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11: 10.2 Definitions

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§10.2(ii) Standard Solutions

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Bessel Functions of the Third Kind (Hankel Functions)

… ►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

12: 3.3 Interpolation

… ►The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … … ►The $(n+1)$-point formula (3.3.4) can be written in the form … ►

3.3.35 $$\left[z_0,z_1,\mathrm{\dots },z_n\right]f=\sum _{k=0}^n\left(f(z_k)/\prod _{\begin{array}{c}0\le j\le n\\ j\ne k\end{array}}(z_k-z_j)\right).$$

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Symbols:

$\left[z_0,z_1,\mathrm{\dots },z_n\right]f$: divided difference, $f(z)$: function and $z_k$: nodes of function

A&S Ref:

25.1.5 (in slightly different form)

Permalink:

http://dlmf.nist.gov/3.3.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(iii), §3.3 and Ch.3

… ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. …

13: Mathematical Introduction

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ►With a few exceptions the adopted notations are the same as those in standard applied mathematics and physics literature. … ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. … ►His genius in the creation of the National Bureau of Standards Handbook of Mathematical Functions paid enormous dividends to the world’s scientific, mathematical, and engineering communities, and paved the way for the development of the NIST Handbook of Mathematical Functions and NIST Digital Library of Mathematical Functions. …

14: 7.20 Mathematical Applications

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Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by $x=C\left(t\right)$, $y=S\left(t\right)$, $t\in [0,\mathrm{\infty })$. Magnify

… ►The normal distribution function with mean $m$ and standard deviation $\sigma$ is given by ►

7.20.1 $$\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{\mathrm{e}}^{-{(t-m)}^2/(2{\sigma }^2)}dt=\frac{1}{2}\mathrm{erfc}\left(\frac{m-x}{\sigma \sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma }\right)=P\left(\frac{x-m}{\sigma }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable, $m$: mean and $\sigma$: standard deviation

A&S Ref:

7.1.22 (slightly modified)

Permalink:

http://dlmf.nist.gov/7.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.20(iii), §7.20 and Ch.7

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15: Notices

… ►Pursuant to Title 17 USC 105, the National Institute of Standards and Technology (NIST), United States Department of Commerce, is authorized to receive and hold copyrights transferred to it by assignment or otherwise. … ►Bulk copying, reproduction, or redistribution in any form is not permitted. … ►

In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

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16: 13.2 Definitions and Basic Properties

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Standard Solutions

►The first two standard solutions are: … ►Another standard solution of (13.2.1) is $U(a,b,z)$, which is determined uniquely by the property … ►

§13.2(iii) Limiting Forms as $z\to 0$

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§13.2(iv) Limiting Forms as $z\to \mathrm{\infty }$

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17: 9.2 Differential Equation

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9.2.1 $$\frac{d^{2}w}{{dz}^2}=zw.$$

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Defines:

$w$: ODE solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

Source:

Olver (1997b, (1.01), p. 392)

A&S Ref:

10.4.1 (in slightly different form)

Referenced by:

§36.8, (9.10.10), (9.10.20), (9.10.21), (9.10.8), (9.10.9), §9.10(iii), (9.11.2), §9.11(i), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), (9.2.16), §9.2(iii), §9.2(vi), (9.8.14), (9.8.16), (9.8.18), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(i), §9.2 and Ch.9

… ►Standard solutions are: … ►

9.2.5 $$\mathrm{Bi}\left(0\right)=\frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}=\mathrm{0.61492\hspace{0.33em}66274}\mathrm{\dots },$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.4 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

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9.2.6 $${\mathrm{Bi}}^{\prime }\left(0\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}=\mathrm{0.44828\hspace{0.33em}83573}\mathrm{\dots }.$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.5 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

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§9.2(vi) Riccati Form of Differential Equation

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18: 18.2 General Orthogonal Polynomials

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§18.2(iii) Standardization and Related Constants

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Standardizations

… ►Two, more specialized, standardizations are: … ►

First Form

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Second Form

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19: Software Index

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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20: 3.7 Ordinary Differential Equations

… ►The remaining two equations are supplied by boundary conditions of the form … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. … ►For $w^{\prime }=f(z,w)$ the standard fourth-order rule reads … ►For $w^{\prime \prime }=f(z,w,w^{\prime })$ the standard fourth-order rule reads …