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MandelbrotSetBoettcher[z]

gives the Böttcher coordinate of z with respect to the Mandelbrot set.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
MaxIterations  
Applications  
Neat Examples  
See Also
Related Guides
History
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MandelbrotSetBoettcher

MandelbrotSetBoettcher[z]

gives the Böttcher coordinate of z with respect to the Mandelbrot set.

Details and Options

  • The Mandelbrot set is the set of all complex numbers for which the sequence does not diverge to infinity when starting with .
  • With the option MaxIterations->m, the sequence will be iterated at most m times to approximate .
  • The default setting is MaxIterations->100.
  • MandelbrotSetBoettcher can be evaluated to arbitrary numerical precision.
  • MandelbrotSetBoettcher automatically threads over lists.

Examples

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Basic Examples  (3)

Get the Böttcher coordinate of :

Wolfram Language code: MandelbrotSetBoettcher[1. + I]

Since is in the Mandelbrot set, the result is undefined:

Wolfram Language code: MandelbrotSetBoettcher[.1]

Make an asymptotic series approximation for the Böttcher function:

Wolfram Language code: Series[MandelbrotSetBoettcher[z], {z, ∞, 3}]

Scope  (3)

MandelbrotSetBoettcher threads itself element-wise over lists:

Wolfram Language code: MandelbrotSetBoettcher[{1., I + 1.}]

MandelbrotSetBoettcher works on all kinds of numbers:

Wolfram Language code: data = MandelbrotSetBoettcher[#]& /@ {-2, -2.1979 + I, -2.9`2, 1 / 3, 0.26, 1 + I, 1 / Pi, N[1 / Pi, 100]}
Wolfram Language code: N /@ data

Evaluate to high precision:

Wolfram Language code: N[MandelbrotSetBoettcher[(1/3)], 50]

Options  (1)

MaxIterations  (1)

Sometimes MaxIterations needs to be increased:

Wolfram Language code: {MandelbrotSetBoettcher[0.250005, MaxIterations -> 1000], MandelbrotSetBoettcher[0.250005, MaxIterations -> 2000]}

Applications  (3)

Use the absolute value to display "equipotential lines" for the Mandelbrot set:

Wolfram Language code: ArrayPlot[Table[Sin[100 Abs[MandelbrotSetBoettcher[x + I * y]]], {y, -1, 1, .01}, {x, -2.6, .5, .01}]]

Use the argument to display "field lines" for the Mandelbrot set:

Wolfram Language code: ArrayPlot[Table[Sin[100 Arg@MandelbrotSetBoettcher[x + I * y]], {y, -1, 1, .01}, {x, -2.6, .5, .01}]]

Use the argument to display the "external ray" with angle :

Wolfram Language code: ttArg = Quiet[Table[If[MandelbrotSetMemberQ[x + I * y], -0.5, Boole[N[Pi 2 / 3 - 0.0004] < Arg[MandelbrotSetBoettcher[x + I * y]] < N[Pi 2 / 3 + 0.0004]]], {y, -1, 1, .01}, {x, -2.6, .5, .01}]]; ArrayPlot[ttArg]

Neat Examples  (1)

Approximate the inverse to the Böttcher function and apply it to the unit circle:

Wolfram Language code: 𝔻to𝕄Series[c_] = Normal[InverseSeries[Series[Series[MandelbrotSetBoettcher[c], {c, Infinity, 300}], {c, ∞, 100}]]];
Wolfram Language code: Show[{MandelbrotSetPlot[], ParametricPlot[ Evaluate[{Re[#], Im[#]}&[𝔻to𝕄Series[Exp[I φ]]]], {φ, -Pi, Pi}, PlotStyle -> Red]}]
Wolfram Research (2014), MandelbrotSetBoettcher, Wolfram Language function, https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html.

Text

Wolfram Research (2014), MandelbrotSetBoettcher, Wolfram Language function, https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html.

CMS

Wolfram Language. 2014. "MandelbrotSetBoettcher." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html.

APA

Wolfram Language. (2014). MandelbrotSetBoettcher. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html

BibTeX

@misc{reference.wolfram_2026_mandelbrotsetboettcher, author="Wolfram Research", title="{MandelbrotSetBoettcher}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_mandelbrotsetboettcher, organization={Wolfram Research}, title={MandelbrotSetBoettcher}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html}, note=[Accessed: 13-June-2026]}