The harmonic conjugate of v=x³-3xy²
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If u is harmonic and v is a conjugate of u, then v is also harmonic. Being “a harmonic conjugate of” is not symmetric. One cannot simply say that u and v are “harmonic conjugates of one another.” If v is a harmonic conjugate of u, then -u is a harmonic conjugate of v: if f = u + iv is analytic, then so is -if = v - iu.
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