Let $u_n$ denote the sample correlation coefficient for $n$ observations from a bivariate, equal variance, complex Gaussian distribution. In this note we derive the exact distribution of $u_n$ by extending a method of Mehta and Gurland to the complex case. The asymptotic behavior of $E|u_n|^k$ as $n \rightarrow \infty$ is determined via the method of steepest descent. Applicability of the results to the analysis of certain estimators of spectral parameters of stationary time series is discussed.
"On the Correlation Coefficient of a Bivariate, Equal Variance, Complex Gaussian Sample." Ann. Math. Statist. 43 (6) 2000 - 2003, December, 1972. https://doi.org/10.1214/aoms/1177690873