The Behrens--Fisher problem revisited

We revisit the two-sample Behrens--Fisher problem -- testing equality of means when two normal populations have unequal, unknown variances -- and derive a compact expression for the null distribution of the classical test statistic. The key step is a Mellin--Barnes factorization that decouples the square root of a weighted sum of independent chi-square variates, thereby collapsing a challenging two-dimensional integral to a tractable single-contour integral. Closing the contour yields a residue series that terminates whenever either sample's degrees of freedom is odd. A complementary Euler--Beta reduction identifies the density as a Gauss hypergeometric function with explicit parameters, yielding a numerically stable form that recovers Student's $t$ under equal variances. Ramanujan's master theorem supplies exact inverse-power tail coefficients, which bound Lugannani--Rice saddle-point approximation errors and support reliable tail analyses. Our result subsumes the hypergeometric density derived by Nel {\etal}, and extends it with a concise cdf and analytic tail expansions; their algebraic special cases coincide with our truncated residue series. Using our derived expressions, we tabulate exact two-sided critical values over a broad grid of sample sizes and variance ratios that reveal the parameter surface on which the well-known Welch's approximation switches from conservative to liberal, quantifying its maximum size distortion.