linear functional
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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 introduced in §10.2(ii), we sometimes use to denote , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and .
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13: Bibliography M
14: 10.2 Definitions
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►The notation denotes , , , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and .
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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 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 .
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►With , the last equations give as the solution of a system of linear equations.
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►(3.11.29) is a system of
linear equations for the coefficients .
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►More generally, let be approximated by a linear combination
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►We take complex exponentials , , and approximate by the linear combination (3.11.31).
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18: 18.38 Mathematical Applications
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►While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations.
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►This gives also new structures and results in the one-variable case, but the obtained nonsymmetric special functions can now usually be written as a linear combination of two known special functions.
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19: 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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20: 16.14 Partial Differential Equations
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►In addition to the four Appell functions there are other sums of double series that cannot be expressed as a product of two
functions, and which satisfy pairs of linear partial differential equations of the second order.
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