# harmonic number

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## 1—10 of 17 matching pages

##### 1: 25.16 Mathematical Applications

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25.16.13
$$\sum _{n=1}^{\mathrm{\infty}}{\left(\frac{h(n)}{n}\right)}^{2}=\frac{17}{4}\zeta \left(4\right),$$

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##### 2: 25.11 Hurwitz Zeta Function

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25.11.33
$$h(n)=\sum _{k=1}^{n}{k}^{-1}.$$

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##### 3: 1.2 Elementary Algebra

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*geometric mean*$G$ and*harmonic mean*$H$ of $n$ positive numbers ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$ are given by …##### 4: Bibliography T

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New congruences for the Bernoulli numbers.
Math. Comp. 48 (177), pp. 341–350.
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Asymptotic estimates of Stirling numbers.
Stud. Appl. Math. 89 (3), pp. 233–243.
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Harmonic Analysis on Symmetric Spaces and Applications. II.
Springer-Verlag, Berlin.
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Hyperspherical elliptic harmonics and their relation to the Heun equation.
Phys. Rev. A 63 (032510), pp. 1–8.
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Representation Theory and Harmonic Analysis.
Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
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##### 5: 1.9 Calculus of a Complex Variable

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###### Harmonic Functions

… ►###### Winding Number

… ►###### Mean Value Property

►For $u(z)$ harmonic, … ►###### Poisson Integral

…##### 6: Bibliography H

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The Theory of Spherical and Ellipsoidal Harmonics.
Cambridge University Press, London-New York.
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An explicit formula for Bernoulli numbers.
Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
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On congruences involving Bernoulli numbers and irregular primes. II.
Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains.
Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
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Bernoulli numbers and polynomials via residues.
J. Number Theory 76 (2), pp. 178–193.
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##### 7: Bibliography M

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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.
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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
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On the evaluation of Bessel functions.
Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.
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On the representation of numbers as a sum of $2r$ squares.
Quarterly Journal of Math. 48, pp. 93–104.
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Asymptotic development of the Stirling numbers of the first kind.
J. London Math. Soc. 33, pp. 133–146.
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##### 8: Errata

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►This release increments the minor version number and contains considerable additions of new material and clarifications.
These additions were facilitated by an extension of the scheme for reference numbers; with “_” introducing intermediate numbers.
These enable insertions of new numbered objects between existing ones without affecting their permanent identifiers and URLs.
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Section 14.30
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Table 26.8.1

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

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

$n$ | $k$ | ||||||||||
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$0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | |

10 | $0$ | $-\mathrm{3\hspace{0.17em}62880}$ | $\mathrm{10\hspace{0.17em}26576}$ | $-\mathrm{11\hspace{0.17em}72700}$ | $\mathrm{7\hspace{0.17em}23680}$ | $-\mathrm{2\hspace{0.17em}69325}$ | $63273$ | $-9450$ | $870$ | $-45$ | $1$ |

*Reported 2013-11-25 by Svante Janson.*

##### 9: Bibliography

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SPHEREPACK 2.0: A Model Development Facility.
NCAR Technical Note
Technical Report TN-436-STR, National Center for Atmospheric Research.
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Congruences of $p$-adic integer order Bernoulli numbers.
J. Number Theory 59 (2), pp. 374–388.
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A harmonic mean inequality for the gamma function.
J. Comput. Appl. Math. 87 (2), pp. 195–198.
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Sharp bounds for the Bernoulli numbers.
Arch. Math. (Basel) 74 (3), pp. 207–211.
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A Centennial History of the Prime Number Theorem.
In Number Theory,
Trends Math., pp. 1–14.
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##### 10: 1.10 Functions of a Complex Variable

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