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Let $u$ be an Harmonic function in $B(0,a)$ in $R^3$

we define $I(x)=x\dfrac{a^2}{|x|^2} $

Let $w(x) = u(I(x))$.

Is there a way to prove that $w$ is harmonic without making too much computation?

If not I will make them my self. Thanks for your help!

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up vote 0 down vote accepted

In fact this are call Kelvin transform, and it work In $R^2$ but it needs a modification for $R^3$

http://en.wikipedia.org/wiki/Kelvin_transform

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