北达科他大学文凭毕业证怎么制作【言正 微aptao168】74b

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11: Bibliography D

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Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.

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Notes:

Revised and with a preface by Hugh L. Montgomery

External Links:

ISBN 0-387-95097-4, MathReview, ZentralBlatt

Referenced by:

§27.12.

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Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96080-5, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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External Links:

ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

Referenced by:

3rd item.

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Novel identities for simple $n$-$j$ symbols. J. Mathematical Phys. 16, pp. 318–319.

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External Links:

Document, MathReview (M. J. Westwater)

Referenced by:

§34.5(vi).

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12: Bibliography E

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Properties of a Raman atom-laser output coupler. J. Phys. B 32 (12), pp. 2935–2950.

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External Links:

FullText

Referenced by:

§12.17.

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The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).

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External Links:

ISSN 0334-2700, FullText, MathReview (J. Németh), ZentralBlatt

Referenced by:

§2.10(i).

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A formula including Legendre’s $E{K}^{\prime }+K{E}^{\prime }-K{K}^{\prime }=\frac{1}{2}\pi$ . Messenger of Math. 33, pp. 31–32.

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Notes:

Errata on p. 45 of the same volume.

External Links:

ZentralBlatt

Referenced by:

§15.16.

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Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.

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ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§18.34(ii).

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Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950. In Sturm-Liouville theory, pp. 45–74.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§1.18(ix), §1.18(x).

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13: 27.11 Asymptotic Formulas: Partial Sums

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27.11.9 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{1}{p}=\frac{1}{\varphi \left(k\right)}\ln \ln x+B+O\left(\frac{1}{\ln x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E9

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Change of Notation (effective with 1.0.10):

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

See also:

Annotations for §27.11 and Ch.27

►where $\left(h,k\right)=1$, $k>0$, and $B$ is a constant depending on $h$ and $k$. … ►The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

14: Bibliography S

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Reihenentwicklungen analytischer Funktionen nach Biorthogonalsystemen spezieller Funktionen. I. Math. Z. 74, pp. 436–470.

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External Links:

ISSN 0025-5874, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§10.23(iii), §28.30(ii).

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Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

Erratum (V1.2.0) Section 18.39.

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:

ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

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Szegő’s Theorem and Its Descendants. Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ.

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External Links:

ISBN 978-0-691-14704-8, MathReview (Harry Dym)

Referenced by:

§18.2(xi).

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

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15: Bibliography W

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Asymptotische Entwicklungen der hypergeometrischen Funktion $F(a,b,c,z)$ für $|c|\to \mathrm{\infty }$ und konstante Werte $a,b$ und $z$ . Demonstratio Math. 21 (2), pp. 441–458 (German).

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External Links:

ISSN 0420-1213, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§15.12(ii), §15.12(ii).

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:

ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear and Nonlinear Waves. John Wiley & Sons, New York.

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Notes:

Reprinted in 1999.

External Links:

ISBN 0471359424, MathReview, ZentralBlatt

Referenced by:

§9.16.

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A presentation of cnoidal wave theory for practical application. J. Fluid Mech. 7 (2), pp. 273–286.

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External Links:

Document, MathReview (T. B. Benjamin), ZentralBlatt

Referenced by:

§21.9.

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A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

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16: 27.2 Functions

… ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. …

17: Bibliography L

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Inequalities for Bessel functions. J. Comput. Appl. Math. 15 (1), pp. 75–81.

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External Links:

ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.14.

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Moment integrals of powers of Airy functions. Z. Angew. Math. Phys. 44 (5), pp. 891–908.

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External Links:

ISSN 0044-2275, Document, MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

(9.11.15), (9.11.18), §9.11(v), §9.11(v).

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On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.

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External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§11.12.

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The confluent hypergeometric functions $M(a,b;z)$ and $U(a,b;z)$ for large $b$ and $z$ . J. Comput. Appl. Math. 233 (6), pp. 1570–1576.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

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New series expansions for the confluent hypergeometric function $M(a,b,z)$ . Appl. Math. Comput. 235, pp. 26–31.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.11, §13.11.

Encodings:

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18: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $ℝ$. … ►Boundary values and boundary conditions for the end point $b$ are defined in a similar way. …Similarly at $b$. … ►The reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results. … ►See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.

19: 3.5 Quadrature

… ►where $h=b-a$, $f\in C^2[a,b]$, and . … ►where . … ►where $h=(b-a)/n$. … ►For the Bernoulli numbers $B_m$ see §24.2(i). … ►For further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960). …

20: 8.17 Incomplete Beta Functions

… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of $a$, $b$, and $x$, and also to complex values. …where, as in §5.12, $\mathrm{B}(a,b)$ denotes the beta function: … ►For a historical profile of ${\mathrm{B}}_x(a,b)$ see Dutka (1981). … ►With $a>0$, $b>0$, and , … ►For $x>(a+1)/(a+b+2)$ or , more rapid convergence is obtained by computing $I_{1-x}(b,a)$ and using (8.17.4). …