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  "body": "\n## Summary\n\nSpheroidal wave functions are solutions of the equations obtained by [[separation of variables|47NFRVB564]] in the [[Helmholtz equation|MNXKTXJ35G]] in prolate or oblate spheroidal coordinates.\nThe angular spheroidal wave functions $S_{mn}(c, x)$ are the solutions bounded on $[-1, 1]$, and they exist only for special eigenvalues $\\lambda_{mn}(c)$.\nFor $c = 0$ they reduce to the [[associated Legendre polynomials|VPJCSVRY6S]], so they deform the spherical case to the spheroidal one.\nThe prolate functions with $m = 0$ are also central in signal processing.\nThere they are the functions of a given bandwidth that are most concentrated in a time interval.\n\n## Explanations\n\n### L1 human\n\nA spheroid is a sphere stretched or squashed along one axis, like a rugby ball or a lentil.\nWaves around a sphere are described with Legendre functions; waves around a spheroid need a deformed family, the spheroidal wave functions.\nThe parameter $c$ measures how far the problem is from the spherical one, and for $c = 0$ the deformation disappears.\nAs with Legendre polynomials, well-behaved solutions exist only for a discrete list of separation constants.\n",
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