{
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  "body": "\n## Statement\n\nLet $z \\in \\mathbb{C}$.\nThen $e^z = 1$ if and only if $z = 2 \\pi i k$ for some [[integer|0C4JS9RF5M]] $k$.\nHere $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.\n\n## Remarks\n\nProof: write $z = x + iy$ with $x, y$ real. Then $e^z = e^x (\\cos y + i \\sin y)$. If $e^z = 1$, then $e^x = |e^z| = 1$, so $x = 0$. Then $\\cos y = 1$ and $\\sin y = 0$, so $y = 2 \\pi k$. The converse follows from [[Euler's formula|04R3CY41VJ]].\nIt follows that $e^z = e^w$ if and only if $z - w = 2 \\pi i k$ for some $k \\in \\mathbb{Z}$.\nThis is the reason why the [[complex logarithm|0X2FY491PF]] has many values.\n\n## Explanations\n\n### L1 human\n\nThe value $1$ has size $1$ and direction $0$.\nSo $e^z = 1$ needs a real part $0$ and an imaginary part that is a whole number of full turns, that is, a multiple of $2\\pi$.\n",
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  "name": "exp z = 1 iff z is an integer multiple of 2πi",
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