10/08/2015, 01:11 PM
(This post was last modified: 10/08/2015, 06:08 PM by sheldonison.)

(10/08/2015, 08:41 AM)tommy1729 Wrote:(10/08/2015, 03:41 AM)sheldonison Wrote:

Yes but this exp ( - n^2 / 4 ) is far from Jay's 2 ^ ( - n (n-1) ) / n !

Its a different base ; exp(- 1/4) =\= 2^(-1).

So this is the worst fit , rather then the best ?

It seems to disprove the conjectures ?!

Or do i need less or more medication ?

Regards

Tommy1729

It is a different base; I interpolated as per your earlier post. I leave it as a home work problem for you to generate the fake function for , which should be closer to Jay's series, in the sense that I think as x gets arbitrarily large, for J(x). But this is a very rough order of magnitude approximation of J(x). If you like, I can post the results for the fake function for Jay's series exactly, but it would be a numerical Taylor series, not a nice closed form function.

Ok, here are the numerical approximations for the ratio of the Gaussian approximation for fake_a_n/a_n, for J(x). I would expect that as n gets arbitrarily large, this would go to a constant, but not get to exactly 1, unlike the case for the fake function for exp(z) or exp^{0.5}(z), where the ratio gets arbitrarily close to 1. The error term is going to be the ratio of the Gaussian approximation over the exact integral, which is complicated, but for this function, we think g'' goes to a constant, instead of getting arbitrarily small. So the error term involves the integral going from -infinity to infinity instead of the exact value of the integral from -pi to pi. Also the exact a_n integral includes all of the derivatives of g(z), not just the g'' approximation with all higher derivatives zero.

My "guess" for the ratio as n goes to infinity: 1.00000000002528325425; this is the limiting ratio as n goes to infinity for the fake function for . Now take the entire function, fake(x)=~f(x); and generate fake2(x) from fake(x), and that is how you get this limiting ratio. More later, only if there is interest.

(10/08/2015, 12:26 PM)tommy1729 Wrote: Sheldon , in your link you apparantly considered similar things.If one includes the full Laurent series with negative terms, then there is a closed form for fake(x)-f(x), which might help with proofs. The negative Laurent series terms, terms cause a singularity at zero, but otherwise quickly become insignificant; elsewhere the full Laurent series converges. One might be able to prove the "guessed ratio" above.

But what is that about Laurent series ?

You mention Laurent and then you drop the negative terms ??

Code:

