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  "body": "\n## Definition\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the [[unit disc|8G5ABXRK85]].\nLet $m \\ge 0$ be an integer.\nLet $(a_n)$ be a finite or infinite sequence of points of $\\mathbb{D} \\setminus \\set{0}$, with repetitions allowed, such that\n$$\\sum_n (1 - \\abs{a_n}) \u003c \\infty.$$\nThe *Blaschke product* with zeros $0$ (of order $m$) and $(a_n)$ is the function $B \\colon \\mathbb{D} \\to \\C$ given by\n$$B(z) = z^m \\prod_n \\frac{\\abs{a_n}}{a_n} \\, \\frac{a_n - z}{1 - \\overline{a_n} z}.$$\nIt is a *finite Blaschke product* when the sequence $(a_n)$ is finite.\n\n## Remarks\n\nUnder the condition above, the product converges uniformly on compact subsets of $\\mathbb{D}$.\nSo $B$ is [[holomorphic|0DG0Z72CQD]] with $\\abs{B} \\le 1$, and its zeros are exactly $0$ and the $a_n$, with multiplicity.\nMoreover the radial limit of $B$ has modulus $1$ at almost every point of the circle, so $B$ is an [[inner function|NN8XH5RJYB]].\nThe factor $\\abs{a_n}/a_n$ makes the $n$-th factor positive at $z = 0$; some authors also allow a constant factor of modulus $1$.\n\nExample: for $a \\in \\mathbb{D} \\setminus \\set{0}$, the single factor $B(z) = \\frac{\\abs{a}}{a} \\frac{a - z}{1 - \\bar a z}$ is a finite Blaschke product.\nFor $\\abs{z} = 1$ one has $\\abs{1 - \\bar a z} = \\abs{\\bar z - \\bar a} = \\abs{a - z}$, so $\\abs{B(z)} = 1$ on the circle.\n\nNon-example: the points $a_n = 1 - 1/n$ for $n \\ge 2$ fail the summability condition, since $\\sum 1/n = \\infty$.\nFor them the partial products tend to $0$ at every point of $\\mathbb{D}$, so no Blaschke product has these zeros.\nIn fact no bounded holomorphic function other than $0$ vanishes at all of them.\n",
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