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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{C}$ be an open set, let $f : U \\to \\mathbb{C}$ be a function, and let $z_0 \\in U$.\nThe function $f$ is complex differentiable at $z_0$ if the limit\n$$f'(z_0) = \\lim_{h \\to 0} \\frac{f(z_0 + h) - f(z_0)}{h}$$\nexists in $\\mathbb{C}$. Here $h$ runs through the nonzero [[complex numbers|09FTAET5Z4]] with $z_0 + h \\in U$.\nThe number $f'(z_0)$ is the complex derivative of $f$ at $z_0$.\nThe limit means: for each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $\\left| \\frac{f(z_0 + h) - f(z_0)}{h} - f'(z_0) \\right| \u003c \\varepsilon$ for all such $h$ with $0 \u003c |h| \u003c \\delta$.\n\n## Remarks\n\nFor example, $f(z) = z^2$ is complex differentiable at each $z_0$ with $f'(z_0) = 2 z_0$, because $\\frac{(z_0 + h)^2 - z_0^2}{h} = 2 z_0 + h$.\nThe number $h$ can go to $0$ from each direction in the plane. This makes the condition much stronger than real differentiability, see [[the conjugate is not holomorphic|0Z8ATBPM5W]].\nIf $f$ is complex differentiable at $z_0$, then $f$ is continuous at $z_0$.\nThe [[Cauchy–Riemann equations|0T71SSYZ39]] describe complex differentiability with partial derivatives.\n\n## Explanations\n\n### L1 human\n\nThe definition is the same as the definition of the derivative of a real function, but $h$ is now a complex number.\nThe quotient must have the same limit when $h$ goes to $0$ along the real axis, along the imaginary axis, or along any other path.\n",
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