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1: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
Other Inverse Functions
Care needs to be taken on the cuts, for example, if 0 < x < then 1 / ( x + i 0 ) = ( 1 / x ) i 0 . … The inverse Gudermannian function is given by …
2: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
§4.37(i) General Definitions
Other Inverse Functions
§4.37(vi) Interrelations
Table 4.30.1 can also be used to find interrelations between inverse hyperbolic functions. …
3: 22.15 Inverse Functions
§22.15 Inverse Functions
The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions4.23). …Each of these inverse functions is multivalued. … Equations (22.15.1) and (22.15.4), for arcsn ( x , k ) , are equivalent to (22.15.12) and also to
§22.15(ii) Representations as Elliptic Integrals
4: 8 Incomplete Gamma and Related
Functions
Chapter 8 Incomplete Gamma and Related Functions
5: William P. Reinhardt
Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
    6: 8.17 Incomplete Beta Functions
    §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. …
    §8.17(ii) Hypergeometric Representations
    §8.17(iv) Recurrence Relations
    §8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties
    7: Gergő Nemes
    Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions. As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. In March 2022, Nemes was named Contributing Developer of the NIST Digital Library of Mathematical Functions.
    8: Wolter Groenevelt
    Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. In July 2023, Groenevelt was named Contributing Developer of the NIST Digital Library of Mathematical Functions.
    9: 33.24 Tables
    §33.24 Tables
  • Abramowitz and Stegun (1964, Chapter 14) tabulates F 0 ( η , ρ ) , G 0 ( η , ρ ) , F 0 ( η , ρ ) , and G 0 ( η , ρ ) for η = 0.5 ( .5 ) 20 and ρ = 1 ( 1 ) 20 , 5S; C 0 ( η ) for η = 0 ( .05 ) 3 , 6S.

  • Curtis (1964a) tabulates P ( ϵ , r ) , Q ( ϵ , r ) 33.1), and related functions for = 0 , 1 , 2 and ϵ = 2 ( .2 ) 2 , with x = 0 ( .1 ) 4 for ϵ < 0 and x = 0 ( .1 ) 10 for ϵ 0 ; 6D.

  • 10: 25.11 Hurwitz Zeta Function
    §25.11 Hurwitz Zeta Function
    The Riemann zeta function is a special case: … For other series expansions similar to (25.11.10) see Coffey (2008). … When a = 1 , (25.11.35) reduces to (25.2.3). … uniformly with respect to bounded nonnegative values of α . …