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If there wasn't that minus sign, the answer would be a wave equation. http://uniquation.com/ was a bust. I asked wolframalpha, and it came back with an answer which looked just like the wave equation with an extra factor of $i$. Does the equation have a more general name?

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    It is a Laplace-type equation.2011-10-06
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    The question came up in a 4D context, specifically: u_tt + u_xx + u_yy + u_zz = 0, so that would be a 4D Laplace equation. The inhomogeneous case would be 4D Poisson. Thanks.2011-10-06
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    I'd like to add that the conic representation of this is an ellipse and the function is elliptic.2011-10-06

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