Elliptic Functions. A Primer

Content: 
Front Matter, Page iii
Copyright, Page iv
EDITOR'S PREFACE, Pages ix-xii
LIST OF TABLES, Page xiii
1 - Double periodicity, Pages 1-2
2 - Lattices, Pages 2-4
3 - Multiples and sub-multiples of periods, Pages 4-6
4 - Fundamental parallelogram, Pages 6-8
5 - Definition of an elliptic function, Pages 8-10
6 - An elliptic function (unless constant) has poles and zeros, Pages 10-11
7 - Residue sum of an elliptic function is zero, Pages 12-13
8 - Derivative of an elliptic function, Pages 13-15
9 - Additive pseudoperiodicity, Pages 15-17
10 - Pole-sum of an elliptic function, Pages 17-18
11 - The mid-lattice points, Pages 19-21
12 - Construction of the function ζkz, Pages 21-23
13 - Construction and periodicity of the Weierstrassian function ζz, Pages 23-26
14 - Zeros of ℘'z, Pages 26-28
15 - Periodicity of the primitive functions, Pages 28-34
16 - pseudoperiodicity of ζz, Pages 34-36
17 - Construction of σz, Pages 36-38
18 - ELLIPTIC FUNCTION: A PRIMER, Pages 38-41
19 - Construction OF AN ELLIPTIC FUNCTION, Pages 41-43
20 - Expression OF AN ELLIPTIC FUNCTION, Pages 44-46
21 - Expression for ℘'2z in terms of ℘z, Pages 46-48
22 - EXPRESSION OF AN ELLIPTIC FUNCTION, Pages 49-51
23 - ELLIPTIC FUNCTIONS ON THE SAME LATTICE, Pages 51-52
24 - The six critical constants pq, Pages 52-54
25 - QUARTER-PERIOD ADDITION, Pages 54-58
26 - THE FUNCTIONS ℘z AND pqz, Pages 58-60
27 - Copolar functions and simultaneous differential equations, Pages 60-62
28 - ADDITION THEOREMS FOR ℘z AND ζz, Pages 62-66
29 - Addition theorems for fjz, jfz and hgz, Pages 67-70
30 - SYMMETRICAL ALGEBRAIC RELATIONS, Pages 70-73
31 - INTEGRATION OF RATIONAL FUNCTIONS, Pages 73-78
32 - THE FUNCTIONS ℘z and pqz, Pages 78-81
33 - STATEMENTS OF THE INVERSION THEOREM, Pages 81-83
34 - THE WEIERSTRASSIAN HALF-PERIODS, Pages 83-86
35 - STANDARDISATION OF AN ELLIPTIC INTEGRAL, Pages 86-89
36 - DEFINITION OF THE JACOBIAN FUNCTIONS, Pages 89-91
37 - PERIODICITY OF pqu, Pages 91-93
38 - PARAMETERS AND MODULI, Pages 93-96
39 - LEADING COEFFICIENTS, Pages 97-98
40 - DERIVATIVES AND DIFFERENTIAL EQUATIONS, Pages 98-100
41 - THE JACOBIAN FUNCTIONS, Pages 100-102
42 - THE JACOBIAN QUARTER-PERIODS, Pages 103-108
43 - ADDITION THEOREMS, Pages 108-110
44 - IMAGINARY AND REAL TRANSFORMATIONS, Pages 110-113
45 - DUPLICATION, Pages 113-118
46 - THE LANDEN TRANSFORMATIONS, Pages 118-122
47 - REDUCTION OF A RATIONAL FUNCTION, Pages 122-124
48 - INTEGRATION OF THE JACOBIAN FUNCTION pqu, Pages 124-126
49 - THE INTEGRATING FUNCTION Pqu, Pages 126-133
50 - INTEGRATION OF A POLYNOMIAL, Pages 133-134
51 - THE FUNCTION IIs (u, a), Pages 134-137
52 - DIFFERENTIATION OF JACOBIAN FUNCTIONS, Pages 138-140
53 - DEGENERATION OF JACOBIAN SYSTEMS, Pages 140-145
54 - The c-DERIVATIVES, Pages 145-158
55 - DIFFERENTIATION OF WEIERSTRASSIAN FUNCTIONS, Pages 158-161
56 - WEIERSTRASSIAN AND ELEMENTARY FUNCTIONS, Pages 161-165
57 - JACOBIAN FUNCTIONS WITH AN axial basis, Pages 165-167
58 - EVALUATION OF THE REAL INTEGRAL, Pages 167-175
59 - REDUCTION OF THE INTEGRALS, Pages 175-186
60 - SIMULTANEOUS UNIFORMISATION, Pages 186-189
APPENDIX A, Page 190
APPENDIX B, Page 191
APPENDIX C, Page 192
EXERCISES, Pages 193-197
ANSWERS TO EXERCISES, Page 198