Function Repository Resource:

# Tetration

Compute the order-k tetration of a number

 ResourceFunction["Tetration"][a,k] computes the order-k tetration of the number a.

## Details and Options

ResourceFunction["Tetration"][a,Infinity] gives a finite value if and only if a is in the convergence interval .
ResourceFunction["Tetration"][a,Infinity] diverges when .
k must be a non-negative integer or Infinity.
a must be a number different from 0.

## Examples

### Basic Examples (4)

Tetration of 2 with order 3:

 In[1]:=
 Out[1]=

Symbolic tetration:

 In[2]:=
 Out[2]=

Infinite tetration of a real number:

 In[3]:=
 Out[3]=

Infinite tetration of :

 In[4]:=
 Out[4]=
 In[5]:=
 Out[5]=

### Scope (3)

Compute infinite tetrations of real numbers inside the convergence interval:

 In[6]:=
 Out[6]=

Compute tetrations of real numbers within the divergence interval :

 In[7]:=
 Out[7]=

Compute tetrations of real numbers within the divergence interval :

 In[8]:=
 Out[8]=

### Properties and Relations (5)

Behavior of the tetration of real numbers inside the divergence interval :

 In[9]:=
 Out[9]=

Behavior of infinite tetration of real numbers inside the convergence interval :

 In[10]:=
 Out[10]=

Evolution and stabilization of the tetration curve inside the convergence interval for increasing k in a single plot:

 In[11]:=
 Out[11]=

Tetration convergence curve:

 In[12]:=
 Out[12]=

Behavior of the tetration of real numbers within the divergence interval :

 In[13]:=
 Out[13]=

The highly divergent property of tetration is easy to see from the previous plots.

### Neat Examples (5)

Real part plot of tetration order 3 for a range of complex values:

 In[14]:=
 In[15]:=
 Out[15]=

Imaginary part plot of tetration order 2:

 In[16]:=
 Out[16]=

Norm plot of tetration order 3:

 In[17]:=
 Out[17]=

Imaginary part of tetration order 2 of a number ez, where z is a complex number:

 In[18]:=
 Out[18]=

Real part of tetration order 2 of a number ez, where z is a complex number:

 In[19]:=
 Out[19]=

### Possible Issues (3)

Tetration of a=0 is Indeterminate:

 In[20]:=
 Out[20]=

Computing tetration for large values of k within the divergence interval may cause overflow, because the tetration exceeded the maximum machine number. You can obtain the maximum k for a given a as such:

 In[21]:=

Maximum tetration for a=12:

 In[22]:=
 Out[22]=

If you want to avoid long execution times, use the function N to get an approximated result:

 In[23]:=
 Out[23]=

Large values of k cause an overflow:

 In[24]:=
 Out[24]=

Infinite tetration does not exists outside the convergence interval:

 In[25]:=

## Publisher

WolframSpecialProjects

## Version History

• 1.0.0 – 22 October 2020