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Nowhere analytic superexponential convergence
#1
I found some surprisingly close similarities in the behavior of tommysexp and the base change function, and may have made some progress on why both are probably nowhere analytic functions. In particular, my hypothesis is that is also a nowhere analytic function, whose convergence in some sense tracks the base change, and tommysexp function. I'm still working out the details, some of which are included today, and some of which will be in a future post.

I calculated the taylor series and error terms and nearest singularities for the base change eta equations, see post, and for the tommysexp equations, see my recent post. Both iterate logarithms from one real valued superexponential superfunction, to generate a particular different real valued base(e) sexp(z) superexponential. The working assumption is that the resulting sexp(e) base change is nowhere analytic. But the intermediate approximations are defined in the complex plane. So here, I generated the series, for both approximations. The similarities between the two are somewhat striking, and lead to some new ideas to explore about nowhere analytic superexponential convergence.

Recall, . So the first two logarithms in the base change are sort of trivial. For the following, I'm comparing five logarithms for the base change, with three logarithms for the 2sinnh superfunction.
  • tommysexp radius of convergence, 0.458. The base change radius of convergence is 0.460
  • The first thirty singularities for both functions are remarkably similar
  • both have similar superexponential convergence. The next iteration delta for both are less than 1E-80, in the region for real(z)>-0.5. This means that doing one more logarithm has an almost negligable difference at the real axis. But for both, one more logarithm changes the radius of convergence from 0.46 to 0.035.
  • I generated the taylor series for both
  • for both, the taylor series matches the function to ~1E-28 for -0.4<z<0.4.
Based on the fact that the singularities gets arbitrarily close to the real axis as the number of logarithm iterations increase, and based on the fact that the singularities get arbitrarily close together, it is assumed that these functions are nowhere analytic, but they are infinitely differentiable.

The next iteration delta is defined as

Let z=sexp(x). Using the chain rule, and first order derivatives, one can derive that the the next iteration convergence delta equations for cheta, and for 2insh superf are approximately the following, which has also been verified empirically.
nextdelta_cheta_e(z)=
nextdelta_tommysexp(z)=
The tommysexp next iteration delta term is roughly the square of the cheta base change next iteration delta term. But that squaring becomes negligible compared to supperexponential converging terms. For both, I suspect this superexponential convergence at the real axis makes it impossible to model the nextdelta functions with a taylor series, which matches the singularities approaching arbitrarily close to the real axis. Below, you can see how very very tiny these next iteration deltas are, with a number like 10^-165653 for the next iteration delta at z=0. For z>0.05, pari-gp overflows, and can't print the the next iteration delta in scientific form. This may also lead to a way to prove that these functions are nowhere analytic, since they are both very close to an analytic function with a radius of 0.46, but adding the tiny nextdelta dramatically reduces the taylor series convergence, because the nextdelta term has such quickly growing derivatives. And for the nextdelta after that, the nextdelta is tinier still, approximately 1/sexp(z+5), with a correspondingly faster growith in the derivatives.

This leads to a natural question. Is this superexponential reciprocal summation also , but nowhere analytic?
Interestingly, the intermediate terms for this summation do not have any actual singularities at radius 0.46, and then at radius 0.035, but it nonetheless behaves like it has a singularities at those radii. More comments comming in a future post! And it has some of the same convergence properties, since for large enough values of sexp(z), one can say that in some sense all of the following are approximately equal, . Then in that same sense, for tomysexp, and for the base change, and for the superexponential reciprocal summation, the nextdelta is approximately the same.
nextdelta(n+4) approximation: 1/sexp(z+4)
nextdelta(n+5) approximation: 1/sexp(z+5)
nextdelta(n+6) approximation: 1/sexp(z+6)
nextdelta(n+7) approximation: 1/sexp(z+7)

