(05/13/2014, 04:23 AM)sheldonison Wrote: [ -> ]

I have an improvement in the equation for a_n

edit: many changes made; more improvements below, stronger theoretical basis.

I implemented this Gaussian approximation with pari-gp, and get the following Taylor series for exp^{0.5}(x), which is accurate to 0.5E-4, for x in the range of 10^12. I like this solution best so far, because it starts out more accurate than my previous approximation was

after scaling, and so it doesn't require any scaling, which might be harder to work with theoretically. If I replace the Gaussian estimation with a numeric integral, accuracy improves dramatically, so that the error term for x=10^12 improves to 1E-28.

The a_n coeffients can be thought of as either the upper bounds, from post#9, or equivalently, as calculating a Cauchy integral around the unit circle at a radius of exp(half(h_n)), for the x^n term. The radius is chosen to be the "best fit" for the x^n term. The d^2(exp^{0.5}) term comes from the fact that the nth term Cauchy unit circle integral can be approximated by an increasingly accurate Gaussian error function! As we go around that unit circle at exp(half(h_n)), where the x^2 Taylor series coefficient at half(h_n) gives the multiplier for that Gaussian error function which turns out to be the envelope for the Cauchy unit circle integral for calculating the a_n coefficient!! See the derivation of the Gaussian approximation below; we can treat Pi as reasonably close to infinity, and the higher order terms x^3, x^4 etc, become increasingly irrelevant.

The conjecture is that the ratio for the Gaussian approximation to the Kneser half iterate approaches 1, as x goes to infinity. I would still like to find an approximation for a_n as n to infinity, that can be expressed in terms of primitive functions like exp and gamma ....

Code:

`Numeric integral for entire half Taylor series; this is the most accurate version`

0.4985632879411144346796190925

+x^ 1* 0.8710111130401411916658936025

+x^ 2* 0.2538290681564146773775546123

+x^ 3* 0.02036897790106364504350768837

+x^ 4* 0.0005733509880744505729717080227

+x^ 5* 0.000006634106505118055642333198577

+x^ 6* 0.00000003523478404131539039591224937

+x^ 7* 9.324782493066663989279567781 E-11

+x^ 8* 1.310819973257249543566613437 E-13

+x^ 9* 1.030445631241455220396533854 E-16

+x^10* 4.726307235474875250341297202 E-20

+x^11* 1.310853283519007878428589500 E-23

+x^12* 2.266720766355745092754180782 E-27

+x^13* 2.509357593978893221893266337 E-31

+x^14* 1.820271304690092996128472955 E-35

+x^15* 8.831750008724271270123393100 E-40

+x^16* 2.919179948947205489411726353 E-44

+x^17* 6.682483996893730450602750368 E-49

+x^18* 1.075358031785365466659192846 E-53

+x^19* 1.233091867410211513631675134 E-58

+x^20* 1.020112248419500535396388387 E-63

+x^21* 6.158191969654037975312149469 E-69

+x^22* 2.741371326342544857122330767 E-74

+x^23* 9.086744322405568705961426660 E-80

+x^24* 2.263024759865679795106323630 E-85

+x^25* 4.270284072518439806922204025 E-91

+x^26* 6.153410152198212590494991930 E-97

+x^27* 6.821113316310223883284516362 E-103

+x^28* 5.856904447459072240001444407 E-109

+x^29* 3.920761917716412776918112254 E-115

+x^30* 2.058813935289383486029897989 E-121

+x^31* 8.529347872122290696613310346 E-128

+x^32* 2.803108465364799002052148786 E-134

+x^33* 7.345799681492560108026378204 E-141

+x^34* 1.542586353075185681435151002 E-147

+x^35* 2.607968946925530945074311843 E-154

+x^36* 3.565596994284168309668197970 E-161

+x^37* 3.958982393573096000878749016 E-168

+x^38* 3.584408766307859670566288919 E-175

+x^39* 2.656548245305104869013412253 E-182

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+x^58* 1.761385400638242415446751924 E-332

