An infinite series formulation for the multivariate alpha-mu joint probability density function with arbitrary correlation matrix and non-identically distributed variates is derived. The expression is exact and general and includes all of the results previously published in the literature concerning the distributions comprised by the alpha-mu distribution. The general expression is then particularized to an indeed very simple, approximate closed-form solution. In addition, a multivariate joint cumulative distribution function is obtained, again in simple, closed-form manner. As an application example, the exact and approximate performances of the selection combining scheme given in terms of the outage probability is shown. Approximate and exact results are very close to each other for small as well as medium values of correlation.914550