bettfunktion

Bettfunktion is not a standard term with a single, universally accepted meaning. In German-language contexts it is often a misspelling or variant of Beta-Funktion (Beta function), a classical special function in analysis. It can also refer, in topology and data analysis, to the Betti function or Betti curve, which relates to Betti numbers in a filtered or evolving space. Because of this ambiguity, the term should be interpreted from context.

Beta-Funktion: The Beta function, denoted Beta(x, y), is defined by the integral Beta(x, y) = ∫_0^1 t^(x−1) (1−t)^(y−1) dt for real parts of x and y positive. It satisfies Beta(x, y) = Γ(x)Γ(y)/Γ(x + y), where Γ denotes the Gamma function, and Beta(x, y) = Beta(y, x). The Beta function arises in probability, statistics (notably in the Beta distribution), and various integral computations, and it connects to many areas of analysis and combinatorics.

Betti function: In algebraic topology and persistent homology, the Betti function B_i(t) describes the i-th Betti number of a space X_t in a filtration X_0 ⊂ X_1 ⊂ ... or as a function of a scale parameter t. The i-th Betti number β_i equals the rank of the i-th homology group H_i and counts independent i-dimensional holes. The Betti function is typically piecewise constant, increasing or decreasing as features appear or disappear with changing t, and is often summarized by persistence diagrams or Betti curves.

Usage and differentiation: In writing, Betti function (or Betti number function) refers to topology-related concepts, while Beta function (with a capital B) is the well-established special function in analysis. Distinguishing them by spelling and context helps avoid confusion.