Sequence of polynomials converging uniformly to $\frac 1z$ on semicircle in $\mathbb{C}$ I wish to construct a sequence of polynomials that converges uniformly on the semicircle $\{z: |z| = 1 , Re(z) \geq 0 \}$ to the function $\frac 1z$. Any help with this would be really appreciated, as I not sure even where to begin. Thanks!
 A: If there were a disk $D_R(a)$ containing the semicircle $K = \{ z : \lvert z\rvert = 1, \operatorname{Re} z \geqslant 0\}$ on which $f(z) = \frac{1}{z}$ is holomorphic, we could just use the Taylor polynomials of $f$ with centre $a$.
But, disks are convex, hence every disk containing $K$ also contains the origin, so it isn't as simple. However, we can approximate $f$ on $K$ easily with functions having their pole in the left half-plane, since for $r > 0$ and $z \in K$, we have
$$\left\lvert \frac{1}{z} - \frac{1}{z+r}\right\rvert = \frac{r}{\lvert z(z+r)\rvert} = \frac{r}{\lvert z+r\rvert} < r.$$
And for all $r > 0$ there are disks containing $K$ but not the pole $-r$ of $\frac{1}{z+r}$. Choosing the centre $a$ of the disk on the positive real axis, $D_R(a)$ contains $K$ but not $-r$ if and only if
$$a^2 + 1 < R^2 \leqslant (a+r)^2,$$
and $a \geqslant \frac{1}{2r}$ is seen to be sufficient.
One can then choose for example $a_k = 2^k$ and $r_k = 2^{-k}$, and for a Taylor polynomial of sufficiently large order $m_k$,
$$P_k(z) = \sum_{n=0}^{m_k}(-1)^n \frac{(z-a_k)^n}{(a_k+r_k)^{n+1}},$$
we have
$$\lvert f(z)-P_k(z)\rvert \leqslant \left\lvert \frac{1}{z} - \frac{1}{z+r_k}\right\rvert + \left\lvert \frac{1}{z+r_k} - P_k(z)\right\rvert < r_k + r_k = 2^{1-k}$$
on $K$. Using $\lvert z-a_k\rvert \leqslant \sqrt{a_k^2+1} < a_k + \frac{1}{2}r_k$, one can explicitly determine sufficiently large $m_k$.
