  It is proposed a closed-form expression of directivity for an arbitrary
volumetric antenna arrays using a general element pattern expression of type
$\sin^u{(\theta)}\cos^v{(\theta)}$, with $v > -\frac{1}{2}$ and $u > -1$, and
$u, v \in \mathbb{Z}$. Variations of this expression for different values of
$v$ and $u$ are analyzed from the analytical and numerical perspectives. The
parameters found in the closed-form expression are related to the order $v$ and
$u$ of the element patterns, the rectangular spatial coordinate of each antenna
element, the magnitude and phase excitation coefficients (complex excitation)
of all elements, and the desired angle in spherical coordinates $(\theta_0,
\phi_0)$. The expression found in this work has been validated by numerical
results, considering distinct configuration scenarios.