+x^59* 4.423493718184744719252398771 E-341

+x^60* 9.606219934933392322191245309 E-350

+x^61* 1.807135789397774896583344421 E-358

+x^62* 2.950079390611926268702307695 E-367

+x^63* 4.186143883211317839842039468 E-376

+x^64* 5.171842056918956608807274076 E-385

+x^65* 5.572129536563015037499004273 E-394

+x^66* 5.243461076186418235561914770 E-403

+x^67* 4.316129021845561706589204069 E-412

+x^68* 3.112377220439181678178609685 E-421

+x^69* 1.968961982906547212687759215 E-430

+x^70* 1.094309435664852668982645759 E-439

+x^71* 5.350523041860647846762344418 E-449

+x^72* 2.304554060572185793854721313 E-458

+x^73* 8.755473947475085919252894579 E-468

+x^74* 2.937844467688804179829547455 E-477

+x^75* 8.717176379165003015992837663 E-487

+x^76* 2.290076343590743636446487727 E-496

+x^77* 5.332962185069965482401108681 E-506

+x^78* 1.102141417377979419944024139 E-515

+x^79* 2.023723645749336434038072025 E-525

+x^80* 3.305173401009004078623330565 E-535

+x^81* 4.806624687997466977313128538 E-545

+x^82* 6.230929915400370391118694177 E-555

+x^83* 7.207527603870478194048469267 E-565

+x^84* 7.447067824925920341109272650 E-575

+x^85* 6.879954261884070418996694979 E-585

+x^86* 5.688712338755421271773128506 E-595

+x^87* 4.213946498415522514357688587 E-605

+x^88* 2.799115985335784071999929037 E-615

+x^89* 1.668828997856926907617005305 E-625

+x^90* 8.938329128856059372616968360 E-636

+x^91* 4.304685213166014098194471219 E-646

+x^92* 1.865721051470808691443267096 E-656

+x^93* 7.283580375885783533990126512 E-667

+x^94* 2.563313842980498623817064008 E-677

+x^95* 8.139100444274145390849055789 E-688

+x^96* 2.333573107131065458867211698 E-698

+x^97* 6.046208392584521828778143914 E-709

+x^98* 1.416777678114006014473790491 E-719

+x^99* 3.004771376670761211508245243 E-730

+x^100*5.772191649076661627264665007 E-741

+x^101*1.005107332953395409129766995 E-751

+x^102*1.587610105165586471491468267 E-762

+x^103*2.276391538527593632197839634 E-773

+x^104*2.965026267574167753439855765 E-784

+x^105*3.510672477453893834286980585 E-795

+x^106*3.781194749940910612332613351 E-806

+x^107*3.707120512552296071449750832 E-817

+x^108*3.310553141937844768111558880 E-828

+x^109*2.694655965903801865391922127 E-839

+x^110*2.000429415385553265088216048 E-850

+x^111*1.355296058493422629926292251 E-861

+x^112*8.385052827601936271862229376 E-873

+x^113*4.740275915051639026129261183 E-884

+x^114*2.450129799251313430222543819 E-895

+x^115*1.158563206764715042176206076 E-906

+x^116*5.014749015804378650861723954 E-918

+x^117*1.988050433351706485020113085 E-929

+x^118*7.222715987266827272057906191 E-941

+x^119*2.406086562703090259789810633 E-952

+x^120*7.353576929945984825943727706 E-964

+x^121*2.062996044866367254648463530 E-975

+x^122*5.315484268416505872622525458 E-987

+x^123*1.258522567866236954382862472 E-998

+x^124*2.739546898498989761925977413 E-1010

+x^125*5.485531653181015540252830319 E-1022

+x^126*1.010882286100850364148000228 E-1033

+x^127*1.715303357604997360378961019 E-1045

+x^128*2.681349668109586179961519367 E-1057

+x^129*3.863215420318792302247556914 E-1069

+x^130*5.132575320637659521483538205 E-1081

+x^131*6.290973901451826499430290031 E-1093

