Derivation of a "Weighted" Mean Value Thm

I am attempting to acquire the list below formula for an analytic function, $f$, specified on the device disc, $D$

\begin{equation*} f(z)=\frac{1}{\pi}\int_{D} \frac{f(w)}{(1- \bar{w}z)^{2}}dA(w). \end{equation*}

Question: Anyone have any kind of suggestions on just how to acquire this formula?

I've had the ability to validate that the above holds true for $z =0$. In trying to acquire for the basic instance, I made use of conformal maps to get to

\begin{equation*} f(z)=\frac{1}{\pi}\int_{D} f(w) \left| \frac{-1+|z|^{2}}{(-1+w\bar{z})^{2}} \right|^{2} dA(w) \end{equation*}

Unfortunately this is not the formula I was attempting to acquire,

2019-05-18 22:51:25
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Answers: 1

First, you are missing out on a variable of $1/\pi$ on the appropriate - hand side, as can be seen by taking $f=1$ and also $z=0$.

To confirm the identification, you can make use of Green is formula, which converted right into intricate variables takes the kind $$\int_C F(w,\bar{w}) dw = 2i \iint \frac{\partial F}{\partial \bar{w}} dA(w).$$. This changes your location indispensable right into a line indispensable around the device circle $C$. To review this line indispensable, make use of that $\bar{w}=1/w$ when $|w|=1$ ; this offers you an indispensable in $w$ alone (no $\bar{w}$) to make sure that you can make use of Cauchy is formula to end up the work.

I've examined that it functions, yet I leave out the information to make sure that I do not ruin your satisfaction also much.: -)

2019-05-21 09:49:38