01/20/2011, 05:23 PM
(This post was last modified: 01/20/2011, 06:09 PM by sheldonison.)

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.

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

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.

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