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21—30 of 759 matching pages

21: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. For further information and examples see §2.5 and Paris and Kaminski (2001, Chapters 5, 6, and 8). …

22: DLMF Project News

error generating summary

23: 11.13 Methods of Computation

… ►For large $|z|$ and/or $|\nu |$ the asymptotic expansions given in §11.6 should be used instead. … ►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. … ►The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. … ►Sequences of values of ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$, with $z$ fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

24: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In what follows $T$ will be taken to be a self adjoint extension of $ℒ$ following the discussion ending the prior sub-section. … ►These eigenvalues will be assumed distinct, i. … ►Spectral expansions of $T$, and of functions $F(T)$ of $T$, these being expansions of $T$ and $F(T)$ in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow: … ►this being a matrix element of the resolvent $F(T)={\left(z-T\right)}^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). … ►Should $q(x)$ be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). …

25: 3.6 Linear Difference Equations

… ►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. … ►However, $w_n$ can be computed successfully in these circumstances by boundary-value methods, as follows. … ►The least value of $N$ that satisfies (3.6.9) is found to be 16. …(It should be observed that for $n>10$, however, the $w_n$ are progressively poorer approximations to ${𝐄}_n\left(1\right)$: the underlined digits are in error.) … ►Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …

26: 24.21 Software

… ►Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. … ►

§24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, and $E_n\left(x\right)$

…

27: 33.13 Complex Variable and Parameters

… ►The functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, and $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ may be extended to noninteger values of $\mathrm{\ell }$ by generalizing $(2\mathrm{\ell }+1)!=\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $U(a,b,z)$ expanded via (13.2.42). ►These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\mathrm{\ell }$. The quantities $C_{\mathrm{\ell }}\left(\eta \right)$, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$, and $R_{\mathrm{\ell }}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that ►

33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

… ►

33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

…

28: 3.9 Acceleration of Convergence

… ►It should be borne in mind that a sequence (series) transformation can be effective for one type of sequence (series) but may not accelerate convergence for another type. It may even fail altogether by not being limit-preserving. … ►Let $k$ be a fixed positive integer. … ►The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: … ►

Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

► ►►►

$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…
9	0.82248 70624 89	0.82246 71865 91	0.82246 70351 34	0.82246 70334 48	0.82246 70334 24
…

►

…

29: 32.8 Rational Solutions

… ► ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … ►

32.8.4 $$w(z;4)=-\frac{1}{z}+\frac{6z^2(z^3+10)}{z^6+20z^3-80}-\frac{9z^5(z^3+40)}{z^9+60z^6+11200}.$$

ⓘ

Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

… ►

$$Q_3(z)=z^6+20z^3-80,$$

… ►Next, let $p_m(z)$ be the polynomials defined by $p_m(z)=0$ for , and … ►

$\alpha =\frac{1}{8}{(2m+1)}^2$, $\beta =-\frac{1}{8}{(2n+1)}^2$, and $\gamma \notin \mathbb{Z}$.

…

30: Ian J. Thompson

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Ian J. Thompson

… ►Ian J. Thompson (b. …