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21: 3.8 Nonlinear Equations

… ►The equation to be solved is … ►If

p=1

and

A<1

, then the convergence is said to be linear or geometric. … ►In this way polynomials of successively lower degree can be used to find the remaining zeros. … ►Newton’s rule can also be used for complex zeros of

p(z)

. … ►for solving fixed-point problems (3.8.2) cannot always be predicted, especially in the complex plane. …

22: Customize DLMF

…Note: Javascript and Cookies must be enabled to support customizations.

23: Bibliography L

… ►

D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

ⓘ

External Links:: FullText, MathReview (W. E. Milne), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

… ►

J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

ⓘ

External Links:: Document, MathReview (R. D. James), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

H. Levine and J. Schwinger (1948) On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.

ⓘ

External Links:: Document, MathReview (C. J. Bouwkamp), ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

N. A. Lukaševič (1968) Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 4(8), pp. 732–735
External Links:: MathReview (C. E. Langenhop), ZentralBlatt
Referenced by:: §32.10(v), §32.8(v).
Encodings:: BibTeX

… ►

Y. L. Luke (1972) Miniaturized tables of Bessel functions. III. Math. Comp. 26 (117), pp. 237–240 and A14–B5.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

…

24: 3.5 Quadrature

… ►If

f\in C^{2m+2}[a,b]

, then the remainder

E_{n}(f)

in (3.5.2) can be expanded in the form …These can be found by means of the recursion … ►For

C^{\infty}

functions Gauss quadrature can be very efficient. … ►can be computed by Filon’s rule. … ►The integrand can be extended as a periodic

C^{\infty}

function on

\mathbb{R}

with period

2\pi

and as noted in §3.5(i), the trapezoidal rule is exceptionally efficient in this case. …

25: 1.12 Continued Fractions

… ►

C_{n}

is called the

n

th approximant or convergent to $C$ . … ►Define … ►A sequence

\{C_{n}\}

in the extended complex plane,

\mathbb{C}\cup\{\infty\}

, can be a sequence of convergents of the continued fraction (1.12.3) iff … ►Conversely,

C

is called an extension of

C^{\prime}

. … ►Then the convergents

C_{n}

satisfy …

26: 2.10 Sums and Sequences

… ►Other examples that can be verified in a similar way are: … ►These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula …For examples see Olver (1997b, Chapters 8, 9). … ►First, the conditions can be weakened. …For example, Condition (b) can be replaced by: …

27: 28.16 Asymptotic Expansions for Large $q$

… ►Let

s=2m+1

m=0,1,2,\dots

, and

\nu

be fixed with

m<\nu<m+1

. … ►

28.16.1

\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.16 and Ch.28

…

28: 18.38 Mathematical Applications

… ►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). … ►The orthogonality relations in §34.3(iv) for the

3j

symbols can be rewritten in terms of orthogonality relations for Hahn or dual Hahn polynomials as given by §§18.2(i), 18.2(iii) and Table 18.19.1 or by §18.25(iii), respectively. … ►The orthogonality relations (34.5.14) for the

6j

symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … ►Similar algebras can be associated with all families of OP’s in the

q

-Askey scheme and the Askey scheme. … ►This gives also new structures and results in the one-variable case, but the obtained nonsymmetric special functions can now usually be written as a linear combination of two known special functions. …

29: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►By repeated differentiation of (3.7.1) all derivatives of

w(z)

can be expressed in terms of

w(z)

and

w^{\prime}(z)

as follows. … ►Let

\mathbf{A}_{P}

be the

(2P)\times(2P+2)

band matrix … ►(

\mathbf{I}

and

\boldsymbol{{0}}

being the identity and zero matrices of order

2\times 2

.) … ►If

q(x)

C^{\infty}

on the closure of

(a,b)

, then the discretized form (3.7.13) of the differential equation can be used. …

30: 1.10 Functions of a Complex Variable

… ►Let

f_{1}(z)

be analytic in a domain

D_{1}

. … ►Let

C

be a simple closed contour consisting of a segment

\mathit{AB}

of the real axis and a contour in the upper half-plane joining the ends of

\mathit{AB}

. Also, let

f(z)

be analytic within

C

, continuous within and on

C

, and real on

\mathit{AB}