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1: 25.16 Mathematical Applications
25.16.13 n = 1 ( h ( n ) n ) 2 = 17 4 ζ ( 4 ) ,
2: 25.11 Hurwitz Zeta Function
25.11.33 h ( n ) = k = 1 n k - 1 .
3: 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 …
4: 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.
  • 5: 1.9 Calculus of a Complex Variable
    Harmonic Functions
    Winding Number
    Mean Value Property
    For u ( z ) harmonic, …
    Poisson Integral
    6: 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.
  • 7: 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.
  • 8: Errata
  • Changes


    • In Equation (19.20.11),

      19.20.11
      R J ( 0 , y , z , p ) = 3 2 p z ln ( 16 z y ) - 3 p R C ( z , p ) + O ( y ln y ) ,

      as y 0 + , p ( 0 ) real, we have added the constant term - 3 p R C ( z , p ) and the order term O ( y ln y ) , and hence was replaced by = .

    • In Paragraph Prime Number Theorem in §27.12, the largest known prime, which is a Mersenne prime, was updated from 2 43 , 112 , 609 - 1 (2009) to 2 82 , 589 , 933 - 1 (2018).

    • Originally Equation (35.7.8) had the constraint ( c ) , ( c - a - b ) > 1 2 ( m - 1 ) . This constraint was replaced with 0 < T < I ; 1 2 ( j + 1 ) - a for some j = 1 , , m ; 1 2 ( j + 1 ) - c and c - a - b - 1 2 ( m - j ) for all j = 1 , , m .

    • Several biographies had their publications updated.

  • Section 14.30


    In regard to the definition of the spherical harmonics Y l , m , the domain of the integer m originally written as 0 m l has been replaced with the more general | m | l . Because of this change, in the sentence just below (14.30.2), “tesseral for m < l and sectorial for m = l ” has been replaced with “tesseral for | m | < l and sectorial for | m | = l ”. Furthermore, in (14.30.4), m has been replaced with | m | .

    Reported by Ching-Li Chai on 2019-10-05

  • Other Changes


    • In (5.11.14), the previous constraint ( b - a ) > 0 was removed, see Fields (1966, (3)).

    • In Paragraph Confluent Hypergeometric Functions in §7.18(iv), a note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind U on the right-hand side of (7.18.10) was inserted.

    • In regard to (25.14.1), the previous constraint a 0 , - 1 , - 2 , , was removed. A clarification regarding the correct constraints for Lerch’s transcendent Φ ( z , s , a ) has been added in the text immediately below. In particular, it is now stated that if s is not an integer then | ph a | < π ; if s is a positive integer then a 0 , - 1 , - 2 , ; if s is a non-positive integer then a can be any complex number.

  • Table 26.8.1


    Originally the Stirling number s ( 10 , 6 ) was given incorrectly as 6327. The correct number is 63273.

    n k
    0 1 2 3 4 5 6 7 8 9 10
    10 0 - 3 62880 10 26576 - 11 72700 7 23680 - 2 69325 63273 - 9450 870 - 45 1

    Reported 2013-11-25 by Svante Janson.

  • Equation (26.7.6)

    26.7.6
    B ( n + 1 ) = k = 0 n ( n k ) B ( k )

    Originally this equation appeared with B ( n ) in the summation, instead of B ( k ) .

    Reported 2010-11-07 by Layne Watson.

  • 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.) …