Glossary
Notations
Good notation is crucial, and the tutorial reflects the way scientists and engineers actually use. When there is a descrepancy in notation used by mathematical community and physists and engineers, we join the latter. In particular, we denote by j (or 𝕁) the complex imaginary unit instead of ⅈ ⅈ, and usualy use asterisk for complex conjugate instead of overline.
List of Symbols
| 𝔽 | field (usualy either ℚ or ℝ or ℂ) | |
| ℝ | field of real numbers | |
| ℚ | field of rational numbers | |
| ℤ | the set of integers is a collection of whole numbers that includes positive numbers, negative numbers, and zero. | |
| ℕ₁ | the set of nonnegative integers 1, 2, 3, … | |
| ℕ | the set of natural numbers ℕ = {0, 1, 2, … } | |
| ℂ | field of complex numbers | |
| j or ⅉ | imaginary unit (the vector in the positive vertical direction on complex plane ℂ), so j² = −1 or ⅉ² = −1. | |
| z* | complex conjugate: z* = (𝑎 + j b)* = 𝑎 − j b | |
| \( \displaystyle \overline{z} \) | complex conjugate: \( \displaystyle \overline{z} = \overline{(a + {\bf j}\, b)} = a - {\bf j}\, b \) | |
| A* | adjoint matrix: \( \displaystyle \overline{{\bf A}^{\mathrm{T}}} , \) transpose and complex conjugate. | |
| AT | transpose matrix is obtained by changing the rows into columns. | |
| O(g(n)) | big-oh is also called Bachmann–Landau notation or asymptotic notation: \( \displaystyle \left\vert f(n) \right\vert \le M\,\left\vert g(n) \right\vert \) as n → ∞. | |
| o(g(n)) | little=oh means \( \displaystyle \lim_{n\to\infty} \frac{f(n)}{g(n)} = 0 . \) | |
| n! | factorial: 1·2·3· ⋯ ·n | |
| n!! | double factorial of a number n is the product of all integers from 1 or 2 up to n that share the same odd or even property (parity) as n. | |
| (2k)!! | = 2k k! | |
| (2k−1)!! | \( \displaystyle \quad = \frac{(2k)!}{(2k)!!} \quad \) double factorial: (2n+1)!! = (2n+1) · (2n−1) · (2n−3) · ⋯ · 1 | |
| \( \displaystyle n^{\underline{m}} \) | falling factorial \( \displaystyle n^{\underline{m}} = n\left( n-1 \right)\left( n-2 \right) \cdots \left( n-m+1 \right) \) | |
| \( \displaystyle n^{\overline{m}} \) | rising factorial (or Pochhammer symbol) \( \displaystyle n^{\overline{m}} = n \left( n+1 \right)\left( n+2 \right) \cdots \left( n+m-1 \right) \) | |
| \( \displaystyle \binom{n}{k} \) | binomial coefficient: \( \displaystyle \binom{n}{k} = \frac{n^{\underline{k}}}{k!} , \) where k ∈ ℕ | |
| (𝑎, b) | open interval on ℝ (𝑎 or b or both can be infinity) | |
| [𝑎, b] | closed interval | |
| |𝑎, b| | any interval with endpoints |𝑎 and b; it can be closed, open, or semi-closed | |
| A∩B | intersection of two sets | |
| A∪B | union of two sets | |
| \( \displaystyle \overline{\Omega} \) | closure of set Ω | |
| ∂Ω | boundary of set Ω | |
| ⇀ | weak convergence: fn ⇀ f iff ⟨ u | fn ⟩ → ⟨ u | f ⟩ for any u ∈ ℌ | |
| ⤳ | asymptotic behavior | |
Vector Spaces
| 𝔽n | direct product of n fields 𝔽×𝔽× ⋯; ×𝔽. | |
| ℝn | real Cartesian product | |
| ℂn | complex Cartesian product | |
| 𝔽m,n | set of all m × n matrices | |
| 𝔽[x] | Set of polynomials of variable x over field 𝔽, also denoted by ℘, ℘ | |
| 𝔽≤n[x] | Set of polynomials over field 𝔽 of degree less than or equal to n. | |
| ker(A) | Kernel or Null space of matrix A, so ker(A) = 𝒩(A). | |
| coker(A) | Cokernel of matrix A is the kernel of adjoint matrix A*. | |
| ⟨ f , g ⟩ | inner product (in mathematics) | |
| ⟨ f | g ⟩ | inner product (in physics) | |
| ∥·∥ | norm in a normed space | |
| ℓ¹ | (pronounced "ell-one") is the set of all infinite sequences of numbers where the sum of the absolute values of the terms is finite. | |
