GLOSSARY OF SYMBOLS AND NOTATIONS
The symbols and notations used in this handbook were chosen so as to permit reference to most standard textbooks while still maintaining consistency throughout the handbook.This glossary lists generally useful symbols whose definitions may not appear in their immediate context; each entry gives the handbook section or sections in which the symbol is defined.
Scalars and Matrices
α, β, . . . represent scalar (numerical) quantities, Chaps. 5, 6, and 12 to 16. α* is the complex conjugate of α, and |α| is the absolute value of α, 1.3-2.
In Chaps. 13 and 14, A, B, . . . represent matrices, most frequently square matrices, with A ≡ [αik], 13.2-1.
Vectors and Vector Components
a, b, . . . and x, y, . . . represent vectors, 5.1-1, 12.4-1, 14.2-1, 16.2-1, 16.7-3
u, unit vector, 5.2-5, 14.2-5, 16.8-1
i, j, k, right-handed rectangular cartesian base vectors, 5.2-3
ei, ei, base vectors, 6.3-3, 14.2-4, 16.6-1
ui, unit base vectors, 6.3-2, 16.8-3;
orthogonal unit base vectors, 6.4-1, 14.7-4
position vectors in three-dimensional Euclidean (space, Chaps. 5, 15, and 17
a = α1e1 + α2e2 + . . . , vector, represented by the column matrix of the components: 
14.2-4, 14.5-2
a • b, scalar product of vectors a, b, 5.2-6, 16.8-1
|a| ≡ (a • a)½, absolute value (norm) of a, 5.2-5, 16.8-1
(a, b), general inner product, 14.2-6
||a|| ≡ (a, a)½, norm of a, 14.2-7
(ƒ, h), inner product of functions, 15.2-1
||ƒ|| ≡ (ƒ,ƒ)½, norm of a function, 15.2-1
||A||, ||A||I, ||A||II, ||A||p, ||A||1, ||A||2 ||x||∞, matrix norms, 13.2-1
a Χ b, vector product of three-dimensional vectors a, b, 5.2-7, 16.8-4 [abc], scalar triple product, 5.2-8, 16.8-4
See also Sec. 16.1-3 for dummy-index notation, and Sec. 14.7-7 for a comparison of notations.
Linear Operators and Tensors
L,B |
linear differential operators operating on functions y(x), y(t), φ(x), φ(V, x2, . . . , xn), 9.3-1, 10.4-2, 15.2-7, 15.4-1 |
K |
linear-integral-transformation operator, 15.3-1 |
|
transpose or adjoint of A, B, . . . ; L, K, 14.4-6, 15.3-1, 15.4-3 |
A†: B†, . . ; L†, K† |
hermitian conjugate of A, B, . . . ; L, K, 14.4-3, 15.3-1, 15.4-3 |
D |
derivative operator, 20.4-2 |
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(del or nabla), vector differential operator, 5.5-2, 16.10-7 |
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Laplacian operator, 5.5-5, 16.10-7 |
E |
shift operator, 20.4-2 |
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forward-difference operator, 20.4-2 |
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backward-difference operator, 20.4-2 |
σ |
central-difference operator, 20.4-2 |
μ |
central-mean operator, 20.4-2 |
Expected Values (Mean Values) and Averages
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expected value, ensemble average, 18.3-3, 18.4-4, 18.4-8, 18.9-2 |
<x> |
t average (a random variable), 18.9-4 |
|
statistical sample average (a random variable), 19.2-3 |
Mean {x} |
mean value over a group, 12.2-12 |
arcsin z, arccos z, arctan z |
inverse trigonometric functions, 21.2-4 |
arg z |
argument of z, 1.3-2 |
Bk, Bk(n) |
Bernoulli numbers, 21.5-2 |
Bk(n)(x) |
Bernoulli polynomial, 21.5-2 |
berw z, beim z, |
|
C(x), |
Fresnel integral, 21.3-2 |
Ci x |
cosine integral, 21.3-1 |
cn z |
(cosinus amplitudinis), elliptic function, 21.6-7 |
cos z |
cosine function, 21.2-1 |
cosh z |
hyperbolic cosine, 21.2-5 |
cosh-1 z |
inverse hyperbolic cosine, 21.2-8 |
dn |
(delta amplitudinis), elliptic function. 21.6-7 |
det [aik] |
determinant, 1.5-1 |
erf x |
error function, 21.3-2 |
erfc x |
complementary error function, 21.3-2 |
E(k, φ) |
Legendre’s normal elliptic integral of the second kind, 21.6-6 |
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exponential integrals, 21.3-1 |
E(k) |
Legendre’s complete normal elliptic integral of the second kind, 21.6-6 |
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hypergeometric function, 9.3-9, 9.3-11 |
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confluent hypergeometric function, 9.3-10, 9.3-11 |
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Legendre’s normalelliptic integral ofthe first kind, 21.6-6 |
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spectral densities, 18.10-8 |
ensemble spectral densities, 18.10-6 |
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spherical Bessel functions of the third kind, 21.8-8 |
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Hermite polynomials, 21.7-1 |
Hankel functions, 21.8-1 |
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unit imaginary number, 1.3-1 |
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modified Bessel function, 21.8-6 |
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incomplete beta-function ratio, 21.4-5 |
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imaginary part of 2, 1.3-1 |
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greatest lower bound, 4.3-3 spherical Bessel function of the first kind, 21.8-8 |
