Elliptic Functions

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
Original partial preface......Page 12
Acknowledgements......Page 14
1.1 The pit and the pendulum......Page 17
1.3 The energy integral......Page 20
1.4 The Euler and Jacobi normal equations......Page 24
1.5 The classical formal solutions of (1.14)......Page 26
1.6 Rigorous solution of Equation (1.2)......Page 28
1.7 The Jacobian elliptic functions......Page 31
1.8 The imaginary period......Page 34
1.9 Miscellaneous exercises......Page 36
2.1 Introduction: the extension problem......Page 41
2.2 The half-angle formulae......Page 42
2.3 Extension to the imaginary axis......Page 45
2.4 The addition formulae......Page 46
2.5 Extension to the complex plane......Page 51
2.6 Periodic properties associated with K, K +iK and iK......Page 55
2.7 The behaviour of the Jacobian elliptic functions near the origin and near iK......Page 59
2.8 Glaisher’s notation, the lemniscatic integral and the lemniscate functions......Page 62
2.9 Fourier series and sums of two squares......Page 67
2.10 Summary of basic properties of the Jacobian elliptic functions......Page 72
2.10 Miscellaneous exercises......Page 75
3.2 Period modules and lattices......Page 78
3.3 The unimodular group SL(2, Z)......Page 80
3.4 The canonical basis......Page 83
3.5 Four basic theorems......Page 86
3.6 Elliptic functions of order 2......Page 88
4.1 Genesis of the theta functions......Page 91
4.2 The functions theta and theta......Page 94
4.3 The functions theta and theta......Page 95
4.4 Summary......Page 99
4.5 The pseudo-addition formulae......Page 102
4.6 The constant G and the Jacobi identity......Page 105
4.7 Jacobi’s imaginary transformation......Page 111
4.8 The Landen transformation......Page 115
4.9 Summary of properties of theta functions......Page 119
5.1 Extension to arbitrary tau with Im (tau) > 0......Page 123
5.2 The inversion problem......Page 124
5.3 Solution of the inversion problem......Page 127
5.4 Numerical computation of elliptic functions......Page 135
5.5 The Weierstrass…......Page 141
5.6 The eta-function of Jacobi......Page 145
6.1 The elliptic modular function......Page 152
6.2 Return to the Weierstrass…......Page 154
6.3 Generators of SL(2, Z)......Page 159
6.4 The transformation problem......Page 161
6.5 Transformation of theta (z|tau), continued......Page 163
6.6 Dependence on tau......Page 165
6.7 The Dedekind…......Page 167
7.1 Construction of the Weierstrass functions; the Weierstrass sigma and zeta functions; the Weierstrass…......Page 172
7.1.1 The Weierstrass sigma functions......Page 174
7.1.2 The Weierstrass zeta functions......Page 175
7.2 The Laurent expansions: the differential equation satisfied by…......Page 181
7.3 Modular forms and functions......Page 186
7.4.1 Construction of elliptic functions with given zeros and poles......Page 198
7.4.2 The addition theorem for…......Page 199
7.5.1 The field of elliptic functions......Page 204
7.5.2 Elliptic functions with given points for poles......Page 205
7.6 Connection with the Jacobi functions......Page 208
7.7 The modular functions j(tau), lambda (tau) and the inversion problem......Page 219
8.1 Elliptic integrals in general......Page 226
8.2 Reduction in terms of Legendre’s three normal forms......Page 227
8.3 Reduction to the standard form......Page 235
8.3.1 Another way of obtaining the reduction of the general integral......Page 240
8.4 Reduction to the Weierstrass normal forms......Page 245
9.1 Introduction......Page 248
9.2 Fagnano’s Theorem......Page 249
9.3 Area of the surface of an ellipsoid......Page 257
9.4 Some properties of space curves......Page 261
9.5 Poncelet’s poristic polygons......Page 263
9.5.1 Poncelet’s poristic polygons......Page 268
9.6.1 Basic idea of spherical trigonometry......Page 271
9.6.2 Elliptic functions and spherical trigonometry......Page 272
9.7.1 Introduction......Page 276
9.7.2 Formulation of Theorem 9.3......Page 277
9.7.3 Proof of Theorem 9.3......Page 279
9.8.1 Introduction......Page 284
9.8.2 The addition of points on an elliptic curve and the group law......Page 286
9.9 Concluding remarks and suggestions for further reading......Page 288
Introduction......Page 292
10.1 Revision of the quadratic, cubic and quartic equations......Page 293
10.2 Reduction of the general quintic equation to normal form......Page 296
10.3 Elliptic functions and solution of the quintic: outline of the proof......Page 300
10.4 Transformation theory and preparation for application to the modular equation for the quintic......Page 302
10.4 The transformation…......Page 305
10.6 The transformations…......Page 307
10.7 The transformation…......Page 311
10.8 The modular equation and the solution of the quintic......Page 315
10.9 The fundamental domain of the group generated by…......Page 323
10.10 The modular equation for the quintic via elliptic integrals......Page 327
10.11 Solution of the quintic equation......Page 330
11.1 Sums of three squares and triangular numbers......Page 334
11.2 Outline of the proof......Page 336
11.3 Proof of (11.5)......Page 337
11.4 Proof of (11.6)......Page 339
11.5 Completion of the proof of Theorem 11.1......Page 346
Introduction......Page 354
12.1 Euler’s dynamical equations......Page 355
12.2 Planetary orbits in general relativity......Page 358
12.3 The spherical pendulum......Page 361
12.4 Green’s function for a rectangle......Page 366
12.5 A statistical application: correlation and elliptic functions......Page 369
12.6 Numerical analysis and the arithmetic-geometric mean of Gauss......Page 373
12.7 Rational maps with empty Fatou set......Page 380
12.8 A final, arithmetic, application: heat-flow on a circle and the Riemann zeta function......Page 382
A.2 The formula for arg (z)......Page 386
A.4 Euler’s infinite product for the sine......Page 388
A.5 Euler’s infinite product for the cosine......Page 390
References......Page 394
Further reading......Page 399
Index......Page 401