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1: 27.12 Asymptotic Formulas: Primes

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27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

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27.12.7

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§27.12

Permalink:

http://dlmf.nist.gov/27.12.E7

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►The largest known prime (2018) is the Mersenne prime $2^{82,589,933}-1$. …

2: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. $p(𝒟k,n)$ denotes the number of partitions of $n$ into parts with difference at least $k$. …If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. …

3: 33.17 Recurrence Relations and Derivatives

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4: Bibliography

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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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The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau

External Links:

ISBN 978-1-611976-38-0, MathReview Entry

Referenced by:

§18.40(ii).

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:

BibTeX

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

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5: 6.7 Integral Representations

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6.7.3 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{a^2+t^2}dt=\frac{\mathrm{i}}{2a}\left({\mathrm{e}}^a{E}_1\left(a-\mathrm{i}x\right)-{\mathrm{e}}^{-a}{E}_1\left(-a-\mathrm{i}x\right)\right),$$ $a>0$, $x>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

A&S Ref:

5.1.41 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.4 $${\int }_x^{\mathrm{\infty }}\frac{t{\mathrm{e}}^{\mathrm{i}t}}{a^2+t^2}dt=\frac{1}{2}\left({\mathrm{e}}^a{E}_1\left(a-\mathrm{i}x\right)+{\mathrm{e}}^{-a}{E}_1\left(-a-\mathrm{i}x\right)\right),$$ $a>0$, $x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

A&S Ref:

5.1.42 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.5 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{a^2+t^2}dt=-\frac{1}{2a\mathrm{i}}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)-{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$,

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.43 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

… ►Many integrals with exponentials and rational functions, for example, integrals of the type $\int {\mathrm{e}}^{z}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be represented in finite form in terms of the function $E_1\left(z\right)$ and elementary functions; see Lebedev (1965, p. 42). … ►For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).

6: Bibliography T

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The numerical computation of the confluent hypergeometric function $U(a,b,z)$ . Numer. Math. 41 (1), pp. 63–82.

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Notes:

Algol 60 variable-precision procedures are included.

External Links:

ISSN 0029-599X, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§13.29(iv), §13.32(ii).

Encodings:

BibTeX

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A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

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Notes:

Includes Pascal program

External Links:

FullText

Referenced by:

§5.24(ii), §7.25(ii), §8.28(ii).

Encodings:

BibTeX

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Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

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External Links:

ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt

Referenced by:

§24.11, §24.16(i).

Encodings:

BibTeX

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The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.

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Notes:

Collection of articles honoring Alston S. Householder. For corrigendum see same journal v. 18 (1975), no. 8, p. 462

External Links:

ISSN 0001-0782, Document, MathReview (L. Fox), ZentralBlatt

Referenced by:

§19.20(i), §19.20(iv), §19.36(iii).

Encodings:

BibTeX

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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7: Bibliography K

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Pascal program for generating tables of $\mathrm{SU}(3)$ Clebsch-Gordan coefficients. Comput. Phys. Comm. 85 (1), pp. 82–88.

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External Links:

ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

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External Links:

ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt

Referenced by:

§16.23(iii).

Encodings:

BibTeX

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

BibTeX

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Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

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Notes:

Translated from the Russian by E. V. Pankratiev

External Links:

ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt

Referenced by:

§10.73(i), §10.73(i), §10.73(i).

Encodings:

BibTeX

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Estimates of trigonometric sums and their applications. Uspehi Mat. Nauk 13 (4 (82)), pp. 185–192 (Russian).

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External Links:

MathReview (R. A. Rankin), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►General references for this subsection include Friedman (1990, pp. 4–6), Shilov (2013, pp. 249–256), Riesz and Sz.-Nagy (1990, Ch. 5, §82). … ►For $T$ to be actually self adjoint it is necessary to also show that $𝒟(T^{\ast })=𝒟(T)$, as it is often the case that $T$ and $T^{\ast }$ have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator $\frac{d}{dx}$. … ►The implicit boundary conditions taken here are that the ${\varphi }_n(x)$ and ${\varphi }_n^{\prime }(x)$ vanish as $x\to \pm \mathrm{\infty }$, which in this case is equivalent to requiring ${\varphi }_n(x)\in L^2\left(X\right)$, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … ►More generally, continuous spectra may occur in sets of disjoint finite intervals $[{\lambda }_a,{\lambda }_b]\in (0,\mathrm{\infty })$, often called bands, when $q(x)$ is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … ►Let $T$ be a linear operator on $V$ with dense domain $𝒟(T)$ and with range $ℛ(T)=\{Tv\mid v\in 𝒟(T)\}$. …

9: 10.32 Integral Representations

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10.32.19 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=\frac{1}{8\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t+\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t+\frac{1}{2}\mu -\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu -\frac{1}{2}\nu \right)}{\mathrm{\Gamma }\left(2t\right)}{(\frac{1}{2}z)}^{-2t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

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10: 12.14 The Function $W(a,x)$

… ►This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. … ►the branch of $\mathrm{ph}$ being zero when $a=0$ and defined by continuity elsewhere. … ►Airy-type uniform asymptotic expansions can be used to include either one of the turning points $\pm 1$. … ►The expansions for the derivatives corresponding to (12.14.25), (12.14.26), and (12.14.31) may be obtained by formal term-by-term differentiation with respect to $t$; compare the analogous results in §§12.10(ii)–12.10(v). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). …