Solving a complex integral (potentially using Taylor's Theorem)

Question

The following is a homework question so please don't give full proof. Some hints would be greatly appreciated.

Part A

Let $\gamma$ be the unit-circle (positively-oriented) and $\phi(\zeta) = 1/\zeta$, calculate the integral \begin{equation} I(z) = \frac{1}{2\pi i} \int_\gamma \frac{\phi(\zeta)}{\zeta - z} d\zeta \end{equation} for (i) $|z|<1$ and (ii) $|z|>1$.

Part B

Moreover, for each $\phi \in \mathcal{C}(\partial\mathbb{D})$ (continuous functions on the boundary of the unit disc), define $\Gamma_\phi \in \mathcal{O}(\mathbb{D})$ (holomorphic functions on the unit disc) by \begin{equation} \Gamma_\phi(z) = \frac{1}{2\pi i} \int_\gamma \frac{\phi(\zeta)}{\zeta - z} d\zeta. \end{equation} Then is the map $\Gamma: \mathcal{C}(\partial\mathbb{D}) \to \mathcal{O}(\mathbb{D})$ (i) injective (ii) surjective?

My Attempt

I've only managed to give a solution to (i) of Part A, but I don't think it is correct. It goes as follows.

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Pick $0 < r < 1$ such that $z \in B_r(0)$.

Then by the developability lemma (R. Remmert, Theory of Complex Functions), $I(z)$ has the power series representation as $I(z) = \sum_{n=0}^{\infty} a_n z^n$, where \begin{equation} a_n = \int_\gamma \frac{\phi(\zeta)}{\zeta^{n+1}}d\zeta. \end{equation}

Now we have \begin{equation} I'(z) = a_1 = \int_\gamma \frac{1}{\zeta^3}d\zeta = 0. \end{equation}

And since $I$ is given by a power series (hence continuous) whose domain is connected, $I$ is constant on $|z|<r$.

An evaluation at $0$ gives $I \equiv 0$ on $|z|<r$.

And since the choice of $r$ is arbitrary, we have $I \equiv 0$ on $|z|<1$.

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The reason this solution doesn't look right to me is that it would imply that the map in Part B is not injective, which is definitely wrong (edit: turns out this is not wrong, see zhw.'s answer below). But I can't see what's wrong with this solution.

For the rest of the problem, mainly I don't have a clue which theorem I should work toward. Possibly Taylor's Theorem would pop up somewhere, but I don't really know how to get there.

Any help would be appreciated. Thanks in advance!

Answer 1

A. i) For $|z|<1,$ you could write

$$\frac{1}{\zeta -z}= \frac{1}{\zeta}\frac{1}{1-(z/\zeta)} = \frac{1}{\zeta}(1+(z/\zeta) + (z/\zeta)^2+\cdots).$$

ii) Similar to the above, but this time factor out $z$ downstairs.

B. i) How could $\Gamma $ be injective given your answer to A. i)?

ii) Verify the estimate

$$ |\Gamma_\varphi (z)| \le \|\varphi\|_\infty\frac{1}{1-|z|}.$$

But aren't there functions in $\mathcal O(\mathbb D)$ that grow faster than that?