Definition: A function of the form
$$ z(x,y) = kx^{\alpha}y^{1-\alpha}, (x>0, y>0)$$
where $k>0$, $0 \leq \alpha \leq 1$, is called a Cobb-Douglas function.

Remark 1: Sometimes, also functions of the form $z(x,y)=kx^{\alpha}y^{\beta}$, with $k>0$, $\alpha>0$ and $\beta >0$ are called Cobb-Douglas functions. Given this definition also $z(x,y) = 4x^2y^4$ is a Cobb-Douglas function for example.

Remark 2: Cobb-Douglas functions often appear in microeconomics dues to desirable properties.

Example: An example of a Cobb-Douglas function is $U(x,y) = 2x^{\tfrac{1}{3}}y^{\tfrac{2}{3}}$.