normal equations
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41—50 of 74 matching pages
41: 28.14 Fourier Series
42: 8.18 Asymptotic Expansions of
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8.18.3
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8.18.13See (5.11.3).
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8.18.14
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►For asymptotic expansions for large values of and/or of the -solution of the equation
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43: Bibliography W
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Asymptotic expansions for second-order linear difference equations with a turning point.
Numer. Math. 94 (1), pp. 147–194.
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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Solutions of the fifth Painlevé equation. I.
Hokkaido Math. J. 24 (2), pp. 231–267.
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The generalised product moment distribution in samples from a normal multivariate population.
Biometrika 20A, pp. 32–52.
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On the central connection problem for the double confluent Heun equation.
Math. Nachr. 195, pp. 267–276.
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44: 10.22 Integrals
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10.22.72
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45: 20.13 Physical Applications
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►The functions , , provide periodic solutions of the partial differential equation
…with .
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►These two apparently different solutions differ only in their normalization and boundary conditions.
…Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281).
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►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.
46: 30.4 Functions of the First Kind
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►They are normalized by the condition
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30.4.5
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►Normalization of the coefficients is effected by application of (30.4.1).
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47: 29.6 Fourier Series
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29.6.2
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►This solution can be constructed from (29.6.4) by backward recursion, starting with and an arbitrary nonzero value of , followed by normalization via (29.6.5) and (29.6.6).
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29.6.17
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29.6.32
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29.6.47
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48: 28.4 Fourier Series
49: 28.6 Expansions for Small
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Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.
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►where is the unique root of the equation
in the interval , and .
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►Leading terms of the power series for the normalized functions are:
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§28.6(i) Eigenvalues
… ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … ►… |