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 paradox, accurate taylor series half iterate of eta not analytic at e sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/02/2011, 04:58 PM (This post was last modified: 06/03/2011, 02:12 PM by sheldonison.) For base exp(1/e), there have been many posts that the half-iterate of $\exp_\eta(z)$ is not analytic at z=e. Below I am posting a paradoxical accurate 30 term Taylor series, for that non-analytic half iterate of $\exp_\eta(z)$, developed at z=e. This series is double precision accurate, out to a radius of around 1. An example, with z=e-0.6, and this series puts out 2.1507815747789682, which is the correct result for the half iterate, generated via $\text{sexp}_\eta(\text{sexp}_\eta^{-1}(e-0.6)+0.5)=\text{sexp}_\eta(5.5342534224332571+0.5)$. The puzzle, is how is it possible to develop such a paradoxical accurate Taylor series, for a function which is not even analytic? A further complication, is that this Taylor series is required to seamlessly stitch together two different functions, the half iterate generated using $\text{sexp}_\eta(z)$, and the half iterate generated using the upper entire superfunction, $\text{cheta}(z)$. This is also the explanation for how it is possible. For a small enough series radius, it turns out the two half iterates can be seamlessly stitched together, with very little discontinuity, since as imag(z) increases, $\text{sexp}_\eta(z)$ exponentially converges to $\text{cheta}(z+k)$. And the imag(z) stitching value gets larger as the Taylor series radius gets smaller. I sampled this series at a radius of 1. $y=\text{sexp}^{-1}_\eta(e+i)=-3.3628841572099938 + 4.9027399771826196i$. At imag(y)=4.9, the two functions half iterates are already consistent to an accuracy of approximately 15 digits, which allows for the merged Taylor series. For real(z)=e, we use the half iterate generated from $\text{cheta}(z)$. The singularity and misbehavior occurs for the half iterate of cheta(z), when real(z)<=e, and similar misbehavior occurs for the half iterate of $\text{sexp}_\eta(z)$ for real(z)>=e. At e itself, both functions have singularities, but both functions agree that the half iterate of e=e. For real(z)=e, at smaller values of imag(z), paradoxically, the stitch is occurring for larger values of imag(y), where the stitch becomes more and more seamless. For the half iterate of $y=\text{sexp}^{-1}_\eta(e+0.5i)=-3.5937424498587124+10.344418312596874i$, where the two functions are consistent to 31 decimal digits! But at a larger sampling radius of r=1.5, the two half iterates are only consistent to approximately 11 decimal digits. So, within acceptable accuracy limits, it turns out it is possible to develop a Taylor series for the half iterate of $\text{sexp}_\eta(z)$, at e, where the function is not even analytic. Code:a0=   2.7182818284590452 a1=   1.0000000000000000 a2=   0.091969860292860588 a3=   0.0028194850674294418 a4=  -8.0085798390301461 E-18 a5=   0.0000047696976272632850 a6=  -0.00000051177982848104645 a7=   0.0000000038423117936633581 a8=   0.000000014046730882359691 a9=  -0.0000000030441687473700918 a10= -1.0220786581293779 E-11 a11=  1.6377605039633389 E-10 a12= -2.8311894064410302 E-11 a13= -8.1773414171527632 E-12 a14=  4.0125808678932662 E-12 a15=  3.0797537350253160 E-13 a16= -5.8574107605901771 E-13 a17=  3.8771237610746716 E-14 a18=  1.0016443539438077 E-13 a19= -2.2477472152925032 E-14 a20= -2.0211321002183050 E-14 a21=  8.7613198763649902 E-15 a22=  4.7138857780993590 E-15 a23= -3.4998961878228463 E-15 a24= -1.2239509050193108 E-15 a25=  1.5383321198806298 E-15 a26=  3.3098494797748605 E-16 a27= -7.5476712927521906 E-16 a28= -7.8877856873676488 E-17 a29=  4.1188557816475547 E-16 a30=  3.5020724883949410 E-18 « Next Oldest | Next Newest »