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Let $H(u,v)$, $\, u:=u(x,y)$, and $\, v:=v(x,y)$ be real valued functions.

I am having trouble taking the second partial derivatives $H_{xx}$ and $H_{yy}$ of this function composition using the chain rule. The first partial derivatives are easy, for example, $$H_x(u,v)=H_u u_x+ H_v u_x,$$ and similarly for $H_y$.

How would we compute $H_{xx}$ and $H_{yy}$? I'm trying to prove that if $f(z)=u(x,y)+v(x,y)$ is complex analytic and $H$ harmonic, then $H(f(z))=H(u,v)$ is harmonic, so I'm checking the second partials of $H$.

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Hint

I think you forgot a $j$ (or $i$ if you are a mathematician) in the question. In any case the procedure is exactly the same. There are "cleaner" ways to do it, but essentially it comes from this (i'll use $\partial_x$ and $\partial_y$ because it is late :P): $$ \partial_xH := \left(\partial_xu \cdot \partial_u + \partial_xv \cdot \partial_u\right)H \\ \partial_{xx}H := \partial_x \partial_xH:= \left(\partial_xu \cdot \partial_u + \partial_xv \cdot \partial_u\right) \left(\partial_xu \cdot \partial_u + \partial_xv \cdot \partial_u\right)H$$ Which after some fiddling reduces to a nicer result

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