| ℓ∞ | (pronounced "ell infinity") is the vector space of all infinite sequences of real or complex numbers that are bounded, meaning there exists a real number M > 0 such that the absolute value of every term in the sequence is less than or equal to M | |
| ℓ² | or ℓ₂ is the set of sequence with norm \( \displaystyle \| {\bf x} \|_2 = \left( \sum_{i\ge 0} |x_i |^2 \right)^{1/2} \) | |
| 𝔏²[𝑎, b] | set of square integrable (Lebesgue) functions on the interval [𝑎, b] | |
| 𝔏²([𝑎, b], w) | set of square integrable (Lebesgue) functions with weight w on the interval [𝑎, b] | |
| 𝔏¹ or 𝔏 | space of Lebesgue integrable functions | |
| 𝔏∞[𝑎, b] | (read as "L-infinity space") is the collection of all measurable functions that are essentially bounded, meaning they are bounded everywhere except possibly on a set of measure zero. It forms a fundamental Banach space when paired with the essential supremum norm. | |
| ℭ[𝑎, b] | set of continuous functions on interval [𝑎, b] | |
| ℭm[Ω] | m-times continuously differentiable functions Ω → ℂ | |
| 𝒮(ℝ) | Schwartz functions (smooth functions with rapid decay), also denoted by S(ℝ) | |
| 𝒮*(ℝ) | set of tempered distributions, also denoted by 𝒮'(ℝ) or S'(ℝ) | |
Operators
| \( \displaystyle \texttt{D} \) | differential or derivative operator in Euler's notation: \( \displaystyle \texttt{D} = \frac{\text d}{{\text d}x}\) with respect to variable x | |
| \( \displaystyle \frac{{\text d}y}{{\text d}x}\) | derivative of function y in Leibniz's notation | |
| y' | derivative of function y in Lagrange's notation | |
| \( \displaystyle \dot{y} \) | derivative of function y in Newton's notation with respect to time variable: \( \displaystyle \dot{y} = {\text d}y/{\text d}t \) | |
| ∂ | partial derivative | |
| ∂xu | partial derivative of u with respect to variable x, also denoted as ux or \( \displaystyle \frac{\partial u}{\partial x} \) | |
| \( \displaystyle \hat{p} \) | momentum operator: \( \displaystyle \hat{p} = -{\bf j}\,\hbar\,\partial , \) where ħ is Planck's reduced constant | |
| ∇ | gradient operator | |
| Δ | Laplace operator \( \displaystyle \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \) | |
| □ | d'Alembert operator \( \displaystyle \square = \frac{\partial^2}{\partial t^2} - c^2 \nabla^2 = \frac{\partial^2}{\partial t^2} - c^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) \) | |
| S[f] | (formal) Fourier series of function f in either exponential form or trigonometric form | |
| S*[f] | conjugate Fourier series | |
| SN(f; x) | N-th partial Fourier sum \( \displaystyle \sum_{n=-N}^N \hat{f}(n) \,e^{{\bf j} n\pi x/\ell} = \frac{a_0}{2} + \sum_{k=1}^N a_k \cos \frac{k\pi x}{\ell} + b_k \sin \frac{k\pi x}{\ell} \) | |
| I f ≫ | list of Fourier coefficients either in complex or trigonometric form | |
| \( ℱ\left[ f \right] \) | Fourier transform \( \displaystyle {\hat {f}} \) or \( ℱ\left[ f \right] \) or \( f^F . \) | |
| \( ℱ^{-1}\left[ f^F \right] \) | inverse Fourier transform | |
| f★g | convolution: \( f\star g (x) = \int f(y)\,g(x-y)\,{\text d} y \) | |
| ℒ[ f ] | Laplace transform \( \displaystyle f^L (\lambda ) \) or \( ℒ\left[ f \right] \) or \( f^L = \int_0^{\infty} f(t)\,e^{-\lambda t}{\text d} t \) | |
| ℒ−1[ fL ] | inverse Laplace transform \( \displaystyle ℒ^{-1}\left[ f^L \right] = \mbox{P.V.} \frac{1}{2\pi{\bf j}} \int_{h-{\bf j}\infty}^{h+{\bf j}\infty} f^L (\lambda )\,e^{\lambda\,t} {\text d}\lambda \) | |
Functions
| lnx | natural logarithm with base e | |
| Γ(ν) | gamma function \( \displaystyle \Gamma (\nu ) = \int_0^{\infty} t^{\nu -1} e^{-t} {\text d} t \) | |
| χA | characteristic (or indicator) function of a set A | |
| δ(x) | delta function of Dirac | |
| H(t) | Heaviside function: \( \displaystyle H(t) = \begin{cases} 1, & \quad t> 0, \\ ½ , & \quad t = 0, \\ 0, & \quad t < 0 . \end{cases} \) | |