  In this paper, on $\mathbb{D}$ we define Cowen-Douglas function introduced by
Cowen-Douglas operator $M_\phi^*$ on Hardy space $\mathcal{H}^2(\mathbb{D})$
and we give a sufficient condition for Cowen-Douglas function, where
$\phi\in\mathcal{H}^\infty(\mathbb{D})$. Moreover, we give some applications of
Cowen-Douglas function on chaos, such as application on the inverse chaos
problem for $\phi(T)$, where $\phi$ is a Cowen-Douglas function and $T$ is the
backward shift operator on $\mathcal{L}^2(\mathbb{N})$.
