Toronto Math Forum
APM346-2016F => APM346--Lectures => Chapter 8 => Topic started by: Shentao YANG on November 12, 2016, 10:05:03 PM
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Why harmonic polynomial of $deg=n$ must also be homogeneous polynomial of $deg=n$?
Say, $\Delta ({x^2} - {y^2} + z) = 2 - 2 + 0 = 0$, but we do not count $ ({x^2} - {y^2} + z)$ as harmonic polynomial of $deg=2$.
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In general NO, but we are looking at homogeneous polynomials. Obviously if a polynomial is harmonic, then all it homogeneous components are also are harmonic.