Here are some computed results, with the tommysexp function on the left, and the base change function on the right. Results are included for closest singularities, next iteration delta, taylor series, and taylor series accuracy. The Taylor series for both was sampled around at a radius of 0.4 around about the origin, z=0, with 200 samples. Both have a singularity at a radius of 0.46. For the superfunction of 2sinh, the singularities occur whenever the superfunction is first equal to n*2*Pi*i. For the base change, the singularities occur where the superfunction of is first equal to e*(1+n*2*Pi*i).
- Sheldon
Code:
tommysexp=superfunction(2sinh)          cheta base change, upper sexp(eta)
first 30 singularities                  first 30 singularities
1      0.007675615 + 0.7110234703*I     1     -0.007046864 + 0.7086921323*I
2      0.156738931 + 0.4901228794*I     2      0.153425672 + 0.4931023030*I
3      0.234549231 + 0.4054846386*I     3      0.233763209 + 0.4081718531*I
4      0.285107736 + 0.3580209487*I     4      0.285306052 + 0.3602436661*I
5      0.321752934 + 0.3267267408*I     5      0.322435529 + 0.3285711201*I
6      0.350104098 + 0.3041253984*I     6      0.351057363 + 0.3056753164*I
7      0.373010280 + 0.2868168927*I     7      0.374127093 + 0.2881352237*I
8      0.392097990 + 0.2730084001*I     8      0.393318661 + 0.2741410912*I
9      0.408376026 + 0.2616549026*I     9      0.409664722 + 0.2626359532*I
10     0.422509625 + 0.2521013150*I     10     0.423843619 + 0.2529563952*I
11     0.434958701 + 0.2439136307*I     11     0.436322954 + 0.2446625228*I
12     0.446053388 + 0.2367915138*I     12     0.447437616 + 0.2374497535*I
13     0.456038030 + 0.2305196835*I     13     0.457435017 + 0.2310996796*I
14     0.465098132 + 0.2249392776*I     14     0.466502705 + 0.2254510910*I
15     0.473377582 + 0.2199301716*I     15     0.474785971 + 0.2203820687*I
16     0.480990056 + 0.2153996258*I     16     0.482399477 + 0.2157984789*I
17     0.488026798 + 0.2112747481*I     17     0.489435178 + 0.2116263308*I
18     0.494562072 + 0.2074973457*I     18     0.495967852 + 0.2078065544*I
19     0.500657065 + 0.2040203194*I     19     0.502059070 + 0.2042913418*I
20     0.506362740 + 0.2008050829*I     20     0.507760082 + 0.2010415280*I
21     0.511721960 + 0.1978196799*I     21     0.513113970 + 0.1980246795*I
22     0.516771090 + 0.1950373863*I     22     0.518157264 + 0.1952136753*I
23     0.521541225 + 0.1924356567*I     23     0.522921193 + 0.1925856373*I
24     0.526059152 + 0.1899953187*I     24     0.527432641 + 0.1901211119*I
25     0.530348090 + 0.1876999500*I     25     0.531714907 + 0.1878034376*I
26     0.534428293 + 0.1855353899*I     26     0.535788306 + 0.1856182493*I
27     0.538317521 + 0.1834893535*I     27     0.539670648 + 0.1835530860*I
28     0.542031431 + 0.1815511233*I     28     0.543377630 + 0.1815970775*I
29     0.545583889 + 0.1797113001*I     29     0.546923147 + 0.1797406922*I
30     0.548987229 + 0.1779616016*I     30     0.550319558 + 0.1779755319*I
600000 1.002503983 + 0.0347283022*I     600000 1.002441996 + 0.0350126288*I

