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  "body": "\n## Summary\n\nAn [[entire function|0419794R0T]] is of exponential type if its modulus grows at most like $e^{\\tau \\abs{z}}$ for some constant $\\tau$.\nThe least such $\\tau$ is the type of the function.\nFourier and Laplace transforms of functions or measures with bounded support are of exponential type, and the Paley–Wiener theorem gives a converse.\nThey are the natural setting of the [[Paley–Wiener theorem|THY12GPZ8C]] and of [[Carlson's theorem|6F5B8KPEGH]].\n\n## Explanations\n\n### L1 human\n\nThe exponential $e^{\\tau z}$ grows like $e^{\\tau \\abs{z}}$ in the direction where $z$ is positive real.\nA function of exponential type never grows faster than this in any direction, for some fixed $\\tau$.\nPolynomials, $\\sin z$, $\\cos z$ and $e^{3z}$ are of exponential type, with types $0$, $1$, $1$ and $3$.\nThe function $e^{z^2}$ grows like $e^{\\abs{z}^2}$ along the real axis and is not of exponential type.\n",
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