Chapter 23 Weierstrass Elliptic and Modular Functions

W. P. Reinhardt University of Washington, Seattle, Washington. P. L. Walker American University of Sharjah, Sharjah, United Arab Emirates.

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Acknowledgements:

This chapter is based in part on Abramowitz and Stegun (1964, Chapter 18) by T. H. Southard.

Notes:

The main references used in writing this chapter are Lawden (1989, Chapters 6, 7, 9), McKean and Moll (1999, Chapters 1–5), Walker (1996, Chapter 7 and §§3.4, 8.4), and Whittaker and Watson (1927, Chapter 20 and §21.7). For additional bibliographic reading see Apostol (1990, Chapters 1–6), Armitage and Eberlein (2006), Copson (1935, Chapters 13 and 15), Erdélyi et al. (1953b, §§13.12–13.15 and 13.24), and Koblitz (1993, Chapters 1–4).The main references used in writing this chapter are Lawden (1989, Chapters 6, 7, 9), McKean and Moll (1999, Chapters 1–5), Walker (1996, Chapter 7 and §§3.4, 8.4), and Whittaker and Watson (1927, Chapter 20 and §21.7). For additional bibliographic reading see Apostol (1990, Chapters 1–6), Armitage and Eberlein (2006), Copson (1935, Chapters 13 and 15), Erdélyi et al. (1953b, §§13.12–13.15 and 13.24), and Koblitz (1993, Chapters 1–4).

Referenced by:

§32.2(i), New Associate Editors for the DLMF, Profile Peter L. Walker, Profile Victor Moll, Profile William P. Reinhardt, Profile William P. Reinhardt, § ‣ Chapter Authors, § ‣ Chapter Authors, § ‣ Associate Editors, § ‣ Associate Editors, § ‣ Associate Editors, Erratum (V1.0.1) for References, § ‣ Software Cross Index

Permalink:

http://dlmf.nist.gov/23

Addition (effective with 1.0.1):

Armitage and Eberlein (2006) was added as a general reference for this chapter.

   

• Notation
  1. 23.1 Special Notation
• Weierstrass Elliptic Functions
  1. 23.2 Definitions and Periodic Properties
  2. 23.3 Differential Equations
  3. 23.4 Graphics
  4. 23.5 Special Lattices
  5. 23.6 Relations to Other Functions
  6. 23.7 Quarter Periods
  7. 23.8 Trigonometric Series and Products
  8. 23.9 Laurent and Other Power Series
  9. 23.10 Addition Theorems and Other Identities
  10. 23.11 Integral Representations
  11. 23.12 Asymptotic Approximations
  12. 23.13 Zeros
  13. 23.14 Integrals

• Modular Functions
  1. 23.15 Definitions
  2. 23.16 Graphics
  3. 23.17 Elementary Properties
  4. 23.18 Modular Transformations
  5. 23.19 Interrelations
• Applications
  1. 23.20 Mathematical Applications
  2. 23.21 Physical Applications
• Computation
  1. 23.22 Methods of Computation
  2. 23.23 Tables
  3. 23.24 Software

22.22 Software23.1 Special Notation

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