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examples of characteristic functions
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(Example)
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In this entry, we give some examples of characteristic functions written in closed form (i.e., not piecewise defined).
We use the following conventions:
Below is a chart of characteristic functions:
| $X$ |
$A$ |
$\chi_A(x)$ |
| $\mathbb{Z}$ |
evens |
$\ds \frac{1+(-1)^x}{2}$ |
| $\mathbb{Z}$ |
odds |
$\ds \frac{1-(-1)^x}{2}$ |
| $\mathbb{R}$ |
positive reals |
$\ds \frac{|x|+x}{2}$ |
| $\mathbb{R}$ |
negative reals |
$\ds \frac{|x|-x}{2}$ |
| $\mathbb{R}$ |
$\mathbb{Z}$ |
$\lfloor \cos^2(\pi x) \rfloor$ |
| $\mathbb{R}$ |
$\mathbb{R}\setminus\mathbb{Z}$ |
$\lceil \sin^2(\pi x) \rceil$ |
| $\mathbb{C}\setminus\{0\}$ |
$\mathbb{R}\setminus\{0\}$ |
$\ds \left\lfloor \frac{\real{x}}{|x|} \right\rfloor$ |
| $\mathbb{C}\setminus\{0\}$ |
$\mathbb{C}\setminus\mathbb{R}$ |
$\ds \left\lceil \frac{\ipart{x}}{|x|} \right\rceil$ |
| $\mathbb{N}$ |
positive squarefree integers |
$\mu^2(x)$ |
| $\mathbb{N}\setminus\{1\}$ |
primes |
$\ds \left\lfloor \frac{\varphi(x)+1}{x} \right\rfloor$ |
| $\order_K\setminus\{0\}$ |
units of $\order_K$ |
$\ds \left\lfloor \frac{1}{|\norm_K(x)|} \right\rfloor$ |
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Cross-references: units, primes, squarefree, negative, reals, evens, norm, number field, ring of integers, Euler totient function, Möbius function, integers, positive, imaginary part, complex number, real part, ceiling function, floor function, absolute value, distance, subset, characteristic function, piecewise
This is version 8 of examples of characteristic functions, born on 2008-03-14, modified 2008-03-14.
Object id is 10401, canonical name is ExamplesOfCharacteristicFunctions.
Accessed 963 times total.
Classification:
| AMS MSC: | 26-00 (Real functions :: General reference works ) | | | 26A09 (Real functions :: Functions of one variable :: Elementary functions) | | | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) | | | 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers) |
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Pending Errata and Addenda
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