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\section*{Problem Set 2}

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\item Show that the mappings of a disk onto a parallel strip, or onto a half strip with two right angles, can be obtained as special cases of the Schwarz-Christoffel formula. [You may want to think about Exercises 1 and 2 on p. 238 of Alhfors, which come before this, first.]
\item Show that $F(w)=\int_{0}^w (1-w^n)^{-2/n}dw$ maps $|w|<1$ onto the interior of a regular polygon with $n$ sides.
\item Read the section on triangle functions in Ahlfors. In each of the three cases, determine the configuration of triangles in a parallelogram spanned by the periods. See of you can determine what is special about the torus $\mathbb{C}/\Lambda$ ($\Lambda$ the period lattice) in these cases. [Hint: automorphisms.]
\item Let $w=u+iv=f(z)=z+\log(z)$ for $z\in \mathfrak{H}$. Prove that $f$ gives an isomorphism of $\mathfrak{H}$ with the open set $\mathcal{U}$ obtained from $\mathfrak{H}$ by deleting the half-line of numbers $u+i\pi$ with $u\leq -1$. [Hint: use Theorem B (Lect.~4) applied to the path consisting of the segment from $R$ to $\epsilon$, the small semicircle in $\mathfrak{H}$ from $\epsilon$ to $-\epsilon$, the segment from $-\epsilon$ to $-R$, and the large semicircle in $\mathfrak{H}$ from $-R$ to $R$. Note that if we write $z=re^{i\theta}$, then $f(z)=re^{i\theta}+\log r+i\theta$.] 
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