= Solution
{c}
\b[There is no such map for the domain printed in the paper.] The planar <domain> $A=\mathbb C\setminus[a,b]$ is not <simply connected>. For example, a sufficiently large positively oriented <circle> $\gamma$ surrounds $[a,b]$ and lies in $A$, while
$$
\int_\gamma\frac{dz}{z-a}=2\pi i\ne0.
$$
The integrand is <holomorphic> on $A$. If $A$ were <simply connected>, the <Cauchy integral theorem> would force this <integral> to be zero. A <conformal bijection> is a <homeomorphism> and preserves <simple connectivity>, whereas $H$ is <simply connected>. Thus a <conformal bijection> $A\to H$ is impossible.
The explicit construction also pinpoints the missing point. Define
$$
T(z)=\frac{z-a}{z-b}.
$$
This <Möbius transformation> sends the removed <line segment> to the nonpositive <real axis>, together with infinity. More precisely, its <image> on the stated planar <domain> is
$$
T(A)=\bigl(\mathbb C\setminus(-\infty,0]\bigr)\setminus\{1\},
$$
since $T(z)=1$ has no finite solution when $a\ne b$. Conversely, its inverse $z=(bw-a)/(w-1)$ is defined for every $w$ in this set and cannot lie on $[a,b]$, proving this image description exactly.
Take the <principal square root> of $T(z)$, characterized by positive <real part>, and then multiply by $i$. The result is the <conformal map>
$$
\boxed{F(z)=i\sqrt{\frac{z-a}{z-b}}:\ A\longrightarrow H\setminus\{i\}.}
$$
Each stage is <injective> and onto the indicated <image>; the <square root> maps the slit plane to the <right half-plane>, with the point $1$ omitted here. Hence this is a <conformal bijection> onto the punctured <upper half-plane>. Its inverse is
$$
z=\frac{a+bF^2}{1+F^2}.
$$
If the source is instead $(\mathbb C\cup\{\infty\})\setminus[a,b]$ on the <Riemann sphere>, the same map extends with $F(\infty)=i$ and becomes a <conformal bijection> onto all of $H$. Indeed, in the local coordinate $u=1/z$ it has expansion $F=i+\tfrac i2(b-a)u+O(u^2)$, whose <derivative> at $u=0$ is nonzero. This gives the sphere version of the requested construction without silently changing the printed planar <domain>.
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