In my current project, I need to find a conformal mapping $f$ such that is maps the interior of the ellipse (say, $\frac{x^2}{a^2} + y^2 = 1$ for some $a >1$, which in the complex plane can be written as $r = r(\theta) = \frac{a}{\sqrt{\cos^2\theta + a^2\sin^2\theta}}$) to the interior of the unit circle ($r = 1$), $f$ maps the boundary of the ellipse to the boundary of the unit disk. Moreover, $f$ is required to map a small part of (symmetric) boundary arc around the point $(a,0)$ on the boundary of the ellipse to a small part of (again symmetric) boundary arc around the point $(1,0)$ on the boundary of the unit disk. I already find that without the additional boundary behavior, the map $f$ is given by some complicated expressions involving Jacobi elliptic $\mathrm{sn}$ function. However in that case it seems impossible to check the prescribed boundary behavior I want. Any suggestions or help will be greatly appreciated!
Edit: I am wondering whether the comments below can be expanded a little bit (but anyway, thank you guys very much!!!) Say I want $f$ to map $\{(\theta,r(\theta)) \in E \mid |\theta| \leq \delta\}$ (where $E$ is boundary of the ellipse I described) to something like $\{e^{i\theta} \mid |\theta| \leq \varepsilon\}$. Is it possible to deduce an explicit formula for $\varepsilon$ in terms of $f$ and $\delta$ (since $f$ is a priori compilcated, I guess to express $\varepsilon$ in terms of $f$ and $\delta$ will also be complicated...)?
Edit (continued): If $a>1$ and $a$ is very close to 1, then the ellipse considered can be viewed as a perturbation of the unit disk, and belongs to the class of "nearly circular domains" in the literature. However, it seems that the conformal map $f$ sending E to the unit disc (which also sends the boundary of $E$ to the boundary of the unit disk $D$) does not reduce to the identity map as $a \to 1$ (at least this is not obvious, as the map $f$ given by the $\mathrm{sn}$ function "blows up" in the sense of vanishing denominator...)