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1: 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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2: 4.42 Solution of Triangles

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§4.42(i) Planar Right Triangles

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Figure 4.42.1: Planar right triangle. Magnify

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3: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …

4: About the Project

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Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). The front row, from left to right: Ronald F. …The back row, from left to right: William P. …

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5: 25.19 Tables

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Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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6: 32.15 Orthogonal Polynomials

… ►For this result and applications see Fokas et al. (1991): in this reference, on the right-hand side of Eq. …

7: 16.5 Integral Representations and Integrals

… ►In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when . In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector ; compare §16.2(iii). Lastly, when $p>q+1$ the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. …

8: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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9: 2.3 Integrals of a Real Variable

… ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … ►When $p(t)$ is real and $x$ is a large positive parameter, the main contribution to the integral ►

2.3.13 $$I(x)={\int }_a^b{\mathrm{e}}^{-xp(t)}q(t)dt$$

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Defines:

$I(x)$: integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: real function and $q(t)$: function

Referenced by:

item (c)

Permalink:

http://dlmf.nist.gov/2.3.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

… ►When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in ►

2.3.19 $$I(x)={\int }_a^b{\mathrm{e}}^{\mathrm{i}xp(t)}q(t)dt$$

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Defines:

$I(x)$: integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $p(t)$: real function, $a$: left endpoint, $b$: right endpoint and $q(t)$: function

Referenced by:

§2.3(iv)

Permalink:

http://dlmf.nist.gov/2.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iv), §2.3 and Ch.2

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10: 2.4 Contour Integrals

… ►Let $𝒫$ denote the path for the contour integral ►

2.4.10 $$I(z)={\int }_a^b{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Defines:

$I(z)$: contour integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

… ►

2.4.14 $$I(z)={\int }_{t_0}^b{\mathrm{e}}^{-zp(t)}q(t)dt-{\int }_{t_0}^a{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $I(z)$: contour integral, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iv), §2.4 and Ch.2

►and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of $p(t)$ and $q(t)$ at $t=t_0$. …Thus the right-hand side of (2.4.14) reduces to the error terms. …