• Source: Dirichlet function

In mathematics, the Dirichlet function is the indicator function





1



Q





{\displaystyle \mathbf {1} _{\mathbb {Q} }}

of the set of rational numbers




Q



{\displaystyle \mathbb {Q} }

, i.e.





1



Q



(
x
)
=
1


{\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=1}

if x is a rational number and





1



Q



(
x
)
=
0


{\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=0}

if x is not a rational number (i.e. is an irrational number).






1



Q



(
x
)
=


{



1


x
∈

Q





0


x
∉

Q









{\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}


It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of a pathological function which provides counterexamples to many situations.




Topological properties






Periodicity


For any real number x and any positive rational number T,





1



Q



(
x
+
T
)
=


1



Q



(
x
)


{\displaystyle \mathbf {1} _{\mathbb {Q} }(x+T)=\mathbf {1} _{\mathbb {Q} }(x)}

. The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of




R



{\displaystyle \mathbb {R} }

.




Integration properties






See also


Thomae's function, a variation that is discontinuous only at the rational numbers




References

Kata Kunci Pencarian:

• Teori bilangan analitik
• Bilangan irasional
• Fungsi phi Euler
• Fungsi zeta Riemann
• Fungsi pembangkit
• Integral Riemann
• Integral
• Hipotesis Riemann
• Bilangan prima
• Turunan kedua
• Dirichlet function
• Dirichlet L-function
• Dirichlet eta function
• Dirichlet beta function
• Peter Gustav Lejeune Dirichlet
• Thomae's function
• Periodic function
• Dirichlet series
• Nowhere continuous function
• Generating function