Our analysis of the asymptotic power of tests for signal detection is based on a study of the asymptotic properties of the likelihood ratio processes { L ( h ; λ ) ; h ∈ ( R $^{+}$) $^{r}$} and { L ( h ; µ ) ; h ∈ ( R $^{+}$) $^{r}$} . First, we will focus on the key terms in the expres- sions (4) and (5), which are the integrals over the unitary group. These in- tegrals are special cases of the complex hypergeometric function $_{0}$F (1) 0 ( A, B ) = ∫ U ( p $_{)}$e tr ( AGBG − 1 ) (d G ) , where A and, possibly, B are rank-deficient. In the next section, we derive a formula for $_{0}$F ( α ) 0 ( A, B ) with rank-deficient A and B that links this function to a hypergeometric functions of full-rank matrix arguments of lower dimensions. We do not restrict attention to the case α = 1 because, as discussed in the introduction, other cases constitute independent interest. Let us first provide a necessary background on hypergeometric functions. Let A and B be Hermitian p × p matrices over real, complex, or quaternion division algebra. The eigenvalues of such matrices are real and we will denote them as a = ( a$_{1}$, ..., a$_{p}$ ) and b = ( b$_{1}$, ..., b$_{p}$ ). The hypergeometric function $_{0}$F ( α ) 0 ( A, B ) is defined as (see, for example, Koev and Edelman, 2006) where C ( α ) κ ( A ) = C ( α ) κ ( a ) , C ( α ) κ ( B ) = C ( α ) κ ( b ) and C ( α ) κ ( I$_{p}$ ) = C ( α ) κ (1 , ..., 1) are normalized Jack polynomials (Macdonald, 1995, chapter VI, § 10), and the inner sum runs over all partitions κ of k, that is over all non-increasing sequences of non- negative integers κ = ( κ$_{1}$, κ$_{2}$, ... ) such that κ$_{1}$ + κ$_{2}$ + ... = k. The normalization of 9 $_{0}$F ( α ) 0 ( A, B ) = ∞ ∑ k =0 ∑ κ ⊢ k 1 k ! C ( α ) κ ( A ) C ( α ) κ ( B ) C ( α ) κ ( I$_{p}$ ) , (6) 3 Contour integral representation for $_{0}$F ( α ) 0 ( A, B )