linear functional
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1: 1.16 Distributions
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βΊA mapping is a linear functional if
… is called a distribution, or generalized function, if it is a continuous linear functional on , that is, it is a linear functional and for every in ,
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βΊA tempered distribution is a continuous linear functional
on .
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βΊA distribution in is a continuous linear functional on .
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βΊTempered distributions are continuous linear functionals on this space of test functions.
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2: 15.19 Methods of Computation
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βΊFor 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 .
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βΊThis is because the linear transformations map the pair onto itself.
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3: 35.11 Tables
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βΊEach table expresses the zonal polynomials as linear combinations of monomial symmetric functions.
4: Howard S. Cohl
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βΊCohl 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 -series.
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5: 21.8 Abelian Functions
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βΊFor every Abelian function, there is a positive integer , such that the Abelian function can be expressed as a ratio of linear combinations of products with factors of Riemann theta functions with characteristics that share a common period lattice.
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6: 2.6 Distributional Methods
7: Bibliography G
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The Computation of Special Functions by Linear Difference Equations.
In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. GyΕri, and G. Ladas (Eds.),
pp. 213–243.
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8: 7.21 Physical Applications
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βΊFried and Conte (1961) mentions the role of in the theory of linearized waves or oscillations in a hot plasma; is called the plasma dispersion
function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954).
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9: 15.8 Transformations of Variable
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§15.8(i) Linear Transformations
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15.8.12
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βΊA quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation.
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βΊThe transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).
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10: Bibliography D
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Complex zeros of linear combinations of spherical Bessel functions and their derivatives.
SIAM J. Math. Anal. 4 (1), pp. 128–133.
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions.
Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
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Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions.
Stud. Appl. Math. 113 (3), pp. 245–270.
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