Elliptic Functions : A Primer

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. 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