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1: 25.12 Polylogarithms
The special case z = 1 is the Riemann zeta function: ζ ( s ) = Li s ( 1 ) .
Integral Representation
Further properties include …and … In terms of polylogarithms …
2: Bibliography G
  • W. Gautschi (1973) Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.
  • W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
  • W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
  • N. Gray (2002) Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.
  • W. Groenevelt (2007) Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.
  • 3: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • 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.
  • 4: Bibliography F
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.
  • F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.
  • 5: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • A. J. MacLeod (2002b) The efficient computation of some generalised exponential integrals. J. Comput. Appl. Math. 148 (2), pp. 363–374.
  • J. W. Meijer and N. H. G. Baken (1987) The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.
  • G. F. Miller (1960) Tables of Generalized Exponential Integrals. NPL Mathematical Tables, Vol. III, Her Majesty’s Stationery Office, London.
  • S. C. Milne (1985c) A new symmetry related to 𝑆𝑈 ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • 6: 20.11 Generalizations and Analogs
    With the substitutions a = q e 2 i z , b = q e 2 i z , with q = e i π τ , we have … As in §20.11(ii), the modulus k of elliptic integrals19.2(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 τ = τ ( k ) of the theta functions via (20.9.2). … For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …
    7: 36.2 Catastrophes and Canonical Integrals
    Canonical Integrals
    Ψ 1 is related to the Airy function (§9.2): … … Addendum: For further special cases see §36.2(iv)
    §36.2(iv) Addendum to 36.2(ii) Special Cases
    8: Bibliography B
  • G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..
  • G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • T. H. Boyer (1969) Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.
  • P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.
  • 9: Software Index
  • Research Software.

    This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • Software Associated with Books.

    An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.

  • 10: Bibliography
  • F. S. Acton (1974) Recurrence relations for the Fresnel integral 0 exp ( c t ) d t t ( 1 + t 2 ) and similar integrals. Comm. ACM 17 (8), pp. 480–481.
  • D. E. Amos (1980b) Computation of exponential integrals. ACM Trans. Math. Software 6 (3), pp. 365–377.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.