# complex analysis – Understanding proof of analyticity of \$overline{fleft (overline zright )}\$

I’m reading Marsden and Hoffman’s book of Complex Analysis, and I’m struggling to understand the following example

Let $$A$$ be an open subset of $$mathbb{C}$$ and $$A^*={z|overline zin A}$$. Suppose $$f$$ is analytic on $$A$$, and define a function $$g$$ on $$A^*$$ by $$g(z)=overline{fleft (overline zright )}$$. Show that $$g$$ is analytic on $$A^*$$.

The proof provided by the book is

If $$f(z)=u(x,y)+iv(x,y)$$, then $$g(z)=overline{fleft (overline zright )}=u(x,-y)-iv(x,-y)$$.
We check the Cauchy-Riemann equations for $$g$$ as follows:$$dfrac{partial}{partial x}(operatorname{Re}g)=dfrac{partial}{partial x}u(x,-y)=left .dfrac{partial u}{partial x}right |_{(x,-y)}=left .dfrac{partial v}{partial y}right |_{(x,-y)}=dfrac{partial}{partial y}(-v(x,-y))=dfrac{partial}{partial y}(operatorname{Im}g).$$

I’m failing to understand why $$left .dfrac{partial v}{partial y}right |_{(x,-y)}=dfrac{partial}{partial y}(-v(x,-y))$$.
When I use the chain rule with $$h(x,y)=(x,-y)$$ I get $$dfrac{partial}{partial y}(v(x,-y))=dfrac{partial}{partial y}(v(h(x,y)))=left .dfrac{partial (vcirc h)}{partial y}right |_{(x,y)}$$ andbegin{align*}begin{pmatrix}dfrac{partial}{partial x}(vcirc h)&dfrac{partial}{partial y}(vcirc h)end{pmatrix} & =D(vcirc h) \ & =D(v)D(h) \ & =begin{pmatrix}dfrac{partial}{partial x}v&dfrac{partial}{partial y}vend{pmatrix}begin{pmatrix}dfrac{partial}{partial x}h_1&dfrac{partial}{partial y}h_1\dfrac{partial}{partial x}h_2&dfrac{partial}{partial y}h_2end{pmatrix} \ & =begin{pmatrix}dfrac{partial}{partial x}v&dfrac{partial}{partial y}vend{pmatrix}begin{pmatrix}1&0\0&-1end{pmatrix} \ & =begin{pmatrix}dfrac{partial}{partial x}v&-dfrac{partial}{partial y}vend{pmatrix}. end{align*}Hence $$dfrac{partial}{partial y}(v(x,-y))=left .dfrac{partial (vcirc h)}{partial y}right |_{(x,y)}=-left .dfrac{partial v}{partial y}right |_{(x,y)}$$.
So $$left .dfrac{partial v}{partial y}right |_{(x,y)}=dfrac{partial}{partial y}(-v(x,-y))=left .dfrac{partial v}{partial y}right |_{(x,-y)}$$, but the last one doesn’t feel right.

Can anyone help me to prove that $$left .dfrac{partial v}{partial y}right |_{(x,y)}=dfrac{partial}{partial y}(-v(x,-y))$$ or at least point out my mistake in the chain rule?

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