Elements of the theory of elliptic functions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
1990
|
| Series: | Translations of mathematical monographs
79. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | VII, 237 S. |
| ISBN: | 0821845322 |
Staff View
MARC
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| 240 | 1 | 0 | |a Elementy teorii elliptičeskich funkciǐ |
| 245 | 1 | 0 | |a Elements of the theory of elliptic functions |c N. I. Akhiezer |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 1990 | |
| 300 | |a VII, 237 S. | ||
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| 490 | 1 | |a Translations of mathematical monographs |v 79. | |
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Record in the Search Index
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|---|---|
| adam_text | Table of Contents
Foreword to the second Russian edition vii
CHAPTER 1. General Theorems on Elliptic Functions 1
§1. On periods of single valued analytic functions 1
§2. Proof of the Jacobi theorem 2
§3. Theta functions 4
§4. Liouville s theorems 6
§5. The Weierstrass function p(u) 9
§6. The differential equation of the function p(u) 12
CHAPTER 2. Modular Functions 15
§7. Invariants 15
§8. Modular forms 18
§9. Fundamental regions of the group Z 22
§10. The modular function J(r) 27
§11. Inversion of elliptic integrals of the first kind 32
CHAPTER 3. Weierstrass Functions 35
§12. The Weierstrass function £(w) 35
§13. The Weierstrass function a(u) 37
§14. Expression of an arbitrary elliptic function by means of a(u)
and by means of £(w) 38
§15. The addition theorems for Weierstrass functions 40
§16. Representation of every elliptic function in terms of the
functions p(u) and p (u) 43
§17. Elliptic integrals 45
CHAPTER 4. Theta Functions 49
§ 18. Representations of theta functions by infinite products 49
§19. The connection between sigma functions and theta functions 52
§20. Expansion of the functions £(«) and p(w) in simple series 54
iii
iv TABLE OF CONTENTS
§21. Expressions for e ,ei, and e s in terms of the zero values of
the theta functions 55
§22. Transformation of theta functions 56
§23. The modular function A(t) 58
CHAPTER 5. Jacobi Functions 65
§24. The elliptic integral of the first kind in the forms of Jacobi
and Riemann 65
§25. The Jacobi functions 67
§26. Differentiation of the Jacobi functions 70
§27. The Jacobi function Z(w) 71
§28. The Euler theorem 72
§29. Normal elliptic integrals of the second and third kinds in
Jacobi form 75
§30. Complete elliptic integrals of the first kind 77
§31. Complete elliptic integrals of the second kind 80
§32. Degeneracy of elliptic functions 82
§33. The simple pendulum 84
CHAPTER 6. Transformation of Elliptic Functions 87
§34. The problem of transformation of elliptic functions 87
§35. Reduction of the general problem 89
§36. The first principal first degree transformation 93
§37. The second principal first degree transformation 94
§38. Landen s transformation 95
§39. Gauss s transformation 96
§40. Principal transformations of «th degree 98
CHAPTER 7. Additional Facts About Elliptic Integrals 101
§41. Elliptic curves of general form 101
§42. The function p(u) with real invariants 104
§43. Reduction of elliptic integrals to Jacobi normal form in the
real case 106
§44. Complete elliptic integrals as hypergeometric functions 109
§45. Computation of h from a given modulus k 114
§46. The arithmetico geometric mean 115
CHAPTER 8. Some Conformal Mappings 119
§47. A conformal mapping of a rectangle onto a half plane 119
§48. A conformal mapping of a doubly connected polygonal
domain onto a circular annulus 127
§49. Examples of conformal mappings 133
TABLE OF CONTENTS v
CHAPTER 9. Extremal Properties of Fractions to Which a
Transformation of Elliptic Functions Reduces 141
§50. Statement of the problems 141
§51. Solution of Problem C 147
CHAPTER 10. Generalization of Tchebycheff Polynomials 151
§52. Polynomials deviating least from zero 151
§53. Orthogonal polynomials on two intervals 156
CHAPTER 11. Various Supplements and Applications 163
§54. Abel s theorem 163
§55. The Green s function for a circular annulus 169
§56. The Dirichlet problem for a circular annulus 171
§57. Elliptic coordinates 175
§58. The Laplace equation in elliptic coordinates 181
§59. The Lame equation 182
§60. The Picard theorems on meromorphic functions 187
§61. The Landau theorem 188
§62. On meromorphic functions having an algebraic addition
theorem 190
§63. On Fourier series of analytic functions 191
Tables of the Most Important Formulas 197
Tables of Values of Elliptic Integrals 217
Bibliography 233
Subject Index 235
|
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| id | DE-604.BV004276339 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:10:48Z |
| institution | BVB |
| isbn | 0821845322 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002659127 |
| oclc_num | 246828753 |
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| owner_facet | DE-384 DE-12 DE-703 DE-739 DE-355 DE-BY-UBR DE-824 DE-91G DE-BY-TUM DE-83 DE-29T DE-188 |
| physical | VII, 237 S. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Achiezer, Naum I. 1901-1980 Verfasser (DE-588)122043014 aut Elementy teorii elliptičeskich funkciǐ Elements of the theory of elliptic functions N. I. Akhiezer Providence, RI American Mathematical Society 1990 VII, 237 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 79. Aus d. Russ. übers. Elliptische Funktion Elliptische Funktion (DE-588)4134665-8 gnd rswk-swf Elliptische Funktion (DE-588)4134665-8 s DE-604 Translations of mathematical monographs 79. (DE-604)BV000002394 79. HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002659127&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Achiezer, Naum I. 1901-1980 Elements of the theory of elliptic functions Translations of mathematical monographs Elliptische Funktion Elliptische Funktion (DE-588)4134665-8 gnd |
| subject_GND | (DE-588)4134665-8 |
| title | Elements of the theory of elliptic functions |
| title_alt | Elementy teorii elliptičeskich funkciǐ |
| title_auth | Elements of the theory of elliptic functions |
| title_exact_search | Elements of the theory of elliptic functions |
| title_full | Elements of the theory of elliptic functions N. I. Akhiezer |
| title_fullStr | Elements of the theory of elliptic functions N. I. Akhiezer |
| title_full_unstemmed | Elements of the theory of elliptic functions N. I. Akhiezer |
| title_short | Elements of the theory of elliptic functions |
| title_sort | elements of the theory of elliptic functions |
| topic | Elliptische Funktion Elliptische Funktion (DE-588)4134665-8 gnd |
| topic_facet | Elliptische Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002659127&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
| work_keys_str_mv | AT achiezernaumi elementyteoriiellipticeskichfunkcii AT achiezernaumi elementsofthetheoryofellipticfunctions |