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  "body": "\n## Statement\n\nLet $z_0 \\in \\C$, $w_0 \\in \\C^n$, and let $F$ be a function from a neighbourhood of $(z_0, w_0)$ in $\\C \\times \\C^n$ to $\\C^n$ that is [[holomorphic|0NAHXR1FRH]] in all its variables.\nThen there is $r \u003e 0$ such that the [[initial value problem|0FXDW992DS]]\n$$w'(z) = F(z, w(z)), \\qquad w(z_0) = w_0$$\nhas a holomorphic solution $w$ on the disc $\\abs{z - z_0} \u003c r$.\nAny two holomorphic solutions on a disc around $z_0$ agree on the smaller disc.\n\n## Remarks\n\nWriting $w(z) = \\sum_{k \\ge 0} a_k (z - z_0)^k$ and differentiating the equation repeatedly, each $a_k$ is a polynomial with nonnegative coefficients in $a_0 = w_0$ and the Taylor coefficients of $F$.\nSo the Taylor series of any holomorphic solution is determined by the equation, which gives uniqueness.\nCauchy's method of majorants proves convergence.\nOne replaces $F$ by a function with larger Taylor coefficients in absolute value, for which the equation can be solved explicitly.\nThe solution of this new equation dominates the formal [[power series|0MZB06ST2B]].\nThe real version holds as well: if $F$ is real [[analytic|081WKEDK6C]], the solution is real analytic.\n\nExample: $w' = w^2$, $w(0) = 1$ has the holomorphic solution $w(z) = \\frac{1}{1 - z} = \\sum_{k \\ge 0} z^k$ on $\\abs{z} \u003c 1$.\nThe radius $r$ cannot be taken arbitrarily large, since this solution has a pole at $z = 1$.\n\nAnalyticity of $F$ is needed for an analytic solution: for a smooth but not analytic function $g$ on $\\R$, the solution $w(x) = \\int_0^x g$ of $w' = g(x)$, $w(0) = 0$, is not analytic.\n\n## Explanations\n\n### L1 human\n\nIf the right side of a differential equation is given by convergent power series, then so is the solution.\nOne can compute the Taylor coefficients of the solution one by one from the equation.\nThe theorem says that the resulting series really converges near the starting point.\n\n### L2 human\n\nThis is the case of ordinary differential equations in the general Cauchy–Kovalevskaya theorem for partial differential equations.\nIt is the starting point of the theory of differential equations in the complex domain.\nUnlike the Picard–Lindelöf theorem, it gives analyticity of the solution, but it needs analytic data.\nFor linear equations, the solution even extends along every path in the domain where the coefficients are holomorphic, which leads to [[monodromy|C8HD62AB6Q]].\n",
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