reflection%20properties%20in%20q

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21: 25.12 Polylogarithms

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§25.12(i) Dilogarithms

… ►For graphics see Figures 25.12.1 and 25.12.2, and for further properties see Maximon (2003), Kirillov (1995), Lewin (1981), Nielsen (1909), and Zagier (1989). … ►

§25.12(ii) Polylogarithms

… ►(In the latter case (25.12.11) becomes (25.5.1)). ►Further properties include …

22: 36 Integrals with Coalescing Saddles

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23: Bibliography N

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D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:: ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

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G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:: ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

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E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

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24: 20.11 Generalizations and Analogs

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§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

… ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus

k

of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in

q

-series via (20.9.1). However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). … ►Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …

25: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

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Curtis (1964a) tabulates $P_{\ell}(\epsilon,r)$ , $Q_{\ell}(\epsilon,r)$ (§33.1), and related functions for $\ell=0,1,2$ and $\epsilon=-2(.2)2$ , with $x=0(.1)4$ for $\epsilon<0$ and $x=0(.1)10$ for $\epsilon\geq 0$ ; 6D.

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26: Peter L. Walker

… ► 1942 in Dorchester, U. …He began his academic career in 1964 at the University of Lancaster, U. … ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►Walker is now retired and living in Cheltenham, UK. … ►

20 Theta Functions

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27: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. Tables of primes (§27.21) reveal great irregularity in their distribution. … ►Other examples of number-theoretic functions treated in this chapter are as follows. … ►In the following examples,

a_{1},\dots,a_{\nu\left(n\right)}

are the exponents in the factorization of

n

in (27.2.1). …

28: 27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product

m

has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

29: 4.37 Inverse Hyperbolic Functions

… ►Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect

\pm 1

. … ►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … ►

§4.37(iii) Reflection Formulas

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§4.37(v) Fundamental Property

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30: 25.4 Reflection Formulas

§25.4 Reflection Formulas

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25.4.1

\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (12), p. 259)
Referenced by:: §25.10(i), §5.17
Permalink:: http://dlmf.nist.gov/25.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

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25.4.2

\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (13), p. 259)
A&S Ref:: 23.2.6
Permalink:: http://dlmf.nist.gov/25.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

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25.4.3

\xi\left(s\right)=\xi\left(1-s\right),

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Symbols:: $\xi\left(\NVar{s}\right)$ : Riemann’s $\xi$ -function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, p. 260)
Permalink:: http://dlmf.nist.gov/25.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

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