1 Algebraic and Analytic Methods Notation 1 Algebraic and Analytic Methods 1.2 Elementary Algebra

§1.1 Special Notation

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See also:: Annotations for Ch.1

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

	real variables.
	complex variable in §§1.2(i), 1.9–1.11, real variable in §§1.5–1.6.
	complex variable in §§1.9–1.11.
$j,k,\ell$	integers.
	nonnegative integers, unless specified otherwise.
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
$L^{2}\left(X,\,\mathrm{d}\alpha\right)$	the space of all Lebesgue–Stieltjes measurable functions on which are square integrable with respect to $\,\mathrm{d}\alpha$ .
$\phi$	a testing function.
$\left\langle\Lambda,\phi\right\rangle$	action of distribution $\Lambda$ on test function $\phi$ .
$\deg$	degree.
primes	derivatives with respect to the variable, except where indicated otherwise.
$\mathbf{u}$ , $\mathbf{v}$	column vectors.
$\mathbf{E}_{n}$	the space of all -dimensional vectors.
$\mathbf{A}$	or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$ .
${\mathbf{A}}^{-1}$	inverse of the square matrix $\mathbf{A}$
$\mathbf{I}$	identity matrix
$\det(\mathbf{A})$	determinant of the square matrix $\mathbf{A}$
$\operatorname{tr}(\mathbf{A})$	trace of the square matrix $\mathbf{A}$
$\operatorname{etr}\left(\mathbf{A}\right)$	exponential of $\operatorname{tr}(\mathbf{A})$
${\mathbf{A}}^{*}$	adjoint of the square matrix $\mathbf{A}$
$\overline{\mathbf{A}}$	complex conjugate of the matrix $\mathbf{A}$
$\mathbf{A}^{\mathrm{T}}$	transpose of the matrix $\mathbf{A}$
${\mathbf{A}}^{{\rm H}}$	Hermitian conjugate of the matrix $\mathbf{A}$
$\mathcal{L}$	linear operator defined on a manifold $\mathcal{M}$
${\mathcal{L}}^{*}$	adjoint of $\mathcal{L}$ defined on the dual manifold ${\mathcal{M}}^{*}$

In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{*}$ , being a complex number or a matrix; the Hermitian conjugate of $\mathbf{A}$ is usually being denoted $\mathbf{A}^{{\dagger}}$ .

1 Algebraic and Analytic Methods 1.2 Elementary Algebra

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