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 \( \text{sexp}_\eta(z) \), and 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.
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 \( \text{sexp}_\eta(z) \), and 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