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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $U \\subseteq \\C^n$ be an open set, and let $f \\colon U \\to \\C$ be continuous.\nAssume that $f$ is holomorphic in each variable separately: for each $j$ and each fixed value of the other variables, $z_j \\mapsto f(z_1, \\dots, z_n)$ is [[holomorphic|0DG0Z72CQD]].\nThen $f$ is holomorphic on $U$.\nThat is, near each point $c \\in U$, $f$ is the sum of a convergent [[power series|0VP56B4EMX]] in $z_1 - c_1, \\dots, z_n - c_n$.\n\n## Remarks\n\nProof idea: on a closed polydisc $\\prod_j \\set{\\abs{z_j - c_j} \\le r_j}$ in $U$, apply the one-variable [[Cauchy integral formula|0DG10TVYPN]] in each variable in turn.\nContinuity makes the iterated integral an integral over the torus $\\prod_j \\set{\\abs{\\zeta_j - c_j} = r_j}$.\nExpanding the kernel $\\prod_j (\\zeta_j - z_j)^{-1}$ as a geometric series gives the power series.\nHartogs's theorem on separate analyticity shows that the continuity hypothesis can be dropped, but its proof is much harder.\nFor real variables the analogue fails: $xy/(x^2 + y^2)$, set to $0$ at the origin, is real-analytic in each variable but not continuous.\n\n## Explanations\n\n### L1 human\n\nIn several complex variables a function can be tested one variable at a time.\nIf it is continuous and holomorphic in each variable when the others are frozen, it is holomorphic in all of them together.\nThis makes it easy to check that many functions of several variables are holomorphic.\n\n### L2 human\n\nOsgood's lemma is often the first step in a course on several complex variables.\nWith it, the Cauchy formula on polydiscs gives power series expansions, Cauchy estimates and Weierstrass convergence theorems in $\\C^n$.\nIt also lets one define holomorphic functions of several variables either by power series or by separate holomorphy plus continuity.\n",
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