What are the different notations for the Fourier transform?

There are four main notations: (1) Hat notation f̂(ξ), common in pure mathematics. (2) Calligraphic F notation 𝓕{f(t)}, common in engineering. (3) Uppercase notation F(ω), common in physics. (4) Bracket notation 𝓕[f](ξ), used to avoid ambiguity. In MathJax/LaTeX, these are written as \hat{f}, \mathcal{F}\{f(t)\}, F(\omega), and \mathcal{F}[f](\xi).

How do you write vector arrows in MathJax?

Use \vec{a} for an arrow above a single letter. For bold notation, use \boldsymbol{a}. The \vec command is best for single-letter vectors, while \boldsymbol is useful when distinguishing vectors from matrices or tensors.

How do you write partial derivative symbols in LaTeX?

The partial derivative symbol ∂ is written as \partial. For a fraction form, use \frac{\partial f}{\partial x}. The nabla operator (used for gradient, divergence, curl) is written as \nabla. Examples: gradient \nabla f, divergence \nabla \cdot \vec{F}, curl \nabla \times \vec{F}.

Math Notation and Symbols - Advanced Guide

About This Page

This page introduces advanced mathematical symbols for science and engineering students.
For basic symbols, please see the university-level mathematics page.

Table of Contents

Vector Calculus
Complex Analysis
Fourier Analysis
Laplace Transform
Partial Differential Equations
Tensors
Special Functions
Usage Examples in Reading Notes

Vector Calculus

Nabla Operator

Code:

$\nabla$

Display:

$\nabla$

Laplacian

Code:

$\nabla^2 f = \Delta f$

Display:

$\nabla^2 f = \Delta f$

Complex Analysis

Complex Function

Code:

$f(z) = u(x,y) + iv(x,y)$

Display:

$f(z) = u(x,y) + iv(x,y)$

Cauchy-Riemann Equations

Code:

$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Display:

$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Residue

Code:

$\text{Res}(f, z_0)$

Display:

$\text{Res}(f, z_0)$

Contour Integral

Code:

$\displaystyle\oint_C f(z) \, dz$

Display:

$\displaystyle\oint_C f(z) \, dz$

Residue Theorem

Code:

$\displaystyle\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, z_k)$

Display:

$\displaystyle\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, z_k)$

Fourier Analysis

Fourier Series

Code:

$\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)$

Display:

$\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)$

Fourier Transform

Code:

$\displaystyle F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$

Display:

$\displaystyle F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$

Fourier Transform Notation Variants

Hat notation (math):

$\hat{f}(\xi)$

Display:

$\hat{f}(\xi)$

Calligraphic F (engineering):

$\mathcal{F}\{f(t)\}$

Display:

$\mathcal{F}\{f(t)\}$

Uppercase (physics):

$F(\omega)$

Display:

$F(\omega)$

Bracket notation (unambiguous):

$\mathcal{F}[f](\xi)$

Display:

$\mathcal{F}[f](\xi)$

Inverse Fourier Transform

Code:

$\displaystyle f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$

Display:

$\displaystyle f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$

Convolution

Code:

$\displaystyle (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau$

Display:

$\displaystyle (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau$

Laplace Transform

Code:

$\displaystyle \mathcal{L}[f(t)] = F(s) = \int_0^{\infty} f(t) e^{-st} dt$

Display:

$\displaystyle \mathcal{L}[f(t)] = F(s) = \int_0^{\infty} f(t) e^{-st} dt$

Inverse Laplace Transform

Code:

$\mathcal{L}^{-1}[F(s)] = f(t)$

Display:

$\mathcal{L}^{-1}[F(s)] = f(t)$

Differentiation Theorem

Code:

$\mathcal{L}[f'(t)] = sF(s) - f(0)$

Display:

$\mathcal{L}[f'(t)] = sF(s) - f(0)$

Partial Differential Equations

Wave Equation

Code:

$\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Display:

$\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Heat Equation

Code:

$\displaystyle\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

Display:

$\displaystyle\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

Laplace's Equation

Code:

$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

Display:

$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

Poisson's Equation

Code:

$\nabla^2 u = f$

Display:

$\nabla^2 u = f$

Tensors

Tensor Notation

Code:

$T^{ij}$ or $T_{ij}$

Display:

$T^{ij}$ or $T_{ij}$

Kronecker Delta

Code:

$\delta_{ij} = \begin{cases} 1 & (i = j) \\ 0 & (i \neq j) \end{cases}$

Display:

$\delta_{ij} = \begin{cases} 1 & (i = j) \\ 0 & (i \neq j) \end{cases}$

Levi-Civita Symbol

Code:

$\epsilon_{ijk}$

Display:

$\epsilon_{ijk}$

Metric Tensor

Code:

$g_{ij}$

Display:

$g_{ij}$

Special Functions

Gamma Function

Code:

$\displaystyle\Gamma(n) = \int_0^{\infty} t^{n-1} e^{-t} dt$

Display:

$\displaystyle\Gamma(n) = \int_0^{\infty} t^{n-1} e^{-t} dt$

Beta Function

Code:

$\displaystyle B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1} dt$

Display:

$\displaystyle B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1} dt$

Bessel Function

Code:

$J_n(x)$

Display:

$J_n(x)$

Legendre Polynomial

Code:

$P_n(x)$

Display:

$P_n(x)$

Usage Examples in Reading Notes

Example 2: Fourier Transform

Code:

The Fourier transform of a rectangular wave $f(t)$ is $\displaystyle F(\omega) = \frac{2\sin(\omega T)}{\omega}$

Display:

The Fourier transform of a rectangular wave $f(t)$ is $\displaystyle F(\omega) = \frac{2\sin(\omega T)}{\omega}$

Example 3: Partial Differential Equation

Code:

The solution to the 1D wave equation $\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ is $u(x,t) = f(x-ct) + g(x+ct)$

Display:

The solution to the 1D wave equation $\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ is $u(x,t) = f(x-ct) + g(x+ct)$

Commonly Used Symbols

\nabla → Nabla operator ∇
\Delta → Laplacian Δ
\mathcal{L} → Script L (for Laplace transform, etc.)
\mathcal{F} → Script F (for Fourier transform, etc.)
\Gamma → Gamma function Γ
\epsilon → Epsilon ε
\delta → Delta δ (Kronecker delta, etc.)
\omega → Angular frequency ω
\tau → Tau τ (time delay, etc.)
\quad → Space
^{ij} → Superscript (contravariant tensor)
_{ij} → Subscript (covariant tensor)