particular%20solutions

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11: 28.33 Physical Applications

… ►

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►The wave equation …The general solution of the problem is a superposition of the separated solutions. … ►In particular, the equation is stable for all sufficiently large values of

\omega

. … ►However, in response to a small perturbation at least one solution may become unbounded. …

12: Bruce R. Miller

… ►In particular, he developed the LaTeXML system used for converting the LaTeX source documents into XML and MathML from which the DLMF website is constructed. …

13: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►For certain combinations of the parameters,

\mbox{P}_{\mbox{\scriptsize II}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. … ►with solution … ►which has the solution … ►

14: 20 Theta Functions

Chapter 20 Theta Functions

…

15: Notices

… ►” NIST makes NO WARRANTY OF ANY TYPE, including no warranties of merchantability or fitness for a particular purpose. … ►

Index of Selected Software Within the DLMF Chapters

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

…

16: 8.23 Statistical Applications

… ►Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …

17: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters

\mu

and

\nu

the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. …

18: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To compute a particular value

p\left(n\right)

it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). …

19: 2.11 Remainder Terms; Stokes Phenomenon

… ►In particular, … ►In particular, on the ray

\theta=\pi

greatest accuracy is achieved by (a) taking the average of the expansions (2.11.6) and (2.11.7), followed by (b) taking account of the exponentially-small contributions arising from the terms involving

h_{2s}(\theta,\alpha)

in (2.11.15). … ►

§2.11(v) Exponentially-Improved Expansions (continued)

… ►

2.11.19

w_{j}(z)=e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{n-1}\frac{a_{s,j}}{z^{s}}+R_{n% }^{(j)}(z),

j=1,2

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $R_{n}^{(j)}(z)$ : remainder, $a_{s,j}$ : coefficients, $\lambda_{j}$ : roots, $\mu_{j}$ : quantities and $w_{j}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/2.11.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §2.11(v), §2.11 and Ch.2

… ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

20: 9.12 Scorer Functions

… ►The general solution is given by …where

A

and

B

are arbitrary constants,

w_{1}(z)

and

w_{2}(z)

are any two linearly independent solutions of Airy’s equation (9.2.1), and

p(z)

is any particular solution of (9.12.1). Standard particular solutions are … ►

§9.12(iv) Numerically Satisfactory Solutions

… ►In

\mathbb{C}

, numerically satisfactory sets of solutions are given by …