+x^132*7.117034858844710792253182914 E-1105

+x^133*7.434963186913381616699570660 E-1117

+x^134*7.175535883707921136701383295 E-1129

+x^135*6.400595385142868802218878845 E-1141

+x^136*5.279212684551819192356274183 E-1153

+x^137*4.028009534268085274427661553 E-1165

+x^138*2.844276211365647877674429606 E-1177

+x^139*1.859508005039463803122793853 E-1189

+x^140*1.126035533626274114177059774 E-1201

+x^141*6.318505947802323797222722837 E-1214

+x^142*3.286724100651648951381637663 E-1226

+x^143*1.585531100527665483167915782 E-1238

+x^144*7.096152409897556355115618752 E-1251

+x^145*2.947675400321429892810975447 E-1263

+x^146*1.136880407531572331200690980 E-1275

+x^147*4.072828729047575634268426013 E-1288

+x^148*1.355781226961727800775025176 E-1300

+x^149*4.195262231654102499876908828 E-1313

+x^150*1.207165060933645530811450970 E-1325

+x^151*3.231263707211930861107870173 E-1338

+x^152*8.048865420729444233874864997 E-1351

+x^153*1.866422498230798292333708820 E-1363

+x^154*4.030445298086469258814308037 E-1376

+x^155*8.108058640520878451816788866 E-1389

+x^156*1.520032260920878101686380056 E-1401

+x^157*2.656503670689144383021561731 E-1414

+x^158*4.329499475128981565124536774 E-1427

+x^159*6.582349863516344984855565238 E-1440

+x^160*9.338668990553129883640703208 E-1453

+x^161*1.236781366859692389297678707 E-1465

+x^162*1.529489353130209794964280670 E-1478

+x^163*1.766793050594006027434806539 E-1491

+x^164*1.906993632878006517639988197 E-1504

+x^165*1.923863751515074897909419206 E-1517

+x^166*1.814665656457886373509359798 E-1530

+x^167*1.600849411868116034890257045 E-1543

+x^168*1.321200755442445354926834067 E-1556

+x^169*1.020429875486358289345036855 E-1569

+x^170*7.377746585147470422447048797 E-1583

+x^171*4.994814787113353646500336149 E-1596

+x^172*3.167361931007179005199974312 E-1609

+x^173*1.881849643653924045494284771 E-1622

+x^174*1.047866808778147260099053198 E-1635

+x^175*5.469970010811979820209209681 E-1649

+x^176*2.677589772452488813768643305 E-1662

+x^177*1.229432523769072880965526166 E-1675

+x^178*5.296476577303116987951193452 E-1689

+x^179*2.141460568563315391686316082 E-1702

+x^180*8.128147745800165253090960394 E-1716

+x^181*2.896998396963448961657560720 E-1729

+x^182*9.698300513209975888283389430 E-1743

+x^183*3.050339590552378843333948030 E-1756

+x^184*9.016111786083733120644214602 E-1770

+x^185*2.505077839762538013844591105 E-1783

+x^186*6.544330883519596549898149683 E-1797

+x^187*1.607909371744948895134530835 E-1810

+x^188*3.716373541858010751920509452 E-1824

+x^189*8.082520263145333164606871495 E-1838

+x^190*1.654440197610463210395411899 E-1851

+x^191*3.188147258600726509182831325 E-1865

+x^192*5.785140801075633800421509267 E-1879

+x^193*9.887398433329089225495816951 E-1893

+x^194*1.592010837665992733213999094 E-1906

+x^195*2.415507012130397350124062001 E-1920

+x^196*3.454382188898361752203626924 E-1934

+x^197*4.657285603659041661380254890 E-1948

+x^198*5.921005627893362617368461077 E-1962

+x^199*7.099978465660990342355250150 E-1976

+x^200*8.031848704412873538835534480 E-1990

Evaluation and comparison of Gaussian approximation, with Kneser half iterate, and numeric integration.

Code:

