# linear

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## 1—10 of 72 matching pages

##### 1: 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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##### 2: Howard S. Cohl

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►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series.
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##### 3: 6.17 Physical Applications

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►Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

##### 4: 21.8 Abelian Functions

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►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice.
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##### 5: 35.11 Tables

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►Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

##### 6: 3.2 Linear Algebra

###### §3.2 Linear Algebra

… ►###### §3.2(iii) Condition of Linear Systems

… ► ►The*$p$-norm of a matrix*$\mathbf{A}=[{a}_{jk}]$ is … ►Then we have the

*a posteriori*error bound …

##### 7: 15.19 Methods of Computation

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►For $z\in \mathbb{R}$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$.
►For $z\in \u2102$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$.
This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself.
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►When $\mathrm{\Re}z>\frac{1}{2}$ it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8).
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##### 8: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I.
Methods Appl. Anal. 2 (2), pp. 173–197.
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On the calculation of Stokes multipliers for linear differential equations of the second order.
Methods Appl. Anal. 2 (3), pp. 348–367.
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On the asymptotic and numerical solution of linear ordinary differential equations.
SIAM Rev. 40 (3), pp. 463–495.
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Hyperasymptotic solutions of second-order linear differential equations. II.
Methods Appl. Anal. 2 (2), pp. 198–211.
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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.
Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
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##### 9: 24.5 Recurrence Relations

###### §24.5 Recurrence Relations

…##### 10: 35.10 Methods of Computation

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►These algorithms are extremely efficient, converge rapidly even for large values of $m$, and have complexity linear in $m$.