Definition:Thomae Function
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Definition
The Thomae function $D_M: \R \to \R$ is the real function defined as:
- $\forall x \in \R: \map {D_M} x = \begin {cases} 0 & : x = 0 \text { or } x \notin \Q \\ \dfrac 1 q & : x = \dfrac p q : p \perp q, q > 0 \end {cases}$
where:
- $\Q$ denotes the set of rational numbers
- $p \perp q$ denotes that $p$ and $q$ are coprime (that is, $x$ is a rational number expressed in canonical form)
Also known as
The Thomae function is also seen styled as Thomae's Function.
It has several names in the literature:
- the modified Dirichlet function after Johann Peter Gustav Lejeune Dirichlet
- the (small) Riemann function after Georg Friedrich Bernhard Riemann
- the popcorn function
- the raindrop function
- the countable cloud function
- the ruler function (although strictly speaking the ruler function is a restriction of this to the dyadic rationals)
- Stars over Babylon (coined by John Horton Conway)
Also see
- Results about the Thomae function can be found here.
Source of Name
This entry was named for Carl Johannes Thomae.
Sources
- Weisstein, Eric W. "Dirichlet Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletFunction.html