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1: 20 Theta Functions

Chapter 20 Theta Functions

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2: 28 Mathieu Functions and Hill’s Equation

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3: 8.17 Incomplete Beta Functions

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8.17.6 $$I_x(a,a)=\frac{1}{2}{I}_{4x(1-x)}(a,\frac{1}{2}),$$ $0\le x\le \frac{1}{2}$.

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Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter and $x$: variable

Referenced by:

§8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(i), §8.17 and Ch.8

►For a historical profile of ${\mathrm{B}}_x(a,b)$ see Dutka (1981). … ►

8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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4: 23 Weierstrass Elliptic and Modular
Functions

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5: 36 Integrals with Coalescing Saddles

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6: 8 Incomplete Gamma and Related
Functions

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7: Gergő Nemes

Profile
Gergő Nemes

… ►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. …

8: Wolter Groenevelt

Profile
Wolter Groenevelt

… ►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. …

9: Bibliography D

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From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.

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Referenced by:

Profile Peter A. Clarkson.

Encodings:

BibTeX

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Special Functions. World Scientific Publishing Co., Inc., River Edge, NJ.

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External Links:

ISBN 981-02-4393-6, Document, Link, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

Encodings:

BibTeX

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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External Links:

ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt

Referenced by:

§8.17(i).

Encodings:

BibTeX

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10: Peter L. Walker

Profile
Peter L. Walker

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20 Theta Functions

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