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 :

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

Make an asymptotic series approximation for the Böttcher function:

Scope  (3)

MandelbrotSetBoettcher threads itself element-wise over lists:

MandelbrotSetBoettcher works on all kinds of numbers:

Evaluate to high precision:

Options  (1)

MaxIterations  (1)

Sometimes MaxIterations needs to be increased:

Applications  (3)

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

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

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

Neat Examples  (1)

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

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_2025_mandelbrotsetboettcher, author="Wolfram Research", title="{MandelbrotSetBoettcher}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mandelbrotsetboettcher, organization={Wolfram Research}, title={MandelbrotSetBoettcher}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetBoettcher.html}, note=[Accessed: 22-February-2025 ]}