For the special case (d=2), we may regard (mathbb{R}^{2} cong mathbb{C}), so that (omega) is a complex

Question:

For the special case $d=2$, we may regard $\mathbb{R}^{2} \cong \mathbb{C}$, so that $\omega$ is a complex number and $e$ is a unit complex number. Show that the weighted spherical distribution with weight proportional to the degree $k$ harmonic perturbation

\[
1+\Re\left(a \bar{e}^{k}ight) \sin ^{k} \theta
\]

is transformed to

\[
\frac{\Gamma(d)}{\pi^{d / 2} \Gamma(d / 2)} \frac{d \omega}{\left(1+\|\omega\|^{2}ight)^{d}} \times\left(1+\frac{2 \Re\left(a \bar{\omega}^{k}ight)}{\left(1+|\omega|^{2}ight)^{k}}ight)
\]

and is positive for $|a| \leq 1$.