Mathematical Formulas — LaTeX Rendering Test

A reference post testing every type of mathematical notation used in computer science and machine learning research.

Basic Arithmetic & Algebra

Inline math flows naturally in text. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ and the Pythagorean theorem is $$a^2 + b^2 = c^2$$ .

Block equations get their own space:

$$(a + b)^2 = a^2 + 2ab + b^2$$

\sum_{i=1}^{n} i = \frac{n(n+1)}{2}

\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

Calculus

The definition of a derivative:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The chain rule:

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Definite integral:

\int_a^b f(x)\, dx = F(b) - F(a)

Integration by parts:

\int u\, dv = uv - \int v\, du

Taylor series expansion:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Linear Algebra

Matrix multiplication — for $A \in \mathbb{R}^{m \times k}$ and $B \in \mathbb{R}^{k \times n}$ :

(AB)_{ij} = \sum_{l=1}^{k} A_{il} B_{lj}

Determinant of a 2×2 matrix:

\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Eigenvalue equation:

A\mathbf{v} = \lambda \mathbf{v}

The dot product and cosine similarity:

\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta

L2 norm:

\|\mathbf{x}\|_2 = \sqrt{\sum_{i=1}^{n} x_i^2}

Singular Value Decomposition:

A = U \Sigma V^T

Probability & Statistics

Bayes’ theorem:

P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}

Normal distribution (Gaussian):

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

Expected value and variance:

\mathbb{E}[X] = \sum_{x} x \cdot P(X = x)

\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2

KL Divergence:

D_{KL}(P \| Q) = \sum_{x} P(x) \log \frac{P(x)}{Q(x)}

Machine Learning

Loss Functions

Mean Squared Error:

\mathcal{L}_{\text{MSE}} = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2

Binary Cross-Entropy:

\mathcal{L}_{\text{BCE}} = -\frac{1}{n}\sum_{i=1}^{n} \left[ y_i \log(\hat{y}_i) + (1 - y_i)\log(1 - \hat{y}_i) \right]

Activation Functions

Sigmoid:

\sigma(x) = \frac{1}{1 + e^{-x}}

Softmax:

\text{softmax}(x_i) = \frac{e^{x_i}}{\sum_{j=1}^{K} e^{x_j}}

ReLU (inline): $\text{ReLU}(x) = \max(0, x)$

Gradient Descent

Weight update rule:

\theta \leftarrow \theta - \alpha \nabla_\theta \mathcal{L}(\theta)

Backpropagation — gradient of loss w.r.t. weights:

\frac{\partial \mathcal{L}}{\partial W^{(l)}} = \delta^{(l)} \cdot (a^{(l-1)})^T

Attention Mechanism (Transformers)

\text{Attention}(Q, K, V) = \text{softmax}\!\left(\frac{QK^T}{\sqrt{d_k}}\right) V

Information Theory

Entropy:

H(X) = -\sum_{x} P(x) \log_2 P(x)

Mutual Information:

I(X; Y) = \sum_{x,y} P(x,y) \log \frac{P(x,y)}{P(x)P(y)}

Big-O Complexity (Inline)

Common complexities as inline math: $$O(1)$$ , $O(\log n)$ , $$O(n)$$ , $O(n \log n)$ , $$O(n^2)$$ , $$O(2^n)$$ , $$O(n!)$$

Master theorem for divide and conquer — if $$T(n) = aT(n/b) + f(n)$$ :

T(n) = \begin{cases} O(n^{\log_b a}) & \text{if } f(n) = O(n^{\log_b a - \varepsilon}) \\ O(n^{\log_b a} \log n) & \text{if } f(n) = O(n^{\log_b a}) \\ O(f(n)) & \text{if } f(n) = \Omega(n^{\log_b a + \varepsilon}) \end{cases}

Greek Letters Reference

Commonly used in ML and math: $\alpha$ (alpha), $\beta$ (beta), $\gamma$ (gamma), $\delta$ (delta), $\epsilon$ (epsilon), $\theta$ (theta), $\lambda$ (lambda), $\mu$ (mu), $\sigma$ (sigma), $\tau$ (tau), $\phi$ (phi), $\psi$ (psi), $\omega$ (omega), $\Sigma$ (Sigma), $\Pi$ (Pi), $\Delta$ (Delta), $\nabla$ (nabla).

If it renders correctly here — calculus, matrices, ML formulas, piecewise functions — it will render correctly in any post you write.

Chemistry (mhchem)

Water and carbon dioxide: $\ce{H2O}$ and $\ce{CO2}$

Reaction:

\ce{H2SO4 -> 2H+ + SO4^2-}

Equilibrium:

\ce{N2 + 3H2 <=> 2NH3}

Isotope notation: $\ce{^{14}_{6}C}$

Chart.js — Algorithm Complexity

Image with Caption

Binary search repeatedly halves the search space — O(log n) time complexity.

Collapsible Section (inline HTML)

Click to see the full Master Theorem

For $$T(n) = aT(n/b) + f(n)$$ where $a \geq 1$ , $$b > 1$$ :

T(n) = \begin{cases} O(n^{\log_b a}) & \text{if } f(n) = O(n^{\log_b a - \varepsilon}) \\ O(n^{\log_b a} \log n) & \text{if } f(n) = \Theta(n^{\log_b a}) \\ O(f(n)) & \text{if } f(n) = \Omega(n^{\log_b a + \varepsilon}) \end{cases}

ASCII Art

Binary Search — step-by-step on [1,3,5,7,9,11,13,15], target=7

Index:  0   1   2   3   4   5   6   7
Value: [1] [3] [5] [7] [9][11][13][15]
                    ^
              mid=3, found!

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