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 diverges 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.


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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:

Introduced in 2014