Exponential maps ^{[1][2][3] [4]}

tetration fractals ^[5]

Baker domains ^[6][7]

How to compute it

${\displaystyle Z=x+y*i}$

${\displaystyle \exp(Z)=e^{Z}}$

${\displaystyle \mathrm {Real(\exp(Z))} =\exp(x)\cos(y)}$ 

${\displaystyle \mathrm {Imag(\exp(Z))} =\exp(x)\sin(y)}$

What is the continous iteration of ${\displaystyle e^{x}-1}$ ?

"The function

 ${\displaystyle e^{x}-1}$ 

is one of the simpler applications of continuous iteration. The reason why is because regular iteration requires a fixed point in order to work, and this function has a very simple fixed point, namely zero: "^[8]

 ${\displaystyle e^{0}-1=0}$

Images

• commons Category:Exponential maps

See also

• Julia_and_Mandelbrot_sets_for_transcendental_functions by Gertbuschmann

References

1. ↑ THE EXPONENTIAL MAP IS CHAOTIC: AN INVITATION TO TRANSCENDENTAL DYNAMICS by ZHAIMING SHEN AND LASSE REMPE-GILLEN
2. ↑ Dynamics of exponential maps by Lasse Rempe
3. ↑ wikipedia : Exponential map (discrete dynamical systems)
4. ↑ Paper by N Fagella
5. ↑ Paul Bourke fractals tetration
6. ↑ On the Stability of Julia Sets of Functions having Baker Domains by Arnd Lauber ( 2004)
7. ↑ Approximation of Baker domains and convergence of Julia sets by Tania Garfias-Macedo aus Mexiko Stadt, Mexiko
8. ↑ Tetration FAQ by Henryk Trappman Andrew Robbins July 10, 2008