Complex Analysis: Theory and Applications · Problem 3.3.6

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · harmonic-product-not-harmonic

Problem

Let u1(x,y)=x2−y2u_1(x,y) = x^2 - y^2 and u2(x,y)=x3−3xy2u_2(x,y) = x^3 - 3xy^2. Show that u1u_1 and u2u_2 are harmonic functions but that their product u1(x,y)u2(x,y)u_1(x,y)u_2(x,y) is not harmonic.

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