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 et al.}, 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.

翻译：本文重新审视两样本Behrens-Fisher问题——即当两个正态总体方差未知且不等时检验均值相等性——并推导出经典检验统计量零分布的紧凑表达式。关键步骤是通过Mellin-Barnes因子分解，将独立卡方变量加权和平方根解耦，从而将具有挑战性的二维积分简化为可处理的单围道积分。闭合围道得到留数级数，该级数在任一样本自由度为奇数时终止。通过互补的Euler-Beta简化，将密度函数识别为具有显式参数的Gauss超几何函数，得到数值稳定的形式，该形式在方差相等时退化为Student's $t$分布。Ramanujan主定理提供了精确的逆幂尾系数，该系数限定了Lugannani-Rice鞍点近似误差并支持可靠的尾部分析。我们的结果包含了Nel等人推导的超几何密度函数，并通过简洁的累积分布函数和解析尾部展开式进行扩展；他们的代数特例与我们的截断留数级数一致。利用推导的表达式，我们在广泛的样本量和方差比网格上编制了精确的双侧临界值表，揭示了众所周知的Welch近似从保守转向自由的参数曲面，并量化了其最大尺寸畸变。