expansion of arbitrary vector

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1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

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§1.6(i) Vectors

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Unit Vectors

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Cross Product (or Vector Product)

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§1.6(ii) Vectors: Alternative Notations

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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(viii) Mixed Spectra and Eigenfunction Expansions

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Spectral expansions and self-adjoint extensions

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3: 12.20 Approximations

§12.20 Approximations

►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty }$ and $x=0$, respectively. As special cases of these results a Chebyshev-series expansion for $U(a,x)$ valid when follows from (12.7.14), and Chebyshev-series expansions for $U(a,x)$ and $V(a,x)$ valid when $0\le x\le \lambda$ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). Here $\lambda$ denotes an arbitrary positive constant.

4: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

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Linear Operators in Finite Dimensional Vector Spaces

… ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. … ►

Orthonormal Expansions

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5: 12.9 Asymptotic Expansions for Large Variable

§12.9 Asymptotic Expansions for Large Variable

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§12.9(i) Poincaré-Type Expansions

►Throughout this subsection $\delta$ is an arbitrary small positive constant. … ►

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

►Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …

6: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

… ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … ►Here …

7: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

… ►Again, $\delta$ denotes an arbitrary small positive constant. … ►Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). … ►For an asymptotic expansion of $W_{\kappa ,\mu }\left(z\right)$ as $z\to \mathrm{\infty }$ that is valid in the sector $|\mathrm{ph}z|\le \pi -\delta$ and where the real parameters $\kappa$, $\mu$ are subject to the growth conditions $\kappa =o\left(z\right)$, $\mu =o\left(\sqrt{z}\right)$, see Wong (1973a).

8: 18.24 Hahn Class: Asymptotic Approximations

… ►This expansion is in terms of the parabolic cylinder function and its derivative. … ►This expansion is in terms of confluent hypergeometric functions. … ►The first expansion holds uniformly for $\delta \le x\le 1+\delta$, and the second for $1-\delta \le x\le 1+{\delta }^{-1}$, $\delta$ being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions. … ►Taken together, these expansions are uniformly valid for and for $a$ in unbounded intervals—each of which contains $[0,(1-\delta )n]$, where $\delta$ again denotes an arbitrary small positive constant. …

9: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

§8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

… ► $\delta$ again denoting an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion of $E_p\left(z\right)$ see §2.11(iii). ►

§8.20(ii) Large $p$

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10: 13.24 Series

§13.24 Series

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§13.24(i) Expansions in Series of Whittaker Functions

►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). ►

§13.24(ii) Expansions in Series of Bessel Functions

… ►For other series expansions see Prudnikov et al. (1990, §6.6). …