tommy sexp next iteration delta         base change next iteration delta
-1.50 delta est 0.000007290427449       -1.50 delta est 0.001312345791
-1.45 delta est 0.000002442380117       -1.45 delta est 0.0006835256251
-1.40 delta est 0.0000007238057956      -1.40 delta est 0.0003348415946
-1.35 delta est 0.0000001844698859      -1.35 delta est 0.0001519916764
-1.30 delta est 0.00000003903526792     -1.30 delta est 0.00006276903195
-1.25 delta est 0.000000006559199672    -1.25 delta est 0.00002304855068
-1.20 delta est 0.000000000826302550    -1.20 delta est 0.000007307424860
-1.15 delta est 7.238205406 E-11        -1.15 delta est 0.000001925394773
-1.10 delta est 3.988868330 E-12        -1.10 delta est 0.0000004008301483
-1.05 delta est 1.207892248 E-13        -1.05 delta est 0.00000006159403329
-1.00 delta est 1.668575191 E-15        -1.00 delta est 0.000000006364409628
-0.95 delta est 8.101375833 E-18        -0.95 delta est 0.000000000388187413
-0.90 delta est 9.522025197 E-21        -0.90 delta est 1.160608067 E-11
-0.85 delta est 1.570526566 E-24        -0.85 delta est 1.296962680 E-13
-0.80 delta est 1.605157311 E-29        -0.80 delta est 3.610457672 E-16
-0.75 delta est 2.884944478 E-36        -0.75 delta est 1.341207737 E-19
-0.70 delta est 1.232007258 E-45        -0.70 delta est 2.468691750 E-24
-0.65 delta est 4.642126163 E-59        -0.65 delta est 4.418654086 E-31
-0.60 delta est 5.509049535 E-79        -0.60 delta est 4.750549878 E-41
-0.55 delta est 8.595247025 E-110       -0.55 delta est 2.108614749 E-56
-0.50 delta est 9.934995750 E-160       -0.50 delta est 3.264931211 E-81
-0.45 delta est 2.778099893 E-245       -0.45 delta est 1.273717448 E-123
-0.40 delta est 9.415182574 E-402       -0.40 delta est 4.549700447 E-201
-0.35 delta est 4.207952100 E-711       -0.35 delta est 4.463595288 E-354
-0.30 delta est 7.675146626 E-1383      -0.30 delta est 2.440718380 E-686
-0.25 delta est 6.000014213 E-3018      -0.25 delta est 1.028884438 E-1495
-0.20 delta est 1.822210125 E-7599      -0.20 delta est 9.833387813 E-3767
-0.15 delta est 7.913126877 E-22903     -0.15 delta est 2.069121979 E-11367
-0.10 delta est 1.381445672 E-86764     -0.10 delta est 5.267169510 E-43161
-0.05 delta est 2.726837470 E-441691    -0.05 delta est 5.555419728 E-220307
0.000 delta est 1.170808246 E-331305    0.000 delta est 2.729744576 E-165653
0.050 delta est 1.997889711 E-41675401  0.050 delta est 3.431529878 E-20863257

