I am working on deriving the normalization issue and face some challenges in finding the closed-form expression. Assume there is a complex vector $\mathbf{h}\in\mathcal{CN}^{P\times 1}$, where each entry independently follows the same complex Gaussian distribution with zero mean and variance $\gamma$. Is it possible to compute a closed-form solution for \begin{align} \mathbb{E}\left[ \frac{1}{\lVert \mathbf{h} \rVert^2} \right], \end{align} where $\mathbb{E}$ denotes expectation operation and $\lVert \cdot \rVert$ denotes the norm-2 operation, i.e., $\lVert \mathbf{h} \rVert^2 = \lvert h_1\rvert^2+\cdots+\lvert h_P\rvert^2$.
Truly appreciate any comments and suggestions!