# expansion of arbitrary vector

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##### 3: 1.3 Determinants, Linear Operators, and Spectral Expansions
###### Linear Operators in Finite Dimensional Vector Spaces
The adjoint of a matrix $\mathbf{A}$ is the matrix ${\mathbf{A}}^{*}$ such that $\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle$ for all $\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}$. …
##### 4: 12.20 Approximations
###### §12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U\left(a,b,x\right)$ and $M\left(a,b,x\right)$13.2(i)) whose regions of validity include intervals with endpoints $x=\infty$ and $x=0$, respectively. As special cases of these results a Chebyshev-series expansion for $U\left(a,x\right)$ valid when $\lambda\leq x<\infty$ follows from (12.7.14), and Chebyshev-series expansions for $U\left(a,x\right)$ and $V\left(a,x\right)$ valid when $0\leq x\leq\lambda$ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). Here $\lambda$ denotes an arbitrary positive constant.
##### 5: 12.9 Asymptotic Expansions for Large Variable
###### §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 expansions2.1(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 $z\to\infty$ that is valid in the sector $|\operatorname{ph}z|\leq\pi-\delta$ 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 $\delta\leq x\leq 1+\delta$, and the second for $1-\delta\leq x\leq 1+\delta^{-1}$, $\delta$ being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions. … Taken together, these expansions are uniformly valid for $-\infty 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(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).
##### 10: 13.24 Series
###### §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). …