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  "body": "\n## Definition\n\nThe complex exponential is the function $\\exp : \\mathbb{C} \\to \\mathbb{C}$ given by\n$$\\exp z = \\sum_{n=0}^{\\infty} \\frac{z^n}{n!} = 1 + z + \\frac{z^2}{2!} + \\frac{z^3}{3!} + \\cdots$$\nfor each [[complex number|09FTAET5Z4]] $z$. The series converges absolutely for each $z \\in \\mathbb{C}$.\nAnother notation for $\\exp z$ is $e^z$.\n\n## Remarks\n\nFor real $x$, $\\exp x$ is the real exponential function, and $e = \\exp 1$.\nFor $z = x + iy$ with $x, y$ real, $e^z = e^x (\\cos y + i \\sin y)$. The case $x = 0$ is [[Euler's formula|04R3CY41VJ]].\nBy the [[product of exponentials|0WM7DJCM6G]], $e^{z + w} = e^z e^w$. So $e^z \\neq 0$ for each $z$, and $|e^z| = e^{\\operatorname{Re} z}$.\n\n## Explanations\n\n### L1 human\n\nThe power series of $e^x$ makes sense for complex numbers too, so we use it as the definition of $e^z$.\nThe real part of $z$ controls the size of $e^z$, and the imaginary part of $z$ controls its direction.\nFor example, $e^{i\\pi} = -1$: the size is $e^0 = 1$ and the direction is a half turn.\n",
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