Numerically, when I try to minimize the error \( \\[15pt]
{\Delta^2=\sum_{}^{}{(^xa \,-\, a^{^{x-1}a})}} \), I get similar coefficients to the ones posted by Sheldonison.
But I can also make it converge to different values of the first derivative at zero, or force some restriction.
Does that means that there are more than one solution to tetration? (a finite or infinite number of solutions)
There is needed an additional restriction when defining what tetration is?
Here are the derivatives I got for each base:
![[Image: gtOTsls.jpg?1]](http://i.imgur.com/gtOTsls.jpg?1)
For bases a>1, I do ever converge to the same values. The same goes for bases between 0.1 and 0.6.
The other derivatives I get are highly variable, even by sign.
I wonder if in that range there are more than a single "correct" derivative at zero, or if I just get artifacts due to local minima when minimizing the error.
These are the derivatives that I get for bases a>1
\( \\[15pt]
{\frac{\mathrm{d} \,^xa}{\mathrm{d} x}\mid_{x=0} \,=\, a_1 \,=\, \,-\,0.6452164561.a^6 \,+\, 7.6244443943.a^5 \,-\, 37.0680451104.a^4 \,+\, 94.9910696302.a^3 \,-\, 135.6878517026.a^2 \,+\, 103.3197783493.a^ \,-\, 32.5227109180} \)
I wonder if there are some tool on Internet capable of proposing different equations matching it.
For example, this website:
https://isc.carma.newcastle.edu.au/advanced
Finds that the derivative got by Sheldonison for base e (a1=1.09176735125832) is nearly imaginary part of
\( \\[15pt]
{ i-((7- i\,.19)^{\frac{1}{16}}) \,=\, -1.2033454171730380657324537148862244072935639001352 + i \,.\, 1.0917673511359367268632631345613715697821843358013} \)
Meanwhile, this place
http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html
Finds
\( \\[15pt]
{(atan(1/2)*Cahen +exp(-1/2*Pi))/atan(1/2) \,=\, 1.0917673852236835355691837676191405901326411536231} \)
where the Cahen constant is
\( \\[15pt]
{C = \sum\frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.64341054629.} \)
{\Delta^2=\sum_{}^{}{(^xa \,-\, a^{^{x-1}a})}} \), I get similar coefficients to the ones posted by Sheldonison.
But I can also make it converge to different values of the first derivative at zero, or force some restriction.
Does that means that there are more than one solution to tetration? (a finite or infinite number of solutions)
There is needed an additional restriction when defining what tetration is?
Here are the derivatives I got for each base:
For bases a>1, I do ever converge to the same values. The same goes for bases between 0.1 and 0.6.
The other derivatives I get are highly variable, even by sign.
I wonder if in that range there are more than a single "correct" derivative at zero, or if I just get artifacts due to local minima when minimizing the error.
These are the derivatives that I get for bases a>1
Code:
2.718281828 1.09562140557617
2.7 1.09143052409660
2.6 1.06822431122401
2.5 1.04307153925875
2.4 1.01659572039055
2.3 0.98764740646929
2.2 0.95793010260192
2.1 0.92527675767840
2 0.89028921820469
1.9 0.85186626289898
1.8 0.80938670730807
1.7 0.76259264471247
1.6 0.70969066859471
1.5 0.64958552115556
1.444667861 0.61224627377907
1.42 0.59451852096765
1.4 0.57955308252839
1.35 0.53981702850685
1.3 0.49599575015175
1.25 0.44716716706888
1.2 0.39190949073500
1.15 0.32798289733884
1.1 0.25143159620504
1.05 0.15333702806960
1.01 0.04148265011217
1 0.00000000000000\( \\[15pt]
{\frac{\mathrm{d} \,^xa}{\mathrm{d} x}\mid_{x=0} \,=\, a_1 \,=\, \,-\,0.6452164561.a^6 \,+\, 7.6244443943.a^5 \,-\, 37.0680451104.a^4 \,+\, 94.9910696302.a^3 \,-\, 135.6878517026.a^2 \,+\, 103.3197783493.a^ \,-\, 32.5227109180} \)
I wonder if there are some tool on Internet capable of proposing different equations matching it.
For example, this website:
https://isc.carma.newcastle.edu.au/advanced
Finds that the derivative got by Sheldonison for base e (a1=1.09176735125832) is nearly imaginary part of
\( \\[15pt]
{ i-((7- i\,.19)^{\frac{1}{16}}) \,=\, -1.2033454171730380657324537148862244072935639001352 + i \,.\, 1.0917673511359367268632631345613715697821843358013} \)
Meanwhile, this place
http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html
Finds
\( \\[15pt]
{(atan(1/2)*Cahen +exp(-1/2*Pi))/atan(1/2) \,=\, 1.0917673852236835355691837676191405901326411536231} \)
where the Cahen constant is
\( \\[15pt]
{C = \sum\frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.64341054629.} \)