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Introduction to Linear Algebra with Mathematica

Notations


List of Symbols

      field of real numbers
      field of complex numbers
j       imaginary unit (the vector in the positive vertical direction on complex plane ℂ), so j² = −1
z*       complex conjugate:   z* = (𝑎 + jb)* = 𝑎 − jb
\( \displaystyle \overline{z} \)       complex conjugate:   \( \displaystyle \overline{z} = \overline{(a + {\bf j}\, b)} = a - {\bf j}\, b \)
      the set nonnegative integers: 0, 1, 2, …
      the set of all integers: 0, ±1, ±2, ±3, …
𝕋       ℝ/(2πℤ) unit circle or one-dimensional tores.
O(g(n))       big-oh is also called BachmannLandau 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
\( \displaystyle n^{\underline{m}} \)       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)   \( \displaystyle n^{\overline{m}} = n \left( n+1 \right)\left( n+2 \right) \cdots \left( n+m-1 \right) \)
(2n)!!       double factorial:   (2n)!! = (2n) · ⋯ · 2 (2n−2) · (2n−4) ·
(2n+1)!!       double factorial:   (2n+1)!! = (2n+1) · (2n−1) · (2n−3) · ⋯ · 1
\( \displaystyle \binom{n}{k} \)       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
AB       intersection of two sets
AB       union of two sets
\( \displaystyle \overline{\Omega} \)       closure of set Ω
∂Ω       boundary of set Ω
      weak convergence: fnf iff ⟨ u | fn ⟩ → ⟨ u | f ⟩ for any u ∈ ℌ
     
     
     

Vector Spaces

      Hilbert space equiped with inner product
f , g       inner product (in mathematics)
f | g       inner product (in physics)
∥·∥       norm in a normed space
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]
ℭ[𝑎, b]       set of continuous functions on interval [𝑎, b]
m[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
fg       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 \( f^L = \int_0^{\infty} f(t)\,e^{-\lambda t}{\text d} t \)
−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:   \( \displaystyle H(t) = \begin{cases} 1, & \quad t> 0, \\ ½ , & \quad t = 0, \\ 0, & \quad t < 0 . \end{cases} \)