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  "body": "\n## Definition\n\nThe *beta function* is the function $B \\colon (0, \\infty) \\times (0, \\infty) \\to \\R$ given by the [[improper integral|0VCTBJMEJF]]\n$$B(x, y) = \\int_0^1 t^{x - 1} (1 - t)^{y - 1}\\, dt.$$\nThe integral converges for all real $x \u003e 0$ and $y \u003e 0$.\n\n## Remarks\n\nExample: $B(\\frac12, \\frac12) = \\int_0^1 \\frac{dt}{\\sqrt{t(1-t)}}$.\nThe substitution $t = \\sin^2 \\theta$ gives $dt = 2 \\sin\\theta \\cos\\theta\\, d\\theta$ and $\\sqrt{t(1-t)} = \\sin\\theta\\cos\\theta$, so $B(\\frac12, \\frac12) = \\int_0^{\\pi/2} 2\\, d\\theta = \\pi$.\n\nNon-example of a valid argument: for $x = 0$ and $y = 1$ the integral is $\\int_0^1 t^{-1}\\, dt = \\infty$, so $(0, 1)$ is outside the domain.\n\nThe beta function is symmetric, $B(x, y) = B(y, x)$, and it satisfies $B(x, y) = \\frac{\\Gamma(x)\\Gamma(y)}{\\Gamma(x + y)}$, where $\\Gamma$ is the [[gamma function|0V3FZMVB4K]].\nFor positive integers $m, n$ this gives $B(m, n) = \\frac{(m-1)!\\,(n-1)!}{(m + n - 1)!}$.\nThe same integral defines $B(x, y)$ for complex $x, y$ with positive real parts.\n",
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