TITLE: What is the domain of $x^x$ when $ x<0$
QUESTION [6 upvotes]: I know that $x^x$ for all $x>0$ 
but what is negative values for that function which give a real number
for example  $$f(-1)=(-1)^{-1}=-1\in R$$
I try to put sequence for that but i faild 
is there any help 
thanks for all

REPLY [3 votes]: $x^x$ is well defined as a real function for $$(0,\infty) \cup \{ -\frac{m}{2n+1}| m, n \in {\mathbb N} \}$$

REPLY [2 votes]: Set $y = -x$.
Thus, we have:
$x^x = (-y)^{-y}$ when $y \ge 0$
Or equivalently,
$$x^x = \frac{1}{(-y)^{y}}$$
Which can be simplified to:
$$x^x = \frac{1}{(-1)^{y} y^y}$$
Therefore, the domain of $x^x$ consists of both reals and complex numbers depending on the value of $(-1)^y$ or to be more precise depending on the value of $y$.