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Suppose that an analytic function $w= f(z) = u(x,y) + iv(x,y)$ maps a domain $D_z$ in the $z$ plane onto a domain $D_w$ in the $w$ plane. Let a function $h(u,v)$ with continuous first and second partial derivatives be defined on $D_w$. Show using chain rule that if $H(x,y) = h\left( u(x,y) , v(x,y) \right)$ then $$H_{xx} (x,y) + H_{yy} (x,y) = \left( h_{uu}(u,v) + h_{vv}(u,v) \right) |f’(z)|^2.$$ How does it follow from this that $H(x,y)$ is harmonic in $D_z$ when $h(u,v)$ is harmonic in $D_w?$

I feel that this is just a huge messy chain rule using partial derivatives problem, is there another approach to this or at least a way to do this elegantly without creating a big mess of terms? We know that since $f$ is analytic, the Cauchy-Riemann equations hold: $u_x = v_y \text{ and } u_y = -v_x$. Also, the functions $u$ and $v$ satisfy Laplace’s Equation. The continuity conditions on the partial derivatives yield $h_{vu} = h_{uv}$. (I am having trouble putting these together as well.)

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  • $\begingroup$ $h$ harmonic $\Rightarrow H$ harmonic follows immediately, all that is left to do is the messy work of taking partial derivatives and applying the chain rule to deduce $\Delta H=(\Delta h)|f'(z)|^2$ $\endgroup$ – Prototank May 8 '18 at 23:03
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    $\begingroup$ @Prototank So I guess there is no way around the messy work? Also, where do we get $|f'(z)|^2$ in the calculation? $\endgroup$ – Compact May 8 '18 at 23:21
  • $\begingroup$ I've filled in the details. $\endgroup$ – Prototank May 9 '18 at 0:47
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For example:

$\begin{align} H_x&=h_uu_x+h_vv_x\\ \Rightarrow H_{xx}&=(h_uu_x)_x+(h_vv_x)_x\\ &=(h_{uu}u_x+h_{uv}v_x)u_x+h_uu_{xx}+(h_{vv}v_x+h_{uv}u_x)v_x+h_vv_{xx}\\ &=h_{uu}u_x^2+h_{uv}v_xu_x+h_uu_{xx}+h_{vv}v_x^2+h_{uv}u_xv_x+h_vv_{xx} \end{align}$

By symmetry:

$$H_{yy}=h_{uu}u_y^2+h_{uv}v_yu_y+h_uu_{yy}+h_{vv}v_y^2+h_{uv}u_yv_y+h_vv_{yy}$$ change some of the $y$'s to $x$'s with CR: $$H_{yy}=h_{uu}u_y^2-h_{uv}v_xu_x+h_uu_{yy}+h_{vv}v_y^2-h_{uv}u_xv_x+h_vv_{yy}$$ Combine: $$H_{xx}+H_{yy}= h_{uu}(u_x^2+u_y^2)+h_u(u_{xx}+u_{yy})+h_v(v_{xx}+v_{yy})+h_{vv}(v_x^2+v_y^2)$$ But $u$ and $v$ are harmonic, so this simplifies further:$$H_{xx}+H_{yy}= h_{uu}(u_x^2+u_y^2)+h_{vv}(v_x^2+v_y^2)$$ Use CR again: $$H_{xx}+H_{yy}= (h_{uu}+h_{vv})(u_x^2+u_y^2)$$

To see that $(u_x^2+u_y^2)=|f'(z)|^2$, (Complex chain rule for complex valued functions)

$$f'(z)=u_x-iu_y\Rightarrow |f'(z)|^2=u_x^2+u_y^2$$

as desired.

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