frac{(k+1)((k+1)+1)}{2}$

$\displaystyle= \frac{(k+1)((k+1)+1)}{2}$=2(k+1)((k+1)+1)​

---

Product notation

$\displaystyle\text{ for }\lvert q\rvert < 1.$

$\displaystyle\text{ for }\lvert q\rvert < 1.$ for ∣q∣<1.

$= \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},$

$= \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},$=j=0∏∞​(1−q5j+2)(1−q5j+3)1​,

$\displaystyle1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots$

$\displaystyle1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots$1+(1−q)q2​+(1−q)(1−q2)q6​+⋯

---

Inline math[^4]
<span class="math">...</span>

And $k_{n+1} = n^2 + k_n^2 - k_{n-1}$, text.

And $k_{n+1} = n^2 + k_n^2 - k_{n-1}$kn+1​=n2+kn2​−kn−1​​, text.

---

Greek Letters

$\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega$

$\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega$Γ Δ Θ Λ Ξ Π Σ Υ Φ Ψ Ω

$\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi\ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega$

$\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi\ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega$α β γ δ ϵ ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ ϕ χ ψ ω

$\varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi$

$\varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi$ε ϑ ϖ ϱ ς φ

---

Arrows

$\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow$

$\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow$← → ← → ↑ ⇑ ↓ ⇓ ↕ ⇕

$\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow$

$\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow$⇐ ⇒ ↔ ⇔ ↦ ↩

$\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow$

$\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow$↼ ↽ ⇌ ⟵ ⟸ ⟶

$\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup$

$\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup$⟹ ⟷ ⟺ ⟼ ↪ ⇀

LaTeX code:
$\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow$

$\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow$⇁ ⇝ ↗ ↘ ↙ ↖

---

Symbols

$\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup$

$\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup$√ ⊼ ⊻ ⊙ ⊕ ⊗ ⊘ ⊚ ⊡ △

$\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle$

$\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle$▽ † ⋄ ⋆ ◃ ▹ ∠ ∞ ′ △

---

Calculus

$\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx$

$\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx$∫udxdv​dx=uv−∫dxdu​vdx

$f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}$

$f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}$f(x)=∫−∞∞​f^​(ξ)e2πiξx

$\oint \vec{F} \cdot d\vec{s}=0$

$\oint \vec{F} \cdot d\vec{s}=0$∮F ⋅ds =0

---

Lorenz Equations

$\begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned}$

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

Cross Product

$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}$

$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}$V1​×V2​=∣∣∣∣∣∣​i∂u∂X​∂v∂X​​j∂u∂Y​∂v∂Y​​k00​∣∣∣∣∣∣​

---

Accents

$\hat{x}\ \vec{x}\ \ddot{x}$

$\hat{x}\ \vec{x}\ \ddot{x}$x^ x  x¨

---

Stretchy brackets

$\left(\frac{x^2}{y^3}\right)$

$\left(\frac{x^2}{y^3}\right)$(y3x2​)

---

Evaluation at limits

$\left.\frac{x^3}{3}\right|_0^1$

$\left.\frac{x^3}{3}\right|_0^1$3x3​∣∣∣​01​

---

Case definitions

$f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\ 3n+1, & \text{if } n\text{ is odd} \end{cases}$

$f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\ 3n+1, & \text{if } n\text{ is odd} \end{cases}$f(n)={2n​,3n+1,​if n is evenif n is odd​

---

Maxwell’s Equations

$\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$

$\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$∇×B −c1​∂t∂E ​∇⋅E ∇×E +c1​∂t∂B ​∇⋅B ​=c4π​j ​=4πρ=0 =0​

$\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\[1em] \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\[0.5em] \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\[1em] \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$

$\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\[1em] \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\[0.5em] \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\[1em] \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$∇×B −c1​∂t∂E ​∇⋅E ∇×E +c1​∂t∂B ​∇⋅B ​=c4π​j ​=4πρ=0 =0​

---

Statistics

$\frac{n!}{k!(n-k)!} = {^n}C_k$

$\frac{n!}{k!(n-k)!} = {^n}C_k$k!(n−k)!n!​=nCk​

${n \choose k}$

${n \choose k}$(kn​)

---

Fractions on fractions

$\frac{\frac{1}{x}+\frac{1}{y}}{y-z}$

$\frac{\frac{1}{x}+\frac{1}{y}}{y-z}$y−zx1​+y1​​

---

n-th root

$\sqrt[n]{1+x+x^2+x^3+\ldots}$

$\sqrt[n]{1+x+x^2+x^3+\ldots}$n1+x+x2+x3+… ​

---

Matrices

$\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}$

$\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}$⎝⎛​a11​a21​a31​​a12​a22​a32​​a13​a23​a33​​⎠⎞​

$begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}$

$\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}$⎣⎢⎡​0⋮0​⋯⋱⋯​0⋮0​⎦⎥⎤​

---

Punctuation

$(x) = \sqrt{1+x} \quad (x \ge -1)$

$f(x) = \sqrt{1+x} \quad (x \ge -1)$f(x)=1+x ​(x≥−1)

$f(x) \sim x^2 \quad (x\to\infty)$

$f(x) \sim x^2 \quad (x\to\infty)$f(x)∼x2(x→∞)

$f(x) = \sqrt{1+x}, \quad x \ge -1$

$f(x) = \sqrt{1+x}, \quad x \ge -1$f(x)=1+x ​,x≥−1

$f(x) \sim x^2, \quad x\to\infty$

$f(x) \sim x^2, \quad x\to\infty$f(x)∼x2,x→∞

---

$\Gamma$Γ function

$\Gamma(n) = (n-1)!\quad\forall n\in\mathbb N$

$\Gamma(n) = (n-1)!\quad\forall n\in\mathbb N$Γ(n)=(n−1)!∀n∈N

---

Euler integral

$\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.$

$\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.$Γ(z)=∫0∞​tz−1e−tdt.

---

---

---

---

1. katext - ref ↩︎