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1 Algebraic and Analytic MethodsNotation

§1.1 Special Notation

(For other notation see Notation for the Special Functions.)

x,y real variables.
z complex variable in §§1.2(i), 1.91.11, real variable in §§1.51.6.
w complex variable in §§1.91.11.
j,k, integers.
m,n nonnegative integers, unless specified otherwise.
f,g inner, or scalar, product for real or complex vectors or functions.
L2(X,dα) the space of all Lebesgue–Stieltjes measurable functions on X which are square integrable with respect to dα.
ϕ a testing function.
Λ,ϕ action of distribution Λ on test function ϕ.
deg degree.
primes derivatives with respect to the variable, except where indicated otherwise.
𝐮, 𝐯 column vectors.
𝐄n the space of all n-dimensional vectors.
𝐀 or [ai,j] or [aij] matrix with elements ai,j or aij.
𝐀1 inverse of the square matrix 𝐀
𝐈 identity matrix
det(𝐀) determinant of the square matrix 𝐀
tr(𝐀) trace of the square matrix 𝐀
etr(𝐀) exponential of tr(𝐀)
𝐀 adjoint of the square matrix 𝐀
𝐀¯ complex conjugate of the matrix 𝐀
𝐀T transpose of the matrix 𝐀
𝐀H Hermitian conjugate of the matrix 𝐀
linear operator defined on a manifold
adjoint of defined on the dual manifold

In the physics, applied maths, and engineering literature a common alternative to a¯ is a, a being a complex number or a matrix; the Hermitian conjugate of 𝐀 is usually being denoted 𝐀.