harmonic mean

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1: 1.7 Inequalities

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§1.7(iii) Means

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1.7.7 $$H\le G\le A,$$

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Symbols:

$A$: arithmetic mean, $G$: geometric mean and $H$: harmonic mean

A&S Ref:

3.2.1

Permalink:

http://dlmf.nist.gov/1.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

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2: 1.2 Elementary Algebra

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§1.2(iv) Means

… ►The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_1,a_2,\mathrm{\dots },a_n$ are given by … ►

1.2.19 $$\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_1}+\frac{1}{a_2}+\mathrm{\cdots }+\frac{1}{a_n}\right).$$

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Defines:

$H$: harmonic mean (locally)

Symbols:

$n$: nonnegative integer

A&S Ref:

3.1.13

Permalink:

http://dlmf.nist.gov/1.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

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$M(-1)=H,$

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3: Bibliography

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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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Notes:

For corrigendum see same journal v. 90 (1998), no. 2, p. 265

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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4: Bibliography G

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A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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5: 1.9 Calculus of a Complex Variable

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Mean Value Property

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6: 1.15 Summability Methods

… ► $A(r,\theta )$ is a harmonic function in polar coordinates (1.9.27), and …

7: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … …

8: 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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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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External Links:

MathReview Entry

Referenced by:

§18.36(vi).

Encodings:

BibTeX

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:

ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

Encodings:

BibTeX

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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