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  "body": "\n## Summary\n\nAn outer function is a special holomorphic function on the [[unit disc|8G5ABXRK85]].\nIt is determined, up to a constant of modulus $1$, by the modulus of its boundary values.\nIt is the exponential of the holomorphic function whose real part is the Poisson integral of the logarithm of that boundary modulus.\nOuter functions are the other half of the [[inner-outer factorization|X6ZQPAQF76]] of functions in [[Hardy spaces|PNNNFJVF6W]].\n\n## Explanations\n\n### L1 human\n\nLet $\\varphi$ be a positive weight on the circle whose logarithm is integrable.\nThere is a holomorphic function with $\\abs{F} = \\varphi$ on the circle that is as large as possible inside the disc.\nIt has no zeros, and $\\log \\abs{F}$ is the harmonic function with boundary values $\\log \\varphi$.\nThat function is the outer function with modulus $\\varphi$.\nFor example, $F(z) = 2 + z$ is outer: it has no zeros in the closed disc.\nBy contrast $z$ is not outer; it is [[inner|NN8XH5RJYB]], and the outer function with the same boundary modulus is the constant $1$.\n",
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