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  "body": "\n## Statement\n\nFor each $z \\in \\mathbb{C}$, the [[complex exponential|054VXHMCA3]] is [[complex differentiable|0D1C9Y1EXA]] at $z$, and\n$$\\exp'(z) = \\exp z.$$\n\n## Remarks\n\nProof: differentiate the series $\\sum_{n \\ge 0} z^n / n!$ term by term. The derivative of $z^n / n!$ is $z^{n-1} / (n-1)!$ for $n \\ge 1$, so the series of derivatives is the same series.\nWith the chain rule, the derivative of $e^{a z}$ is $a e^{a z}$ for each constant $a \\in \\mathbb{C}$.\nIt follows that $\\exp$ is [[entire|0G2YJN0H5S]].\n\n## Explanations\n\n### L1 human\n\nAs in the real case, the exponential function is its own derivative.\nYou can see it in the power series: differentiating each term gives the term before it, so the whole series does not change.\n",
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