Liouville function

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11: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations …where $f$, $g$, and $h$ are analytic functions in a domain $D\subset ℂ$. …For applications to special functions $f$, $g$, and $h$ are often simple rational functions. … ►

§3.7(iv) Sturm–Liouville Eigenvalue Problems

… ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …

12: Bibliography D

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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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External Links:

ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

Encodings:

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13: 2.7 Differential Equations

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§2.7(iii) Liouville–Green (WKBJ) Approximation

►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: ►

Liouville–Green Approximation Theorem

… ►Here $F(x)$ is the error-control function … ►The first of these references includes extensions to complex variables and reversions for zeros. …

14: 30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ►The Liouville normal form of equation (30.2.1) is ►

30.2.2 $$\frac{d^{2}g}{{dt}^2}+\left(\lambda +\frac{1}{4}+{\gamma }^2{\sin }^{2}t-\frac{{\mu }^2-\frac{1}{4}}{{\sin }^{2}t}\right)g=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.2(iii)

Permalink:

http://dlmf.nist.gov/30.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.2(ii), §30.2 and Ch.30

… ►If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

15: 18.38 Mathematical Applications

… ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …

16: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for $\nu \left(n\right)$, $\mathrm{\Lambda }\left(n\right)$, $p_n$, and $\pi \left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and … ►

27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

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Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

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See also:

Annotations for §27.3 and Ch.27

… ►A function $f$ is completely multiplicative if $f(1)=1$ and …Examples are $\lfloor 1/n\rfloor$ and $\lambda \left(n\right)$, and the Dirichlet characters, defined in §27.8. …

17: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►where $z$ is a real or complex variable and the function $f$ is nonlinear. … ►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: … ►

§3.8(v) Zeros of Analytic Functions

… ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …

18: Bibliography S

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Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§18.36(vi), §18.39(i).

Encodings:

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Liouville-Green approximations via the Riccati transformation. J. Math. Anal. Appl. 116 (1), pp. 147–165.

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External Links:

ISSN 0022-247X, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

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Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.

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External Links:

ISSN 0021-9045, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

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Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.

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External Links:

ISSN 0377-0427, Document, MathReview (Wan Biao Ma), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

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A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 567–577.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

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19: Bibliography P

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Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations F.

Encodings:

BibTeX

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Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.

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External Links:

ISSN 1364-5021, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§10.46.

Encodings:

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Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

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Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.

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External Links:

ISBN 0-19-853415-9, MathReview (L. Greenberg), ZentralBlatt

Referenced by:

§3.7(iv).

Encodings:

BibTeX

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20: 2.6 Distributional Methods

… ►To each function in this equation, we shall assign a tempered distribution (i. …, a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. … ►We have now assigned a distribution to each function in (2.6.10). … ►The Riemann–Liouville fractional integral of order $\mu$ is defined by … ►Also, …