Elliptic Functions
eBook - PDF

Elliptic Functions

A Primer

  1. 212 pages
  2. English
  3. PDF
  4. Only available on web
eBook - PDF

Elliptic Functions

A Primer

Book details
Table of contents
Citations

About This Book

Elliptic Functions: A Primer defines and describes what is an elliptic function, attempts to have a more elementary approach to them, and drastically reduce the complications of its classic formulae; from which the book proceeds to a more detailed study of the subject while being reasonably complete in itself. The book squarely faces the situation and acknowledges the history of the subject through the use of twelve allied functions instead of the three Jacobian functions and includes its applications for double periodicity, lattices, multiples and sub-multiple periods, as well as many others in trigonometry. Aimed especially towards but not limited to young mathematicians and undergraduates alike, the text intends to have its readers acquainted on elliptic functions, pass on to a study in Jacobian elliptic functions, and bring a theory of the complex plane back to popularity.

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Information

Publisher
Pergamon
Year
2014
ISBN
9781483151915
Topic
Mathematics
Subtopic
Calculus
Index
Mathematics

Table of contents

  1. Front Cover
  2. Elliptic Functions: A Primer
  3. Copyright Page
  4. Table of Contents
  5. EDITOR'S PREFACE
  6. LIST OF TABLES
  7. Chapter 1. Double periodicity
  8. Chapter 2. Lattices
  9. Chapter 3. Multiples and sub-multiples of periods
  10. Chapter 4. Fundamental parallelogram
  11. Chapter 5. Definition of an elliptic function
  12. Chapter 6. An elliptic function (unless constant) has poles and zeros Identification of an elliptic function
  13. Chapter 7. Residue sum of an elliptic function is zero
  14. Chapter 8. Derivative of an elliptic function
  15. Chapter 9. Additive pseudoperiodicity
  16. Chapter 10. Pole-sum of an elliptic function
  17. Chapter 11. The mid-lattice points
  18. Chapter 12. Construction of the function...
  19. Chapter 13. Construction and periodicity of the Weierstrassian function...
  20. Chapter 14. Zeros of..z
  21. Chapter 15. Periodicity of the primitive functions
  22. Chapter 16. Construction and pseudoperiodicity of...
  23. Chapter 17. Construction of óz...
  24. Chapter 18. Construction, in terms of... and... of an elliptic function with assigned poles and principal parts
  25. Chapter 19. Construction, in terms of óæ, of an elliptic function with assigned poles and zeros
  26. Chapter 20. Expression of an elliptic function in the form ...
  27. Chapter 21. Expression for.'2z in terms of.z
  28. Chapter 22. Expression of an elliptic function in the form S ...
  29. Chapter 23. Elliptic functions on the same lattice are connected algebraically
  30. Chapter 24. The six critical constants pq
  31. Chapter 25. Quarter-period addition to the argument of a primitive function
  32. Chapter 26. The functions pz and pqz as solutions of differential equations
  33. Chapter 27. Copolar functions and simultaneous differential equations
  34. Chapter 28. Addition theorems for pz and .z and .z
  35. Chapter 29. Addition theorems for fjz, jfz and hgz
  36. Chapter 30. Symmetrical algebraic relations between fjx, fjy, fjz, x + z = 0
  37. Chapter 31. Integration of rational functions of .z and .'z
  38. Chapter 32. The functions .z and pqz as inverted integrals
  39. Chapter 33. Statements of the inversion theorem
  40. Chapter 34. The Weierstrassian half-periods as definite integrals
  41. Chapter 35. Standardisation of an elliptic integral
  42. Chapter 36. Definition of the Jacobian functions
  43. Chapter 37. Periodicity of pqu
  44. Chapter 38. Parameters and moduli
  45. Chapter 39. Leading coefficients
  46. Chapter 40. Derivatives and differential equations
  47. Chapter 41. The Jacobian functions as inverted integrals
  48. Chapter 42. The Jacobian quarter-periods as definite integrals
  49. Chapter 43. Addition theorems for the Jacobian functions
  50. Chapter 44. Jacobi's imaginary and real transformations
  51. Chapter 45. Duplication
  52. Chapter 46. The Landen transformations
  53. Chapter 47. The reduction of a rational function of Jacobian functions
  54. Chapter 48. Integration of the Jacobian function pqu
  55. Chapter 49. The integrating function Pqu
  56. Chapter 50. Integration of a polynomial in the squares of Jacobian functions
  57. Chapter 51. The function IIs (u, a)
  58. Chapter 52. Differentiation of Jacobian functions
  59. Chapter 53. Degeneration of Jacobian systems (c = 0) to circular functions
  60. Chapter 54. The c-derivatives of Kc, Kn, DsKc, DsKn
  61. Chapter 55. Differentiation of Weierstrassian functions with respect to h2, h3
  62. Chapter 56. Weierstrassian and elementary functions with an axial basis
  63. Chapter 57. Jacobian functions with an axial basis
  64. Chapter 58. Evaluation of the real integral...
  65. Chapter 59. Reduction of the integrals...
  66. Chapter 60. Simultaneous uniformisation of two quadratic functions y, z...
  67. Appendix A
  68. Appendix B
  69. Appendix C
  70. Exercises
  71. Answers to Exercises