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  "body": "\n## Statement\n\nFor all complex numbers $z$ and $w$,\n$$e^{z + w} = e^{z} e^{w},$$\nwhere $e^{z} = \\sum_{n=0}^{\\infty} z^n / n!$ is the complex exponential function.\n\n## Remarks\n\nThe proof multiplies the two series and uses the binomial theorem.\nIt follows that $e^{z} \\neq 0$ and $e^{-z} = 1 / e^{z}$ for each $z$.\nWith [[Euler's formula|04R3CY41VJ]], it gives the addition formulas for $\\cos$ and $\\sin$.\n\n## Explanations\n\n### L1 human\n\nThe exponential function turns sums into products, for complex numbers as for real numbers.\nFor example, $e^{i \\theta} e^{i \\varphi} = e^{i (\\theta + \\varphi)}$: turning by $\\theta$ and then by $\\varphi$ is turning by $\\theta + \\varphi$.\n",
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  "name": "Product of exponentials",
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