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11: 28.34 Methods of Computation

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Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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12: Bibliography Y

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Partitions with difference conditions and Alder’s conjecture. Proc. Natl. Acad. Sci. USA 101 (47), pp. 16417–16418.

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External Links:

ISSN 1091-6490, FullText, Document, MathReview (Scott D. Ahlgren), ZentralBlatt

Referenced by:

§26.10(iv).

Encodings:

BibTeX

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13: 14.16 Zeros

… ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)$ if any of the following sets of conditions hold: … ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)+1$ if either of the following sets of conditions holds: … ► $P_{\nu }^{\mu }\left(x\right)$ has exactly one zero in the interval $(1,\mathrm{\infty })$ if either of the following sets of conditions holds: …

14: 20.13 Physical Applications

… ►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). …

15: 19.15 Advantages of Symmetry

… ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

16: Notices

… ►As a condition of using the DLMF, you explicitly release NIST from any and all liabilities for any damage of any type that may result from errors or omissions in the DLMF. …

17: 21.9 Integrable Equations

… ►These parameters, including $𝛀$, are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition $u(x,y,0)$ (Deconinck and Segur (1998)). … ►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). …

18: 3.8 Nonlinear Equations

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§3.8(vi) Conditioning of Zeros

… ►are well separated but extremely ill-conditioned. …

19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Self-adjoint extensions of (1.18.28) and the Weyl alternative

… ► A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …Then, if the linear form $ℬ$ is nonzero, the condition $ℬ(f)=0$ is called a boundary condition at $a$. Boundary values and boundary conditions for the end point $b$ are defined in a similar way. … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. …

20: 3.6 Linear Difference Equations

… ►Let us assume the normalizing condition is of the form $w_0=\lambda$, where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_1^{(N)},w_2^{(N)},\mathrm{\dots },w_{N-1}^{(N)}$; see §3.2(ii). … ►It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution $w_n$ of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. … ►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). … ►Thus the asymptotic behavior of the particular solution ${𝐄}_n\left(1\right)$ is intermediate to those of the complementary functions $J_n\left(1\right)$ and $Y_n\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied. … ►Typically $k-\mathrm{\ell }$ conditions are prescribed at the beginning of the range, and $\mathrm{\ell }$ conditions at the end. …