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  "body": "\n## Statement\n\nLet $x$ and $y$ be complex numbers with $\\operatorname{Re} x \u003e 0$ and $\\operatorname{Re} y \u003e 0$.\nLet $B(x, y) = \\int_0^1 t^{x-1} (1-t)^{y-1}\\, dt$ be the [[beta function|8K9YQ1YGVS]] and let $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nThen\n$$B(x, y) = \\frac{\\Gamma(x)\\, \\Gamma(y)}{\\Gamma(x + y)}.$$\n\n## Remarks\n\nWrite $\\Gamma(x)\\Gamma(y)$ as the double integral $\\int_0^\\infty \\int_0^\\infty u^{x-1} v^{y-1} e^{-u-v}\\, du\\, dv$.\nThe change of variables $u = st$, $v = s(1-t)$ with $s \u003e 0$ and $0 \u003c t \u003c 1$ has Jacobian $s$.\nIt turns the double integral into $\\int_0^\\infty s^{x+y-1} e^{-s}\\, ds \\cdot \\int_0^1 t^{x-1}(1-t)^{y-1}\\, dt = \\Gamma(x+y)\\, B(x, y)$.\nAbsolute convergence, which needs $\\operatorname{Re} x \u003e 0$ and $\\operatorname{Re} y \u003e 0$, justifies the change of order.\n\nExample: $B(\\frac12, \\frac12) = \\Gamma(\\frac12)^2 / \\Gamma(1) = (\\sqrt{\\pi})^2 = \\pi$.\nFor positive integers $m, n$ the formula gives $B(m, n) = \\frac{(m-1)!\\,(n-1)!}{(m+n-1)!}$.\n\nWithout the condition on the real parts the integral defining $B$ diverges, for example at $x = 0$.\nThe right side still makes sense there, and it is used to extend $B$ to a meromorphic function of $x$ and $y$.\n\n## Explanations\n\n### L1 human\n\nThe beta function is an integral over the interval from $0$ to $1$, and the gamma function is an integral over the whole half line.\nThis theorem says that the beta function is a simple quotient of gamma values.\nSo every beta integral can be computed from values of the gamma function.\nFor whole numbers it becomes a formula with factorials.\n\n### L2 human\n\nThe formula is the bridge between the two Euler integrals.\nIt shows at once that $B(x, y) = B(y, x)$ and gives the analytic continuation of $B$.\nWith $t = \\sin^2 \\theta$ it computes integrals $\\int_0^{\\pi/2} \\sin^{a}\\theta \\cos^{b}\\theta\\, d\\theta$, and with $x = y = \\frac12$ it gives $\\Gamma(\\frac12) = \\sqrt{\\pi}$.\nIn probability it gives the normalising constant of the [[beta distribution|0EW824Z7G2]].\n",
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