Phi Symbol in LaTeX: Greek Letters for Math and Physics

The Phi Symbol in LaTeX

Phi (φ, Φ) is one of the most frequently used Greek letters in mathematics and physics.

Typing Phi

Lowercase Phi

Two variants exist:

$\phi$ — straight phi (more common in American texts)

$\varphi$ — curly phi (more common in European texts)

Both are valid—pick one and be consistent.

Uppercase Phi

$\Phi$

Common Uses of Phi

Euler's Totient Function

$\phi(n)$ or $\varphi(n)$

Counts integers less than n that are coprime to n.

The Golden Ratio

$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$

Sometimes denoted as $\varphi$ to distinguish from other uses.

Spherical Coordinates

Physics convention: - $\theta$ — polar angle (from z-axis) - $\phi$ — azimuthal angle (in xy-plane)

Math convention sometimes swaps these.

Electrostatics

Electric potential: $\Phi$ or $\phi$

$\vec{E} = -\nabla\Phi$

Magnetic Flux

$\Phi_B = \int \vec{B} \cdot d\vec{A}$

Quantum Mechanics

Wave function phase: $e^{i\phi}$

State vectors: $|\phi\rangle$

Statistics

Normal distribution (PDF): $\phi(x)$

CDF: $\Phi(x)$

$\Phi(x) = \int_{-\infty}^{x} \phi(t)\,dt$

Phi vs. Varphi

The choice often depends on:

- Field: Physics often uses $\varphi$, pure math uses $\phi$ - Context: $\varphi$ for angles, $\phi$ for functions - Readability: $\varphi$ is more distinct from other letters

Related Greek Letters

$\psi$ (psi) — wave functions, angles $\theta$ (theta) — angles $\chi$ (chi) — chi-squared, susceptibility $\omega$ (omega) — angular frequency

Phi in Complex Analysis

Argument of a complex number:

$z = re^{i\phi}$

$\phi = \arg(z)$

Styling Phi

Bold Phi

$\boldsymbol{\phi}$

Phi with Subscripts

$\phi_0$ — initial phase $\phi_n$ — nth phase $\Phi_B$ — magnetic flux

Phi with Accents

$\dot{\phi}$ — time derivative $\bar{\phi}$ — average $\tilde{\phi}$ — Fourier transform

Example: Golden Ratio Properties

$\phi^2 = \phi + 1$

$\frac{1}{\phi} = \phi - 1$

$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\ddots}}}$

Write Physics Notation with Octree

Octree helps with Greek letters:

- Autocomplete for all Greek symbols - AI suggests appropriate variants - Preview your equations instantly - Export publication-ready PDFs

Try it at https://useoctree.com