modular%20functions

SearchAdvancedHelp

(0.003 seconds)

1—10 of 14 matching pages

1: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

►

23 Weierstrass Elliptic and Modular Functions

…

2: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

►

23 Weierstrass Elliptic and Modular Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

3: 23 Weierstrass Elliptic and Modular
Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

…

4: Bibliography B

… ►

Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

…

5: Software Index

… ► ►►►

	Open Source													With Book						Commercial
…
20 Theta Functions
…

►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

6: 20.7 Identities

… ►

§20.7(v) Watson’s Identities

… ►

§20.7(vi) Landen Transformations

… ►

§20.7(vii) Derivatives of Ratios of Theta Functions

… ►See Lawden (1989, pp. 19–20). … ►These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183). …

7: Bibliography

… ►

Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

… ►

On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

… ►

Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

ⓘ

External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

… ►

Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-97127-0, MathReview, ZentralBlatt

Referenced by:

§17.2(i), §23.10(iv), §23.15(i), §23.17(ii), §23.18, §23.18, §23.19, §23.20(i), §23.20(v), Ch.23, §27.14(iii), §27.14(iv), §27.14(vi), §27.7, Ch.27.

Encodings:

BibTeX

…

8: 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 …

9: 27.2 Functions

§27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►and if $\varphi \left(n\right)$ is the smallest positive integer $f$ such that $a^f\equiv 1(modn)$, then $a$ is a primitive root mod $n$. …This is Jordan’s function. … ►This is Liouville’s function. …

10: Bibliography C

… ►

Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

… ►

Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

… ►

Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

ⓘ

External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

… ►

Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

Encodings:

BibTeX

… ►

Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 0-521-59820-6, MathReview (Joseph H. Silverman), ZentralBlatt

Referenced by:

§23.20(ii).

Encodings:

BibTeX

…