# linear functionals

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## 11—20 of 64 matching pages

##### 11: Bibliography B

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Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations.
The Macmillan Co., New York.
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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions.
SIAM J. Math. Anal. 17 (2), pp. 422–450.
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Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case.
In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.),
Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.
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##### 12: 10.25 Definitions

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►Corresponding to the symbol ${\mathcal{C}}_{\nu}$ introduced in §10.2(ii), we sometimes use ${\mathcal{Z}}_{\nu}\left(z\right)$ to denote ${I}_{\nu}\left(z\right)$, ${\mathrm{e}}^{\nu \pi \mathrm{i}}{K}_{\nu}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu $.
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##### 13: Bibliography M

##### 14: 10.2 Definitions

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►The notation ${\mathcal{C}}_{\nu}\left(z\right)$ denotes ${J}_{\nu}\left(z\right)$, ${Y}_{\nu}\left(z\right)$, ${H}_{\nu}^{(1)}\left(z\right)$, ${H}_{\nu}^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu $.
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##### 15: 2.9 Difference Equations

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►Many special functions that depend on parameters satisfy a three-term linear recurrence relation
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##### 16: William P. Reinhardt

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►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions.
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##### 17: 3.11 Approximation Techniques

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►Also, in cases where $f(x)$ satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the ${c}_{n}$.
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►With ${b}_{0}=1$, the last $q$ equations give ${b}_{1},\mathrm{\dots},{b}_{q}$ as the solution of a system of linear equations.
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►(3.11.29) is a system of $n+1$
linear equations for the coefficients ${a}_{0},{a}_{1},\mathrm{\dots},{a}_{n}$.
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►More generally, let $f(x)$ be approximated by a linear combination
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►We take $n$ complex exponentials ${\varphi}_{k}(x)={\mathrm{e}}^{\mathrm{i}kx}$, $k=0,1,\mathrm{\dots},n-1$, and approximate $f(x)$ by the linear combination (3.11.31).
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##### 18: Bibliography S

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Non-linear integral equations for Heun functions.
Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
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##### 19: 16.14 Partial Differential Equations

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►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_{2}F_{1}$
functions, and which satisfy pairs of linear partial differential equations of the second order.
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##### 20: 9.15 Mathematical Applications

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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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