Function Repository Resource:

# FabiusF

Evaluate the Fabius function

Contributed by: Jan Mangaldan
 ResourceFunction["FabiusF"][x] computes the Fabius function.

## Details and Options

The Fabius function is an example of a nowhere-analytic, infinitely differentiable function.
For dyadic rational arguments, ResourceFunction["FabiusF"] automatically evaluates to exact values.
ResourceFunction["FabiusF"] is only defined for real arguments.
ResourceFunction["FabiusF"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (1)

Plot the Fabius function:

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### Scope (7)

Evaluate at an integer:

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Evaluate at a dyadic rational (a rational number whose denominator is a power of two):

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FabiusF does not automatically evaluate for arguments that are not a dyadic rational number:

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Evaluate numerically:

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Evaluate to arbitrary precision:

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The precision of the output tracks the precision of the input:

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### Properties and Relations (3)

The Fabius function for odd integer arguments can be expressed in terms of ThueMorse:

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Functional differential equation for the Fabius function:

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Higher derivatives of the Fabius function can be expressed in terms of the function itself:

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### Possible Issues (1)

FabiusF is undefined for complex numbers:

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### Neat Examples (1)

Plot a parametric function defined in terms of FabiusF:

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## Version History

• 2.0.0 – 16 March 2020
• 1.0.0 – 22 November 2019

## Author Notes

This submission is based on this original implementation on Stack Exchange: https://mathematica.stackexchange.com/a/154080.
The code is partly based on original code by Vladimir Reshetnikov.