About the Project
NIST

harmonic number

AdvancedHelp

(0.001 seconds)

1—10 of 17 matching pages

1: 25.16 Mathematical Applications
25.16.5 H ( s ) = n = 1 H n n s ,
where H n is given by (25.11.33). …
25.16.13 n = 1 ( H n n ) 2 = 17 4 ζ ( 4 ) ,
2: 25.11 Hurwitz Zeta Function
25.11.32 0 a x n ψ ( x ) d x = ( - 1 ) n - 1 ζ ( - n ) + ( - 1 ) n H n B n + 1 n + 1 - k = 0 n ( - 1 ) k ( n k ) H k B k + 1 ( a ) k + 1 a n - k + k = 0 n ( - 1 ) k ( n k ) ζ ( - k , a ) a n - k , n = 1 , 2 , , a > 0 ,
where H n are the harmonic numbers:
25.11.33 H n = k = 1 n k - 1 .
3: Errata
  • Notation

    Previously the notation h ( n ) was used for the harmonic number H n (defined in (25.11.33)). The more widely used notation H n will now be used throughout the DLMF. In particular, this change was made in (25.11.32), (25.11.33), (25.16.5) and (25.16.13) (Suggested by Gergő Nemes on 2021-08-23).

  • 4: 1.2 Elementary Algebra
    The geometric mean G and harmonic mean H of n positive numbers a 1 , a 2 , , a n are given by …
    5: Bibliography T
  • J. W. Tanner and S. S. Wagstaff (1987) New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.
  • N. M. Temme (1993) Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.
  • A. Terras (1988) Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.
  • O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
  • T. Ton-That, K. I. Gross, D. St. P. Richards, and P. J. Sally (Eds.) (1995) Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
  • 6: 1.9 Calculus of a Complex Variable
    Harmonic Functions
    Winding Number
    Mean Value Property
    For u ( z ) harmonic, …
    Poisson Integral
    7: Bibliography H
  • E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
  • K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
  • K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
  • L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
  • I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.
  • 8: Bibliography M
  • T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
  • Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
  • G. Matviyenko (1993) On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.
  • L. J. Mordell (1917) On the representation of numbers as a sum of 2 r squares. Quarterly Journal of Math. 48, pp. 93–104.
  • L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.
  • 9: Bibliography
  • J. C. Adams and P. N. Swarztrauber (1997) SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.
  • A. Adelberg (1996) Congruences of p -adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.
  • H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.
  • H. Alzer (2000) Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.
  • T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.
  • 10: 1.10 Functions of a Complex Variable
    Harmonic Functions
    If u ( z ) is harmonic in D , z 0 D , and u ( z ) u ( z 0 ) for all z D , then u ( z ) is constant in D . Moreover, if D is bounded and u ( z ) is continuous on D ¯ and harmonic in D , then u ( z ) is maximum at some point on D . … Let α and β be real or complex numbers that are not integers. … (The integer k may be greater than one to allow for a finite number of zero factors.) …