dominated convergence theorem

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11: 13.17 Continued Fractions

… ►This continued fraction converges to the meromorphic function of

z

on the left-hand side for all

z\in\mathbb{C}

. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … ►This continued fraction converges to the meromorphic function of

z

on the left-hand side throughout the sector

|\operatorname{ph}{z}|<\pi

. …

12: 1.4 Calculus of One Variable

… ►

Mean Value Theorem

… ►Absolute convergence also implies convergence. … ►

Fundamental Theorem of Calculus

… ►

First Mean Value Theorem

…

13: 14.13 Trigonometric Expansions

… ►

14.13.2

\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{1/2}2^{\mu}(\sin\theta)^{% \mu}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu% +k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\cos\left% ((\nu+\mu+2k+1)\theta\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : variable
Source:: Cohl et al. (2021, Theorem 6.2)
A&S Ref:: 8.7.2
Referenced by:: §14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)
Permalink:: http://dlmf.nist.gov/14.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.13 and Ch.14

►These Fourier series converge absolutely when

\Re\mu<0

. If

0\leq\Re\mu<\frac{1}{2}

then they converge, but, if

\theta\not=\frac{1}{2}\pi

, they do not converge absolutely. … ►with conditional convergence for each. …

14: 2.11 Remainder Terms; Stokes Phenomenon

… ►Even when the series converges this is unwise: the tail needs to be majorized rigorously before the result can be guaranteed. … ►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of

\operatorname{ph}z=\pi

, is increasingly significant as

\operatorname{ph}z

passes from

\pi

\frac{3}{2}\pi

, and becomes the dominant contribution after

\operatorname{ph}z

passes

\frac{3}{2}\pi

. … ►Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines (

\theta=\pi

in the present example). … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. … ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.

15: 3.6 Linear Difference Equations

… ►For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a). … ►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

Y_{n}\left(1\right)

is dominant and can be computed by forward recursion, whereas

J_{n}\left(1\right)

is recessive and has to be computed by backward recursion. …

16: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

17: 18.18 Sums

… ►See Szegő (1975, Theorems 3.1.5 and 5.7.1). … ►

§18.18(ii) Addition Theorems

►

Ultraspherical

… ►

Legendre

… ►

§18.18(iii) Multiplication Theorems

…

18: 2.9 Difference Equations

… ►If

|\rho_{2}|>|\rho_{1}|

, or if

|\rho_{2}|=|\rho_{1}|

and

\Re\alpha_{2}>\Re\alpha_{1}

, then

w_{1}(n)

is recessive and

w_{2}(n)

is dominant as

n\to\infty

. As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. When

|\rho_{2}|=|\rho_{1}|

and

\Re\alpha_{2}=\Re\alpha_{1}

neither solution is dominant and both are unique. …

19: 2.1 Definitions and Elementary Properties

… ►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►This result also holds with both

O

’s replaced by

o

’s. … ►Let

\sum a_{s}x^{-s}

be a formal power series (convergent or divergent) and for each positive integer

n

, … ►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. … ►It can even happen that a generalized asymptotic expansion converges, but its sum is not the function being represented asymptotically; for an example see §18.15(iii).

20: 27.4 Euler Products and Dirichlet Series

… ►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …if the series on the left is absolutely convergent. In this case the infinite product on the right (extended over all primes

p

) is also absolutely convergent and is called the Euler product of the series. …