Tetration to base 2

  Fig.1. \(y=\mathrm{tet}_b(x) \) for vaious \(b\). See , Fig.17.1.

   Fig.2.   \( u+\mathrm i v=\mathrm{tet}_2(x\!+\!\mathrm i y)\)

Tetration to base 2 (Tet2 or \(\mathrm {tet}_2\) ) is special case of Tetration \(\mathrm {tet}_b\) while the base \( b\!=\!2\).

For real values of argument, tetration to base 2 is compared to tetrations to other bases in Fig.1.

Complex map of tetration to base 2 is shown in Fig.2.

Equations

Tetration to base 2 appears as superfunction of exponential to base 2.

Function \(~ T(z)=\exp_2(z)=\exp(\ln(2)z)=2^z \)

is considered as Transfer function.

The Superfunction \(F\) is solution of the Transfer equation

\( F(z\!+\!1)=T(F(z)) \)

The solution \(F\) is called Tetration to base 2 and denoted \(F=\mathrm{tet}_2\), if it satisfies condition

\(F(0)=1\)

and additional conditions:

\(F(z) \) is real holomorphic in the whole complex \(z\) plane except halfline \(z\le -2\).

\( F(z)\) approaches the fixed point \(L\) of logarithm to base 2 in the upper half plane \(\Im(z)>0\)) and \(L^*\) in the lower half plane. This fixed point can be espressed with function Filog:

\( L=\) Filog\((\ln(2)) \) \(\approx 0.8246785461420740 + 1.5674321238496474 \, \mathrm i \)

This requirement can be written as follows:

\( \displaystyle \lim_{y\rightarrow + \infty} F(x\!+\!\mathrm i y)=L \)
\( \displaystyle \lim_{y\rightarrow - \infty} F(x\!+\!\mathrm i y)=L^* \)

\( \displaystyle \lim_{x\rightarrow -\infty} F(x\!+\!\mathrm i y)=L \)     for \(y\!>\!0\)
\( \displaystyle \lim_{x\rightarrow -\infty} F(x\!+\!\mathrm i y)=L^* \)     for \(y\!<\!0\)

The conditions above are a little bit redundant, some of them follow from the others. Up to year 2020, it is not yet established, which of these conditions should be kept in the definition of tetration, and which of them should be deduced as theorems.

Properties

   Fig.3. \( u+\mathrm i v = \mathrm{tet}_2(x\!+\!\mathrm i y ) - \log_2(2\!+\!x\!+\!\mathrm i y) \)

At integer values of argument, beginning with \(-1\), tetration to base 2 has integer values; in particular,
\(\mathrm{tet}_2(-1)=0\)
\(\mathrm{tet}_2(0)=1\)
\(\mathrm{tet}_2(1)=2\)
\(\mathrm{tet}_2(2)=4\)
\(\mathrm{tet}_2(3)=16\)
\(\mathrm{tet}_2(4)=2^{16}=65536\)
\(\mathrm{tet}_2(5)=2^{65536}\)   is already too large for the decimal representation

At \(z =-2\), function \(\mathrm{tet}_2(z)\) has logarithmic singularity and branch point, but function

\( \tau(z)= \mathrm{tet}_2(z)-\log_2(2\!+\!z) \)

looks smooth at this point; the cut begins only with \(z=-3\), and goes to minus infinity. Complex map of function \(\tau\) is shown in Fig.3.

In the upper halfplane, tetration to base 2 approaches the displaced Kneser superexponential Tek\(_2\),

\( \mathrm{tet}_2 \approx\) Tek\(_2(z-Z_\mathrm k) \) ,   \( \Im(z)\rightarrow \infty \)

For base \(b=2\), the Kneser parameter

\(Z_{\mathrm k} \approx 1.23806941407137 + 1.24816132484493 \,\mathrm i \)

Implementation

The primary implementation of Tetration to base 2 is based on the Cauchi integral along the special rectangular contour

The secondary implementation (faster and more precise) uses the Kneser expansion for large values of imaginary part of the argument and two Taylor expansions for moderate values of the imaginary pat of the argument, in analogy with implementation of the natural tetration

The primitive complex double routine in C++ is loaded as Tet2.cin.

Also, the imlementation in Mathematica, based on the modified Kneser expansion is described by William Paulsen and Samuel Cowgill for various base \(b\) (not only for \(b\!=\!2\) . The online evaluator is arranged

Keywords

Filog, Superfunction, Superfunctions, Tek2, Tet2, Tetration