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1: 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). …

2: 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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3: 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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4: 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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5: 32.15 Orthogonal Polynomials

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

6: 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. …

7: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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8: 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:

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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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9: 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

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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. …

10: 19.32 Conformal Map onto a Rectangle

… ►As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_3)$ is negative imaginary. …