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Let $f$ be a holomorphic function on unit disk $D$. (a). express the Jacobian of the map $f$ in terms of $f$ or $f'$ (b).Give a formula for the area of $f(D)$ in terms of the Taylor Coefficients of $f$. (c).If $f(x)=z+\dfrac{z^{2}}{2}$,find the area of $f(D)$.

My solotion: (a). According to Cauchy-Riemann equation,I compute the Jacobian of the map $f$ as $|f'(x)|^{2}$. （b).From the elementary calculus,the area of $f(D)$ can be computed according to coordinate transform.So $\iint_{f\left(D\right)}dw=\iint_{D}\left|f'\left(D\right)\right|^{2}dz$.But I don't know how to express it in terms of Taylor coefficients. (c).Followed from the result in (b), Area$f(D)$=$\iint_{D}\left|1+re^{i\theta}\right|^{2}drd\theta=\dfrac{4}{3}$

Are my results valid and what about answer for the second question?

Jack
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  • Note that a factor $r$ is missing in your expression for the area, it should be $\iint_{D}\left|1+re^{i\theta}\right|^{2}rdrd\theta$ – Martin R Jul 29 '16 at 05:38

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