variation of parameters
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1—10 of 12 matching pages
1: 1.13 Differential Equations
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βΊ
Variation of Parameters
…2: 10.15 Derivatives with Respect to Order
3: 18.39 Applications in the Physical Sciences
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βΊWhile in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b).
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4: 9.12 Scorer Functions
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βΊSolutions of this equation are the Scorer functions and can be found by the method of variation of parameters (§1.13(iii)).
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5: 10.40 Asymptotic Expansions for Large Argument
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βΊ
10.40.11
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6: 10.17 Asymptotic Expansions for Large Argument
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βΊ
10.17.14
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7: 2.3 Integrals of a Real Variable
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βΊThen
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βΊIn both cases the th error term is bounded in absolute value by , where the variational
operator
is defined by
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βΊThen the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion:
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βΊ(In other words, differentiation of (2.3.8) with respect to the parameter
(or ) is legitimate.)
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βΊWhen the parameter
is large the contributions from the real and imaginary parts of the integrand in
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8: Bibliography M
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On one-parameter families of Painlevé III.
Stud. Appl. Math. 101 (3), pp. 321–341.
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Latent roots and matrix variates: A review of some asymptotic results.
Ann. Statist. 6 (1), pp. 5–33.
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The variation with respect to order of zeros of Bessel functions.
Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.
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The characteristic numbers of the Mathieu equation with purely imaginary parameter.
Phil. Mag. Series 7 8 (53), pp. 834–840.
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Tables of the functions of the parabolic cylinder for negative integer parameters.
Zastos. Mat. 13, pp. 261–273.
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9: 10.23 Sums
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10.23.1
.
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10.23.2
.
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βΊ
10.23.14
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βΊ
10.23.20
βΊprovided that is of bounded variation (§1.4(v)) on an interval with .
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10: 1.14 Integral Transforms
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βΊSuppose that is absolutely integrable on and of bounded variation in a neighborhood of (§1.4(v)).
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βΊIf is absolutely integrable on and of bounded variation (§1.4(v)) in a neighborhood of , then
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βΊSuppose is a real- or complex-valued function and is a real or complex parameter.
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βΊIf is integrable on for all in , then the integral (1.14.32) converges and is an analytic function of in the vertical strip .
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βΊSuppose the integral (1.14.32) is absolutely convergent on the line and is of bounded variation in a neighborhood of .
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