Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$:
$\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, dS}{\int_{S}\, dS} = \frac{1}{4\pi} \int_{\theta=0}^{\theta=\pi}\int_{\phi=0}^{\phi=2\pi} f(\vec{x})\, \sin(\theta)\, d\phi\,d\theta $
where $\phi$ is the azimuth. This is relatively straight-forward for $\vec{x}\in\mathbb{R}_3$.
However, if my vector elements are now complex, $\vec{z}\in\mathbb{C}_3$, is there an equivalent spherical transform? That is, while I can define:
$\begin{align*} z_1 = & r \sin(\theta)\cos(\phi)\\ z_2 = & r \sin(\theta)\sin(\phi)\\ z_3 = & r \cos(\theta) \end{align*}$
where $z_1,z_2,z_3,r,\theta,\phi\in\mathbb{C}$, does the surface element $dS = r^2 \sin(\theta)\, d\theta\, d\phi$ still hold for an integral over the complex sphere $\vec{z}^{\ast}\cdot \vec{z}=R^2$? What would my limits of integration be, given that $\theta,\phi\in\mathbb{C}$?
I am open to any suggestions. Ultimately, I would like to average a real-valued scalar field $f(\vec{z})$ over a complex sphere of $\vec{z}^{\ast}\cdot \vec{z}=R^2$.
Thank you!