`x Gauss series at x Gaussian/Kneser Half numeric intetgral/Kneser `

1/2 1.009600147857 1.007947076464 0.9984707461893

1 1.664777638208 1.011190425535 0.9987841217590

2 3.475227097782 1.012802437685 0.9991092778066

4 9.622858431184 1.011659759461 0.9988701672778

8 37.09830694534 1.008919521341 0.9985195257893

16 209.6008686180 1.006329241170 0.9986170527351

32 1830.759692420 1.004559841264 0.9990581577384

64 26151.47799104 1.003455005276 0.9994965790411

128 648661.8576542 1.002721431338 0.9997710270601

256 29786305.89243 1.002192879433 0.9999063463259

512 2711120116.146 1.001793913224 0.9999651058788

1024 525959987977.7 1.001485094862 0.9999884061876

2048 2.349112269692 E14 1.001241871705 0.9999967699125

4096 2.621071536336 E17 1.001047620358 0.9999993881637

8192 7.965701394346 E20 1.000890626833 1.000000035772

16384 7.224766966614 E24 1.000762421236 1.000000113394

32768 2.153638714039 E29 1.000656756479 1.000000074344

65536 2.335868262933 E34 1.000568948079 1.000000035914

131072 1.026088749416 E40 1.000495432712 1.000000014689

262144 2.043386486924 E46 1.000433465457 1+0.00000000532128446

524288 2.076940129986 E53 1.000380907824 1+0.00000000174075232

1048576 1.220321423874 E61 1.000336076709 1+0.00000000051912754

2097152 4.723250515159 E69 1.000297635079 1+1.418025757259 E-10

4194304 1.381073893616 E79 1.000264511776 1+3.555589248262 E-11

8388608 3.521531572583 E89 1.000235841927 1+8.189070349476 E-12

16777216 9.099903639076 E100 1.000210922178 1+1.731794371046 E-12

33554432 2.788647458979 E113 1.000189176699 1+3.359170814484 E-13

67108864 1.194430932904 E127 1.000170131136 1+5.966788610686 E-14

134217728 8.489581185954 E141 1.000153392461 1+9.685500427077 E-15

268435456 1.197833913454 E158 1.000138633266 1+1.433153817130 E-15

536870912 4.044763362506 E175 1.000125579421 1+1.927446011318 E-16

1073741824 3.972335392963 E194 1.000114000308 1+2.348062402030 E-17

2147483648 1.390244357146 E215 1.000103701046 1+2.580693625797 E-18

4294967296 2.142544151526 E237 1.000094516252 1+2.546713291110 E-19

8589934592 1.812291692626 E261 1.000086305016 1+2.243166560231 E-20

17179869184 1.058083505201 E287 1.000078946824 1+1.749951927294 E-21

34359738368 5.411414353099 E314 1.000072338235 1+1.196160689628 E-22

68719476736 3.105872410678 E344 1.000066390156 1+7.045019712667 E-24

137438953472 2.587656256198 E376 1.000061025608 1+3.465186873779 E-25

274877906944 4.088298834066 E410 1.000056177870 1+1.299802399317 E-26

549755813888 1.616444697079 E447 1.000051788938 1+1.206903030891 E-28

1099511627776 2.132798432202 E486 1.000047808251 1-9.985173164859 E-29

The numeric integral is more complicated than the standard Gaussian error integral, and is derived below. The value h_n was chosen so that a1=n, and then the Cauchy integration multiplier for x^-n exactly cancels the a1 term..... Lets go back to the general case for a Cauchy integral used to calculate the Taylor series coefficients, where we have f(x) instead of exp^{0.5}(x)...

The only practical way to calcute this integral is by using a circle centered on the origin;

radius=1 Cauchy integral formula. but we can use any radius

Now, for an arbitrary radius of r,

Now, replace f(x) with exp^{0.5}(x)

A couple of quick comments on exp^{0.5}. It doesn't follow the Cauchy integral rules, because the circle of radius r usually circles the singularities that are near the origin, at L,L*, and at +/- pi i -0.37. So its not a legal circular integral; there will be a discontinuity at -r, due to the branch singularities. But lets ignore that somewhat inconvenient fact, and evaluate the integral anyway. Now, we are free to choose any value of r we want. So let us choose the value of r, which has derivative for

.

The

is exactly the value I posted earlier in this post and is that radius. Now, we have the the following.

Notice the b1 term conveniently cancels out!

substituting ln( r)=h_n and substituting the equation above

this a_n is the most accurate entire asymptotic Taylor series!

The less accurate Gaussian approximation

Here b2 is the second derivative from the beginning of this post... and this is the Gaussian integral approximation for a_n from the beginning of the post, which works very well. The Gaussian approximation is derived as an approximation of the much more accurate numeric integral. But the Gaussian integral itself works very well, and better than any approximation I had before this post.