Schwarz-Christoffel mapping ($F^{\prime }(z)=C{\prod }_j(z-a_j{)}^{{\alpha }_j-1}$)

A conformal map from the complex upper half-plane to a polygonal domain with finite prevertices $a_j\in \mathbb{R}$ and interior angles $\pi {\alpha }_j$ has the displayed derivative, with upper-half-plane branches of the powers. Integrating supplies the map, up to translation and nonzero scaling. For a slit boundary, repeated geometric vertices are treated as distinct boundary approaches: the two sides at the slit base have their respective angles, and a slit tip has angle $2\pi$, hence exponent one.

The principal square root maps the upper $Z$ half-plane bijectively onto $\Re z>0$ with the ray $[1,\infty )$ removed. The prevertex zero is the slit tip, and $\pm 1$ represent the two ends at infinity. The inverse is the unique root $Z^2=1-z^{-2}$ with positive imaginary part. Pulling back ${\pi }^{-1}\arg ((Z-1)/(Z+1))$ gives the bounded harmonic function with value one on the slit banks and zero on the imaginary axis. The harmonic function $\Im [Z/(1-Z^2)]$ has zero finite boundary values but grows at infinity, explaining the role of boundedness.

The upper half-plane joined to the downward strip $-1<x<1$ has reentrant corners at $\pm 1$. The Schwarz-Christoffel formula gives $z^{\prime }(Z)=2R/(\pi Z)$, with $R\sim Z$ at upper infinity. Since $(R+{\tan }^{-1}(1/R){)}^{\prime }=R/Z$, integration gives the displayed map. The inverse trigonometric function is continued analytically in the upper half-plane so its corner values are $\pm \pi /2$. The origin is the lower channel end and infinity the upper open end. The complex potentials $2Z/\pi$ and $2\log Z/\pi$ respectively normalize unit horizontal velocity in the upper region and unit upward velocity in the channel.

With the square root positive for $Z$ on the positive imaginary axis, the displayed conformal map takes the upper half-plane bijectively to $\{\mathrm{Re}z>0\}\setminus [1,\infty )$. Squaring, subtracting from one, taking the positive-real-part square root, and reciprocating identifies each intermediate slit domain. The boundary prevertices $-1,0,1$ correspond to the lower infinite end, the slit tip at one, and the upper infinite end. The inverse is $\sqrt{1-z^{-2}}$ with positive imaginary part.

A boundary point of the source half-plane that maps to a polygon vertex. If that corner has angle $\pi \alpha$, the Schwarz-Christoffel mapping has local behavior $F(z)-F(a)\sim C(z-a{)}^{\alpha }$ at its prevertex, so the derivative has exponent $\alpha -1$. Infinity may also be a prevertex, expressed in the local coordinate $1/z$.

For $0<k<1$, principal upper-half-plane branches give a conformal map onto the upper half-plane cut along $[0,\tau ]$, where $\tau =2k^k(1-k{)}^{1-k}{e}^{i\pi k}$. Its derivative is $(z-1+2k)(z-1{)}^{k-1}(z+1{)}^{-k}$. The two prevertices $-1,1$ map to the slit base, and $1-2k$ maps to its tip. Boundary traversal goes out along the slit and back, while $g(z)=z+1-2k+O(1/z)$ at infinity. The argument principle gives one preimage of every point off the slit.