Searching for an asymptotic to exp[0.5]

[sheldonison]

...to understand the negative real axis, you mostly have to understand how Kneser's half iterate behaves for positive \( z+\pi i \), due to the equation \( \exp^{0.5}(z)=\exp(\exp^{0.5}(\log(z))) \). And the definition I'm using for negative axis comes down to \( \text{asymptotic}(-z) \approx \exp(\exp^{0.5}(\log(z)+\pi i))+\exp(\exp^{0.5}(\log(z)- \pi i)) \) plus stuff, where stuff can be shown too small to create zeros anywhere else, since halfk(z) can be shown to have a large absolute value at the negative axis, so stuff winds up acting like a small perturbation of the zeros of the approximation.

Before we can understand "stuff", we need an exact converging equation for a_n. Here is the equation, as a converging integral. Of course, using h_n is a pretty good approximation, but the limit converges, and gets more accurate as y goes to infinity. First, we need an equation for the trough of \( \exp^{0.5} \), at a particular real value of y, where we want the local minimum of \( \Re(\exp^{0.5}(y+zi))=\frac{1}{2}(\exp^{0.5}(y+zi)+\exp^{0.5}(y-zi)) \), which is where the derivative has a zero crossing.

\( \text{dhalfi}(y,z)=\frac{d}{dz} (\exp^{0.5}(y+zi)+\exp^{0.5}(y-zi)) \;\;\; mi(y) = \text{dhalfi}^{-1}(y,z)=0 \)

We note that z=0 has a derivative of zero, since the function has a local maximum at zi=0, but we are interested in the local minimum, where the real part at the minimum gets arbitrarily large negative as y gets bigger.

Then we get this equation for all of the Taylor series coefficients of the asymptotic half iterate, f(z). Of course, you can use y=h(n), for our previous "best behaved" radius, but the conjecture is that this integral converges as y gets arbitrarily large, for all values of a_n. This conjecture only depends on the first real minimum getting arbitrarily negative for exp^{0.5}. In this equation, exp(y) is the radius of a circle, where we are wrapping the approximation around the circle multiple times until we get to the local minimum. Each z=pi i corresponds to half way around the circle. For example, for y=2, we get mi(2)~=5.65pi, so we wrap the approximation around the circle 5.65 times, for a radius of exp(2). This limit converges rapidly as y increases.

\( a_n = \lim_{y \to \infty} \frac{1}{2\pi}
\int_{-mi(y)}^{+mi(y)} \exp(\exp^{0.5} (y+iz) - n(y+iz))\;dz \)

A more difficult conjecture is that this equation also converges for all negative values of n, where we are talking about a Laurent Series! Then, here are the golden values of a_n, for the asymptotic half iterate, accurate to 32 decimal digits.

Code:

{half=
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+x^267*  2.0973678925508310140708454839134 E-2977
+x^268*  7.0204513860124048128929361569859 E-2993
+x^269*  2.2435619969851113721406057440386 E-3008
+x^270*  6.8462320075794491042780947822524 E-3024
+x^271*  1.9950930497586676299541608886840 E-3039
+x^272*  5.5530277032422158108999083214016 E-3055
+x^273*  1.4764149510058814271782272044002 E-3070
+x^274*  3.7502078958721485969741426534598 E-3086
+x^275*  9.1017858622372148040390949833395 E-3102
+x^276*  2.1109499323164391347227188665436 E-3117
+x^277*  4.6791248960258165813941772872856 E-3133
+x^278*  9.9138356530228591442513442058349 E-3149
+x^279*  2.0080004529199037968113698528820 E-3164
+x^280*  3.8885278195209745127169291405925 E-3180
+x^281*  7.2004541037708460787761986112735 E-3196
+x^282*  1.2750918172412703104661669248805 E-3211
+x^283*  2.1596492273536361542217699775661 E-3227
+x^284*  3.4989531476600975459124994446056 E-3243
+x^285*  5.4232399623854837069841157428345 E-3259
+x^286*  8.0426211741222782354153066476251 E-3275
+x^287*  1.1413175498182740675299773645312 E-3290
+x^288*  1.5500194377377993981378886128387 E-3306
+x^289*  2.0148416599353000816235310923698 E-3322
+x^290*  2.5070839651532656491438274506800 E-3338
+x^291*  2.9865635421626323775665331571620 E-3354
+x^292*  3.4064341659042264496424959171075 E-3370
+x^293*  3.7205208039357170462514006673813 E-3386
+x^294*  3.8916413340839957809707676312809 E-3402
+x^295*  3.8988520048176433637470666861845 E-3418
+x^296*  3.7416642242029463512766194970626 E-3434
+x^297*  3.4400594322819498160219151491178 E-3450
+x^298*  3.0303222824401450669357416642614 E-3466
+x^299*  2.5578896958096305057557562347395 E-3482
+x^300*  2.0691549879104937795561894240776 E-3498
}

I mentioned "stuff" in a previous post. The 1/x Laurent series is an excellent approximation for -stuff, where here we use a 5 term approximation. If the |z| is fairly large, then we might want to use more terms in the approximation, there is always some stuff left over.
\( f(z) \approx \sum_{k=-2}^{2}
\exp(\exp^{0.5}(\ln(z) + 2k\pi i)) +
\text{stuff} \)

And here are the first 20 terms of the Laurent series, also generated and printed to 32 decimal digits accuracy. Generating the Laurent series is much much more difficult computationally, since there is no optimal h_n value for negative values of n, so extremely high precision calculations are required for the Kneser half iterate. I call this function the half_error term, since it is the dominant term for stuff, for small radius's. By the way, I also conjecture that even though all of these Laurent series terms converge, they don't behave very well as n gets arbitrarily large negative, and the function itself probably doesn't converge anywhere. But the truncated Laurent (1/z) series is very useful for understanding the "stuff" terms of the half iterate. A 20 term truncated Laurent series seems to work pretty well.

"Stuff" in the equation above, can be approximated by -half_error(z), since at any given radius, any function can be approximated arbitrarily well by a Laurent series. We throw away the negative terms for our entire half iterate. So naturally, there is an error term generated by the negative terms we threw away. I will give a more complete thorough analysis of all of the "stuff" near some of the zeros of f(z), and at the positive real axis, in a later post, by using the mi function, the the exp^{0.5}(y+mi(y)) function, and the Taylor/Laurent series.

Code:

{half_error=
+x^ -1*  0.0038981927072789568905822639730044
+x^ -2* -0.0017214881666811347076719510576548
+x^ -3* -0.00013712753070231088880268844304361
+x^ -4*  0.00013907417275118396879544895662740
+x^ -5*  0.000051320769626162064593356837293471
+x^ -6* -0.000022366390692218719076651284271858
+x^ -7* -0.000023771055718461525133758346306702
+x^ -8*  0.0000013731519550483503617729218008457
+x^ -9*  0.000012027038063261925914628822119819
+x^-10*  0.0000039087491754042668374627189270986
+x^-11* -0.0000057892790783328757123356666460684
+x^-12* -0.0000051218889509626574137447251389938
+x^-13*  0.0000019098976291086407044050882321080
+x^-14*  0.0000048547491203337398558987063953054
+x^-15*  0.00000074114812516008642387127350420916
+x^-16* -0.0000038309228524031537802224892859947
+x^-17* -0.0000025748868434074919176037441576404
+x^-18*  0.0000022951530139076283185894597305398
+x^-19*  0.0000036906301942910425788903290199039
+x^-20* -0.00000035885452386419832591366749556753}