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preliminary analysis indicates that the asymptotic power of the likelihood ratio test based on the information contained in λ is close to the asymptotic power envelope. In contrast, we find that the asymptotic powers of various previously proposed tests are well below the envelope. 

The central technical result of this paper is the contour integral representa- tion of the likelihood ratio. To derive such a representation, we establish what we believe to be a novel formula for the hypergeometric functions of two matrix argu- ments $_{0}$F ( α ) 0 ( A, B ) , where a p × p matrix A has rank r < p, so that, without loss of generality, only its upper-left r × r block A is non-zero. Such functions appear as a key term in the explicit expressions for the joint density of the eigenvalues of Wishart matrices with spiked covariance parameter. In Lemma 1, we show that 

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where Z = diag ( z$_{1}$, ..., z$_{r}$ ) is an auxiliary matrix, and ω ( α $^{)}$( · ) is a simple function of A, B, and Z . This formula expresses the hypergeometric function of high- dimensional arguments as a repeated contour integral of a hypergeometric function of low-dimensional arguments, which is convenient for analysis. 

For the special case r = 1 , (1) reduces to the formula that has been recently derived in Mo (2011) and, independently, in Wang (2012) and Onatski et al (2012) (see also Forrester, 2011 for a short derivation). Our method of proof is different from the methods used by these authors. It is based on the orthogonality of Jack polynomials with respect to the torus scalar product (Macdonald (1995), Chapter VI, § 10). 

Although our analysis of signal detection in complex data requires only the formula for $_{0}$F (1) 0 ( A, B ) , we establish (1) for all α = 2 /β, where β is a positive 

