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1: 3.6 Linear Difference Equations

… ►The Weber function ${𝐄}_n\left(1\right)$ satisfies …We apply the algorithm to compute ${𝐄}_n\left(1\right)$ to 8S for the range $n=1,2,\mathrm{\dots },10$, beginning with the value ${𝐄}_0\left(1\right)=-\mathrm{0.56865\hspace{0.33em}\hspace{0.17em}663}$ obtained from the Maclaurin series expansion (§11.10(iii)). ►In the notation of §3.6(v) we have $M=10$ and $ϵ=\frac{1}{2}\times {10}^{-8}$. …The values of $w_n$ for $n=1,2,\mathrm{\dots },10$ are the wanted values of ${𝐄}_n\left(1\right)$. … ►For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).

2: Bibliography C

… ►

Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.

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External Links:

Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.29(iii).

Encodings:

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A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.

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External Links:

ISBN 0-8176-3753-2, MathReview (R. M. M. Mattheij), ZentralBlatt

Referenced by:

§3.6(vii).

Encodings:

BibTeX

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Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:

MathReview (I. I. Hirschman, Jr.), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

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Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.

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External Links:

Document

Referenced by:

§5.20.

Encodings:

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… ►

Chebyshev approximations for the natural logarithm of the gamma function. Math. Comp. 21 (98), pp. 198–203.

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External Links:

FullText, ZentralBlatt

Referenced by:

§5.23(i).

Encodings:

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3: DLMF Project News

error generating summary

4: 33.20 Expansions for Small $|ϵ|$

… ►

$f(0,\mathrm{\ell };r)={(2r)}^{1/2}{J}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right),$

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$h(0,\mathrm{\ell };r)=-{(2r)}^{1/2}{Y}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right)$ , $r>0$,

►

$f(0,\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }+1}{(2|r|)}^{1/2}{I}_{2\mathrm{\ell }+1}\left(\sqrt{8|r|}\right),$

… ►where … ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

5: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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6: 6.14 Integrals

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6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

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6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

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6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

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Symbols:

$dx$: differential of $x$ , $E_1\left(z\right)$: exponential integral, $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

5.1.33

Permalink:

http://dlmf.nist.gov/6.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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6.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{Ci}}^2\left(t\right)dt={\int }_0^{\mathrm{\infty }}{\mathrm{si}}^2\left(t\right)dt=\frac{1}{2}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.31

Permalink:

http://dlmf.nist.gov/6.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

… ►For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).

7: Bibliography

… ►

Tables of $\ln \mathrm{\Gamma }(z)$ for Complex Argument. Pergamon Press, New York.

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External Links:

ZentralBlatt

Referenced by:

§5.22(iii).

Encodings:

BibTeX

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Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

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Referenced by:

§9.16.

Encodings:

BibTeX

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Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.

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External Links:

Document

Referenced by:

6th item.

Encodings:

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Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

Encodings:

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Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

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External Links:

FullText, MathReview (F. Schipp), ZentralBlatt

Referenced by:

§16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.

Encodings:

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8: 27.2 Functions

… ►

§27.2(i) Definitions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. ($\nu \left(1\right)$ is defined to be 0.) … ►The $\varphi \left(n\right)$ numbers $a,a^2,\mathrm{\dots },a^{\varphi \left(n\right)}$ are relatively prime to $n$ and distinct (mod $n$). …It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …

9: 8.19 Generalized Exponential Integral

… ►Most properties of $E_p\left(z\right)$ follow straightforwardly from those of $\mathrm{\Gamma }(a,z)$. For an extensive treatment of $E_1\left(z\right)$ see Chapter 6. … ►Integral representations of Mellin–Barnes type for $E_p\left(z\right)$ follow immediately from (8.6.11), (8.6.12), and (8.19.1). … ►The general function $E_p\left(z\right)$ is attained by extending the path in (8.19.2) across the negative real axis. Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. …

10: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

… ►

§24.2(ii) Euler Numbers and Polynomials

… ►

$E_{2n+1}=0$ ,

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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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