Lemma 27.6.2.label Let $\gamma \in C([a, b]; \complex)$ be a rectifiable curve, $K \subset \complex$ such that $K \cap \gamma([a, b]) = \emptyset$, $f \in C(\gamma([a, b]); \complex)$, and $\eps > 0$, then there exists $R \in \complex(z)$ such that:
- (1)
$R \in H(\complex \setminus \gamma([a, b]); \complex)$.
- (2)
For each $z_{0} \in K$,
\[\abs{\int_{\gamma} \frac{f(z)}{z - z_{0}}dz - R(z)}< \eps\]
Proof, [Lemma VIII.1.5, Con78]. Since the mapping
is continuous, it is uniformly continuous by Proposition 6.4.5. Hence the mappings $\bracs{\varphi(z_0, \cdot)|t \in [a, b]}$ are uniformly equicontinuous. Thus there exists $\delta > 0$ such that for each $s, t \in [a, b]$ with $|s - t| \le \delta$,
for all $z_{0} \in K$.
Let $(P = \seqfz{t_j}) \in \scp([a, b])$ with $\sigma(t) < \delta$, and
then $R \in \complex(z) \cap H(\complex \setminus \gamma([a, b]); \complex)$, and for each $z_{0} \in K$,
$\square$
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