examples of characteristic functions

closed form

In this entry, we give some examples of characteristic functions written in closed form (i.e. (http://planetmath.org/Ie), not piecewise defined).

We use the following conventions:

• $X$ denotes the domain (http://planetmath.org/Domain) of the characteristic function

• $A$ denotes a subset of $X$

• ${\chi }_A$ denotes the characteristic function of $A$

• $|\cdot |$ denotes distance from $0$ (e.g. (http://planetmath.org/Eg) absolute value or modulus (http://planetmath.org/Modulus2))

• $\lfloor \cdot \rfloor$ denotes the floor function

• $\lceil \cdot \rceil$ denotes the ceiling function

• $\mathrm{Re}$ denotes the real part of a complex number

• $\mathrm{Im}$ denotes the imaginary part of a complex number

• $\mathbb{N}$ denotes the positive integers

• $\mu$ denotes the Möbius function

• $\phi$ denotes the Euler totient function

• ${𝒪}_K$ denotes the ring of integers of a number field $K$

• ${\mathrm{N}}_K$ denotes the norm on a number field $K$

Below is a of characteristic functions: