modular functions

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41—44 of 44 matching pages

41: Errata

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The overloaded operator $\equiv$ is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).

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Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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Originally named as a complementary error function, $w\left(z\right)$ has been renamed as the Faddeeva (or Faddeyeva) function.

… ►The Handbook of Mathematical Functions was published, and the Digital Library of Mathematical Functions was released.

42: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

Encodings:

BibTeX

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Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

External Links:

FullText

Referenced by:

§35.7(ii), §35.8(iii).

Encodings:

BibTeX

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Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.

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Referenced by:

§25.21(ii).

Encodings:

BibTeX

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

Encodings:

BibTeX

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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43: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►

§26.10(ii) Generating Functions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►

§26.10(vi) Bessel-Function Expansion

… ►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and …

44: 24.15 Related Sequences of Numbers

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24.15.3 $$\tan t=\sum _{n=0}^{\mathrm{\infty }}{T}_n\frac{t^n}{n!},$$

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Symbols:

$!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer, $t$: real or complex and $T_n$: tangent numbers

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(ii), §24.15 and Ch.24

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24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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