Continued Fractions, Convergence Theory. Vol. 1, 2nd Editions. Loretzen, Waadeland. Atlantis Press. 2008

Recommendations

Info

Contents xi 4.2.3 Finite limits, parabolic case . . . . . . . . . . . . . . . . . . . . . . . . 192 4.2.4 Finite limits, elliptic case . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.2.5 Infinite limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.3 <strong>Continued</strong> fractions with multiple limits . . . . . . . . . . . . . . . . . . . . 203 4.3.1 Periodic continued fractions with multiple limits . . . . . . . . . 203 4.3.2 Limit periodic continued fractions with multiple limits . . . . . 204 4.4 Fixed circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.4.2 Fixed circles for M . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.4.3 Fixed circles and periodic continued fractions . . . . . . . . . . . . 207 4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 5 Numerical computation of continued fractions 217 5.1 Choice of approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.1.1 Fast convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.1.2 The fixed point method . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.1.3 Auxiliary continued fractions . . . . . . . . . . . . . . . . . . . . . . . 223 5.1.4 The improvement machine for the loxodromic case . . . . . . . . 227 5.1.5 Asymptotic expansion of tail values. . . . . . . . . . . . . . . . . . . 232 5.1.6 The square root modification . . . . . . . . . . . . . . . . . . . . . . . 235 5.2 Truncation error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5.2.1 The ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5.2.2 Truncation error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . 240 5.2.3 The Oval Sequence Theorem. . . . . . . . . . . . . . . . . . . . . . . . 243 5.2.4 An algorithm to find value sets for a given continued fraction of form K(a n /1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 5.2.5 Value sets and the fixed point method . . . . . . . . . . . . . . . . . 248 5.2.6 Value sets B(w n n ) for limit 1-periodic continued fractions of loxodromic or parabolic type . . . . . . . . . . . . . . . 255 5.2.7 Error bounds based on idea 3 . . . . . . . . . . . . . . . . . . . . . . . 258 5.3 Stable computation of approximants . . . . . . . . . . . . . . . . . . . . . . . 260 5.3.1 Stability of the backward recurrence algorithm. . . . . . . . . . . 260 5.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 A Some continued fraction expansions 265 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 A.1.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 A.1.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 A.2 Elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 A.2.1 Mathematical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 A.2.2 The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . 268 A.2.3 The general binomial function . . . . . . . . . . . . . . . . . . . . . . 269 A.2.4 The natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 A.2.5 Trigonometric and hyperbolic functions. . . . . . . . . . . . . . . . 272
xii Contents A.2.6 Inverse trigonometric and hyperbolic functions . . . . . . . . . . 273 A.2.7 <strong>Continued</strong> fractions with simple values . . . . . . . . . . . . . . . . 274 A.3 Hypergeometric functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 A.3.1 General expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 A.3.2 Special examples with 0 F 1 . . . . . . . . . . . . . . . . . . . . . . . . . 277 A.3.3 Special examples with 2 F 0 . . . . . . . . . . . . . . . . . . . . . . . . . 277 A.3.4 Special examples with 1 F 1 . . . . . . . . . . . . . . . . . . . . . . . . . 279 A.3.5 Special examples with 2 F 1 . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.3.6 Some integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 A.3.7 Gamma function expressions by Ramanujan . . . . . . . . . . . . 284 A.4 Basic hypergeometric functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 291 A.4.1 General expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 A.4.2 Two general results by Andrews . . . . . . . . . . . . . . . . . . . . . 292 A.4.3 q -expressions by Ramanujan . . . . . . . . . . . . . . . . . . . . . . . 292 Bibliography 295 Index 306
Page 2 and 3: ATLANTIS STUDIES IN MATHEMATICS FOR
Page 4 and 5: Continued Fractions Second edition
Page 6 and 7: Preface to the second edition 15 ye
Page 8: Preface vii For whom? We are aiming
Page 11: x Contents 2.1.6 Value sets . . . .
Page 15 and 16: 2 Chapter 1: Introductory examples
Page 63 and 64:
50 Chapter 1: Introductory examples
Page 65 and 66:
52 Chapter 1: Introductory examples
Page 67 and 68:
54 Chapter 2: Basics 2.1 Convergenc
Page 69 and 70:
56 Chapter 2: Basics Example 1. We
Page 71 and 72:
58 Chapter 2: Basics Proof of the e
Page 73 and 74:
60 Chapter 2: Basics ✤ ✜ Defini
Page 75 and 76:
62 Chapter 2: Basics ✗ ✔ Defini
Page 77 and 78:
64 Chapter 2: Basics Proof : □
Page 79 and 80:
66 Chapter 2: Basics that is, t n =
Page 81 and 82:
68 Chapter 2: Basics ✬ ✩ Coroll
Page 83 and 84:
70 Chapter 2: Basics Therefore ✸
Page 85 and 86:
72 Chapter 2: Basics and the result
Page 87 and 88:
74 Chapter 2: Basics of value sets
Page 89 and 90:
76 Chapter 2: Basics More important
Page 91 and 92:
78 Chapter 2: Basics Proof : Let A
Page 93 and 94:
80 Chapter 2: Basics An equivalence
Page 95 and 96:
82 Chapter 2: Basics is non-restrai
Page 97 and 98:
84 Chapter 2: Basics It will be evi
Page 99 and 100:
86 Chapter 2: Basics • an even pa
Page 101 and 102:
88 Chapter 2: Basics show that {A 2
Page 103 and 104:
90 Chapter 2: Basics and to require
Page 105 and 106:
92 Chapter 2: Basics 4. Periodic ta
Page 107 and 108:
94 Chapter 2: Basics 12. Tail seque
Page 109 and 110:
96 Chapter 2: Basics 22. ♠ Genera
Page 111 and 112:
98 Chapter 2: Basics Prove that b 0
Page 113 and 114:
100 Chapter 3: Convergence criteria
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
172 Chapter 4: Periodic and limit p
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
218 Chapter 5: Computation of conti
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
266 Appendix A: Some continued frac
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
296 Bibliography [Bear04] A. F. Bea
Page 311 and 312:
298 Bibliography [Glai74] J. W. L.
Page 313 and 314:
300 Bibliography [JoTW82] W. B. Jon
Page 315 and 316:
302 Bibliography [Mind69] F. Mindin
Page 317 and 318:
304 Bibliography [ThWa80b] W. J. Th
Page 319 and 320:
Index (a) n Pochhammer symbol, 27 A
Page 321:
308 Index moment problem, 40, 41 od
show all

Continued Fractions, Convergence Theory. Vol. 1, 2nd Editions. Loretzen, Waadeland. Atlantis Press. 2008

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?