`n ratio of fake_a_n/a_n for J(x)`

1 1.048770528303

2 1.013850038715

3 1.005968616357

4 1.003144747621

5 1.001868901379

6 1.001204340672

7 1.000822844603

8 1.000587713520

9 1.000434687401

10 1.000330708190

11 1.000257534636

12 1.000204519322

13 1.000165153630

14 1.000135302082

15 1.000112249218

16 1.000094160750

17 1.000079766664

18 1.000068168339

19 1.000058717592

20 1.000050938776

21 1.000044477428

22 1.000039065682

23 1.000034498530

24 1.000030617249

25 1.000027297641

26 1.000024441575

27 1.000021970809

28 1.000019822446

29 1.000017945527

30 1.000016298466

31 1.000014847101

32 1.000013563191

33 1.000012423249

34 1.000011407636

35 1.000010499843

36 1.000009685927

37 1.000008954052

38 1.000008294132

39 1.000007697532

40 1.000007156832

41 1.000006665634

42 1.000006218397

43 1.000005810309

44 1.000005437176

45 1.000005095334

46 1.000004781568

47 1.000004493054

48 1.000004227300

49 1.000003982105

50 1.000003755518

51 1.000003545807

52 1.000003351428

53 1.000003171005

54 1.000003003307

55 1.000002847233

56 1.000002701791

57 1.000002566091

58 1.000002439331

59 1.000002320786

60 1.000002209802

61 1.000002105785

62 1.000002008196

63 1.000001916547

64 1.000001830392

65 1.000001749326

66 1.000001672978

67 1.000001601010

68 1.000001533112

69 1.000001469001

70 1.000001408415

71 1.000001351117

72 1.000001296886

73 1.000001245519

74 1.000001196830

75 1.000001150646

76 1.000001106809

77 1.000001065171

78 1.000001025596

79 1.000000987958

80 1.000000952139

81 1.000000918031

82 1.000000885534

83 1.000000854553

84 1.000000825000

85 1.000000796795

86 1.000000769861

87 1.000000744128

88 1.000000719529

89 1.000000696003

90 1.000000673491

91 1.000000651940

92 1.000000631299

93 1.000000611520

94 1.000000592559

95 1.000000574374

96 1.000000556926

97 1.000000540177

98 1.000000524094

99 1.000000508642

100 1.000000493793

101 1.000000479516

102 1.000000465784

103 1.000000452571

104 1.000000439854

105 1.000000427608

106 1.000000415814

107 1.000000404449

108 1.000000393494

109 1.000000382932

110 1.000000372744

111 1.000000362915

112 1.000000353428

113 1.000000344269

114 1.000000335424

115 1.000000326879

116 1.000000318622

117 1.000000310641

118 1.000000302925

119 1.000000295462

120 1.000000288242

121 1.000000281255

122 1.000000274493

123 1.000000267945

124 1.000000261604

125 1.000000255462

126 1.000000249511

127 1.000000243743

128 1.000000238151

129 1.000000232729

130 1.000000227471

131 1.000000222370

132 1.000000217420

133 1.000000212616

134 1.000000207953

135 1.000000203425

136 1.000000199028

137 1.000000194756

138 1.000000190606

139 1.000000186573

140 1.000000182653

141 1.000000178842

142 1.000000175136

143 1.000000171532

144 1.000000168026

145 1.000000164616

146 1.000000161296

147 1.000000158066

148 1.000000154921

149 1.000000151859

150 1.000000148878

151 1.000000145973

152 1.000000143144

153 1.000000140388

154 1.000000137702

155 1.000000135084

156 1.000000132532

157 1.000000130044

158 1.000000127618

159 1.000000125251

160 1.000000122943

161 1.000000120692

162 1.000000118495

163 1.000000116351

164 1.000000114258

165 1.000000112215

166 1.000000110221

167 1.000000108274

168 1.000000106372

169 1.000000104515

170 1.000000102701

171 1.000000100928

172 1.000000099196

173 1.000000097503

174 1.000000095849

175 1.000000094232

176 1.000000092651

177 1.000000091105

178 1.000000089594

179 1.000000088116

180 1.000000086670

181 1.000000085255

182 1.000000083871

183 1.000000082517

184 1.000000081192

185 1.000000079896

186 1.000000078626

187 1.000000077384

188 1.000000076167

189 1.000000074976

190 1.000000073810

191 1.000000072667

192 1.000000071549

193 1.000000070453

194 1.000000069379

195 1.000000068327

196 1.000000067296

197 1.000000066286

198 1.000000065296

199 1.000000064325

200 1.000000063374

201 1.000000062441

202 1.000000061527

203 1.000000060630

204 1.000000059751

205 1.000000058889

206 1.000000058043

207 1.000000057213

208 1.000000056399

209 1.000000055601

210 1.000000054817

211 1.000000054048

212 1.000000053294

213 1.000000052553

214 1.000000051826

215 1.000000051113

216 1.000000050412

217 1.000000049724

218 1.000000049049

219 1.000000048386

220 1.000000047735

221 1.000000047095

222 1.000000046467

223 1.000000045850

224 1.000000045244

225 1.000000044648

226 1.000000044063

227 1.000000043488

228 1.000000042923

229 1.000000042368

230 1.000000041822

231 1.000000041286

232 1.000000040759

233 1.000000040240

234 1.000000039731

235 1.000000039230

236 1.000000038737

237 1.000000038253

238 1.000000037777

239 1.000000037308

240 1.000000036848

241 1.000000036394

242 1.000000035949

243 1.000000035510

244 1.000000035079

245 1.000000034655

246 1.000000034237

247 1.000000033826

248 1.000000033422

249 1.000000033024

250 1.000000032632

251 1.000000032247

252 1.000000031867

253 1.000000031494

254 1.000000031126

255 1.000000030764

256 1.000000030408

257 1.000000030057

258 1.000000029712

259 1.000000029372

260 1.000000029037

261 1.000000028707

262 1.000000028382

263 1.000000028062

264 1.000000027746

265 1.000000027436

266 1.000000027130

267 1.000000026828

268 1.000000026532

269 1.000000026239

270 1.000000025951

271 1.000000025667

272 1.000000025387

273 1.000000025111

274 1.000000024839

275 1.000000024571

276 1.000000024307

277 1.000000024047

278 1.000000023790

279 1.000000023537

280 1.000000023287

281 1.000000023042

282 1.000000022799

283 1.000000022560

284 1.000000022324

285 1.000000022092

286 1.000000021863

287 1.000000021637

288 1.000000021414

289 1.000000021194

290 1.000000020977

291 1.000000020763

292 1.000000020552

293 1.000000020344

294 1.000000020138

295 1.000000019936

296 1.000000019736

297 1.000000019539

298 1.000000019344

299 1.000000019152

300 1.000000018963

301 1.000000018776

302 1.000000018591

303 1.000000018409

304 1.000000018229

305 1.000000018052

306 1.000000017877

307 1.000000017704

308 1.000000017533

309 1.000000017365

310 1.000000017198

311 1.000000017034

312 1.000000016872

313 1.000000016712

314 1.000000016554

315 1.000000016398

316 1.000000016244

317 1.000000016092

318 1.000000015942

319 1.000000015793

320 1.000000015647

321 1.000000015502

322 1.000000015359

323 1.000000015218

324 1.000000015078

325 1.000000014941

326 1.000000014804

327 1.000000014670

328 1.000000014537

329 1.000000014406

330 1.000000014276

331 1.000000014148

332 1.000000014022

333 1.000000013897

334 1.000000013773

335 1.000000013651

336 1.000000013530

337 1.000000013411

338 1.000000013293

339 1.000000013177

340 1.000000013062

341 1.000000012948

342 1.000000012836

343 1.000000012725

344 1.000000012615

345 1.000000012506

346 1.000000012399

347 1.000000012293

348 1.000000012188

349 1.000000012084

350 1.000000011982

351 1.000000011880

352 1.000000011780

353 1.000000011681

354 1.000000011583

355 1.000000011486

356 1.000000011390

357 1.000000011295

358 1.000000011202

359 1.000000011109

360 1.000000011017

361 1.000000010927

362 1.000000010837

363 1.000000010748

364 1.000000010661

365 1.000000010574

366 1.000000010488

367 1.000000010403

368 1.000000010319

369 1.000000010236

370 1.000000010154

371 1.000000010073

372 1.000000009993

373 1.000000009913

374 1.000000009834

375 1.000000009756

376 1.000000009679

377 1.000000009603

378 1.000000009528

379 1.000000009453

380 1.000000009379

381 1.000000009306

382 1.000000009234

383 1.000000009162

384 1.000000009091

385 1.000000009021

386 1.000000008952

387 1.000000008883

388 1.000000008815

389 1.000000008748

390 1.000000008681

391 1.000000008615

392 1.000000008550

393 1.000000008485

394 1.000000008421

395 1.000000008358

396 1.000000008295

397 1.000000008233

398 1.000000008172

399 1.000000008111

400 1.000000008051

- Sheldon