tommysexp supref(2sinh(z)) series       base change, cheta/sexp(eta) upper series
first 60/200 terms calculated           first 60/200 terms calculated
0 1.00000000000000                      0 1.00000000000000      
1 1.09146536076840                      1 1.09245822365347      
2 0.273334906394121                     2 0.263554328999923    
3 0.215218479242466                     3 0.205343154888622    
4 0.0652715037680265                    4 0.0904346945443277    
5 0.0391656564308927                    5 0.0505926411913673    
6 0.0171521314068201                    6 -0.00497282375975114  
7 0.0117058806324846                    7 0.0469251697418079    
8 0.00471958861559290                   8 -0.0177179396467206  
9 0.00123667589678222                   9 -0.186320226068563    
10 -0.00226288150336297                 10 0.286274823965696    
11 0.00321559868096087                  11 0.786721506210894    
12 -0.00820154271014946                 12 -1.74904574987150    
13 0.00218777707039554                  13 -4.22035142074436    
14 0.0477372146016102                   14 8.20344527496622    
15 -0.117247563749023                   15 26.9037752819119    
16 -0.0788849220732723                  16 -24.7206037630541    
17 0.883038307995801                    17 -166.586604395346    
18 -0.610166815880536                   18 -43.2725525131210    
19 -5.10470797277627                    19 839.172975959226    
20 7.81612106347109                     20 1336.31403138573    
21 28.6479580354880                     21 -2550.74078072519    
22 -60.0481521891273                    22 -10958.6656809462    
23 -173.314801251505                    23 -4272.50237468355    
24 382.323334608732                     24 50062.8915107697    
25 1156.14068365026                     25 110734.098785714    
26 -2075.65103517000                    26 -54918.2588154844    
27 -8124.15769732862                    27 -671682.734465615    
28 8589.21772341065                     28 -1036994.99812864    
29 56026.7180961308                     29 1303463.94817780    
30 -10973.8915910648                    30 7906099.55967434    
31 -353646.843998969                    31 10264075.6059732    
32 -264336.726812852                    32 -17004741.2263050    
33 1880963.01782064                     33 -88499153.4024295    
34 3669078.97540668                     34 -116329792.315688    
35 -7012765.98982617                    35 167304017.910329    
36 -30701470.8276538                    36 963110565.025803    
37 1548864.00645558                     37 1471157605.81809    
38 184151139.582793                     38 -1166177629.67364    
39 254566511.139843                     39 -10032662178.5555    
40 -702354526.642371                    40 -19039573638.1062    
41 -2569640324.09206                    41 1443219835.56116    
42 -5013267.92728414                    42 95794081305.7755    
43 14864113060.2053                     43 234210144815.370    
44 25072200943.3146                     44 133790170777.420    
45 -41910819127.1582                    45 -773512716992.095    
46 -217305763066.960                    46 -2602252032300.76    
47 -154949067429.982                    47 -3041284384185.33    
48 943505478224.865                     48 4232438021867.08    
49 2649848118455.03                     49 24816333063823.4    
50 -211692701210.664                    50 44988721281848.7    
51 -15217717895592.4                    51 5695511204255.34    
52 -28995187122276.9                    52 -184546634307910.    
53 25950229728157.7                     53 -510468123278426.    
54 213031534180868.                     54 -550701240269359.    
55 298483216448330.                     55 732817220216991.    
56 -540350815684974.                    56 4.36282875951352 E15
57 -2.77584178284245 E15                57 8.84380011906777 E15
58 -3.02896731520363 E15                58 5.81870949781462 E15
59 8.44582062003156 E15                 59 -2.21417549121022 E16

tommysexp                               base change
-0.50 series error -4187.308964         -0.50 series error -612.3595433
-0.45 series error -0.00000314007994    -0.45 series error -0.00000046821885
-0.40 series error -1.967651776 E-16    -0.40 series error -2.995061854 E-17
-0.35 series error -4.969752342 E-28    -0.35 series error -7.246590384 E-29
-0.30 series error 2.916083910 E-29     -0.30 series error 9.542284363 E-30
-0.25 series error 2.414667177 E-29     -0.25 series error 8.897390862 E-30
-0.20 series error 1.721748423 E-29     -0.20 series error 7.931710119 E-30
-0.15 series error 7.733890007 E-30     -0.15 series error 6.533447419 E-30
-0.10 series error -5.131358874 E-30    -0.10 series error 4.558415157 E-30
-0.05 series error -2.240424609 E-29    -0.05 series error 1.828944318 E-30
0.000 series error -4.525864081 E-29    0.000 series error -1.856258370 E-30
0.050 series error -7.482860942 E-29    0.050 series error -6.689333719 E-30
0.100 series error -1.117784941 E-28    0.100 series error -1.278228379 E-29
0.150 series error -1.555621683 E-28    0.150 series error -2.005090791 E-29
0.200 series error -2.035294954 E-28    0.200 series error -2.808322941 E-29
0.250 series error -2.505008111 E-28    0.250 series error -3.608655740 E-29
0.300 series error -2.897231036 E-28    0.300 series error -4.303680442 E-29
0.350 series error -8.215790599 E-28    0.350 series error -2.460314623 E-28
0.400 series error -1.431085214 E-16    0.400 series error -7.232990979 E-17
0.450 series error -0.00000153057824    0.450 series error -0.00000110157473
0.500 series error -1086.697955         0.500 series error -1366.400977
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Messages In This Thread
Nowhere analytic superexponential convergence - by sheldonison - 01/20/2011, 05:23 PM

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