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Jm{z) |
Bessel function of the first kind, 21.8-1 |
K(t) |
Legendre’s complete elliptic integral of the first kind, 21.8-1 |
Km(z) |
modified Hankel function, 21.8-6 |
Kerm z, Keim z |
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Ln(z) |
Laguerre polynomial, 21.7-1 |
Lnm |
associated Laguerre polynomial or generalized Laguerre function, 21.7-5 |
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class of quadratically integrable func tions, 15.2-1 |
li (z) |
logarithmic integral, 21.3-1 |
lim z |
limit, 4.4-1 |
l.i.m. x |
limit-in-mean, 15.2-2 |
loga z |
logarithm, 1.2-3, 21.2-10 |
max x, min x |
maximum and minimum values, 4.3-3 |
nj(z) |
spherical Bessel function of the second kind, 21.8-8 |
Nm(z) |
Neumann’s Bessel function of the second kind, 21.8-1 |
o[g(x)], 0[g(x)] |
asymptotic relations, 4.4-3 |
Pn(z) |
Legendre’s polynomial of the first kind, 21.7-1 |
Pjm(z) |
associated Legendre “polynomial” of the first kind, 21.8-10 |
Qn{z) |
Legendre function of the second kind, 21.7-3 |
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t correlation functions, 18.10-7 |
|
|
Re z |
real part of z, 1.3-1 |
Resf a |
residue of f(z) at z = a, 7.7-1 |
s(t) |
sampling function, 18.10-6 |
S(x) |
Fresnel integral, 21.3-2 |
Sm(n) |
Stirling numbers, 21.5-1 |
Sgn x or sgn x |
sign function, 21.9-1 |
Si(x) |
sine integral, 21.3-1 |
sin z |
sine function, 21.2-1 |
sinh z |
hyperbolic sine, 21.2-5 |
sinh-1 z |
inverse hyperbolic sine, 21.2-8 |
sup x |
least upper bound, 4.3-3 |
Tn(z) |
Chebyshev polynomial of the first kind, 21.7-1 |
tan z |
tangent function, 21.2-1 |
tanh z |
hyperbolic tangent, 21.2-5 |
tanh-1 z |
inverse hyperbolic tangent, 21.2-8 |
Tr[aik] |
trace, 13.2-7 |
Un(z) |
Chebyshev “polynomial” of the second kind, 21.7-4 |
|
spherical surface harmonic, 21.8-12[Nm(z) rather than Ym(z) is used for Neumann functions, 21.8-1] |
Zm(z) |
cylinder function, 21.8-1 |
B(p, q) |
beta function, 21.4-4 |
Bz(p, q) |
incomplete beta function, 21.4-5 |
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gamma function, 21.4-1 |
incomplete gamma function, 21.4-5 |
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impulse functions, 21.9-2 |
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forward difference, 20.4-1 |
backward difference, 20.4-1 |
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central difference, 20.4-1 |
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Weierstrass zeta function, 21.6-3 |
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Jacobi’s theta functions, 21.6-8 |
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central mean, 20.4-1 |
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Weierstrass sigma function, 21.6-3 |
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psi function, 21.4-3 |
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Weierstrass function, 21.6-2 |
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Fourier transform, 4.11-3 |
Fourier cosine and sine transforms 4.11-3 |
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z transform, 8.7-3 |
Laplace transform, 8.2-1 |
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factorial, 1.2-4 |
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binomial coefficient, 21.5-1 |
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jump function, 20.4-5 |
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Jacobian, 4.5-6 |
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cup, 12.8-1 |
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convolution symbol, 4.6-18 |
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summation, 1.2-5 |
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product, 1.2-5 |
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equality symbol, 1.1-3, 12.1-3 |
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identity symbol, 1.1-4 |
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identity by definition, 1.1-4 |
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approximate equality |
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asymptotically equal, 4.4-3 |
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asymptotically proportional, 4.4-3 |
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inequality, inclusion, 1.1-5, 12.6-1 |
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inclusion, 4.3-2, 12.8-3 |
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element of, 4.3-2 |
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such that |
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domain, region |
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surface, boundary surface or hyper-surface |
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C |
curve, boundary curve |
ds, dr |
scalar and vector path elements (see index) |
dA, dA |
scalar and vector surface elements (see index) |
