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Suppose $f(z) = z^2$

This function has the component functions $u(x,y) = x^2 - y^2$, $v(x,y) = 2xy$

And it says in a book I'm reading that $v$ is a harmonic conjugate of $u$. But v is not harmonic as

$v_{xx} = 2$ and $v_{yy} = 2$

So $v_{xx} + v_{yy} = 4 \not= 0$

So how can it say that v is a harmonic conjugate of u? I presume I'm missing something as I don't think the book is wrong.

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up vote 2 down vote accepted

$$\partial_x 2xy = 2y$$$$\partial_x 2y = 0$$Similarly, $\partial_{yy} 2xy = 0$, and sure enough 0 + 0 = 0!

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lol, note to self, write stuff out on paper fully. – Jim_CS Apr 30 '12 at 11:25

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