# examples of characteristic functions

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

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

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

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

• $\mu$ denotes the Möbius function

• $\varphi$ denotes the Euler totient function

• $\mathcal{O}_{K}$ denotes the ring of integers of a number field $K$

• $\operatorname{N}_{K}$ denotes the norm on a number field $K$

Below is a of characteristic functions:

Title examples of characteristic functions ExamplesOfCharacteristicFunctions 2013-03-22 17:54:28 2013-03-22 17:54:28 Wkbj79 (1863) Wkbj79 (1863) 11 Wkbj79 (1863) Example msc 11R04 msc 11A25 msc 26A09 msc 26-00