Question 7.3.3: Using the Schwarz-Christoffel Formula. Use the Schwarz-Chris......
Using the Schwarz-Christoffel Formula
Use the Schwarz-Christoffel formula (6) to construct a conformal mapping
from the upper half-plane onto the polygonal region bounded by the equilateral triangle with vertices w_{1}=0,\;w_{2}=1,\;\mathrm{and}\;w_{3}=\textstyle{\frac{1}{2}}+\textstyle{\frac{1}{2}}\sqrt{3}i.. See Figure 7.25.
f^{\prime}(z)=A\left(z-x_{1}\right)^{(\alpha_{1}/\pi)-1}(z-x_{2})^{(\alpha_{2}/\pi)-1}\cdot\cdot\cdot(z-x_{n})^{(\alpha_{n}/\pi)-1}, (6)
The region bounded by the equilateral triangle is a bounded polygonal region with interior angles α_{1} = α_{2} = α_{3} = π/3. As mentioned on page 413, we can find a desired mapping by using the Schwarz-Christoffel formula (6) with n − 1 = 2 of the interior angles. After selecting x_{1} = 0 and x_{2} = 1, (6) gives
f^{\prime}(z)=A z^{-2/3}\left(z-1\right)^{-2/3}. (12)
There is no antiderivative of the function in (12) that can be expressed in terms of elementary functions. However, f^{\prime} is analytic in the simply connected domain y > 0, and so, from Theorem 5.8 of Section 5.4, an antiderivative f does exist in this domain. The antiderivative is given by the integral formula
f(z)=A\int_{0}^{z}{\frac{1}{s^{2/3}\left(s-1\right)^{2/3}}}d s+B, (13)
where A and B are complex constants. Requiring that f(0) = 0 allows us to solve for the constant B. Since \int_{0}^{0}{} =0, we have
f(0)=A\int_{0}^{0}{\frac{1}{s^{2/3}\left(s-1\right)^{2/3}}}d s+B=0+B=B,and so f(0) = 0 implies that B = 0. If we also require that f(1) = 1, then
f(1)=A\int_{0}^{1}{\frac{1}{s^{2/3}\left(s-1\right)^{2/3}}}d s=1.Let Γ denote value of the integral
\Gamma=\int_{0}^{1}{\frac{1}{s^{2/3}\left(s-1\right)^{2/3}}}d s.Then A = 1/Γ and f can be written as
f(z)={\frac{1}{\Gamma}}{\int_{0}^{z}{\frac{1}{s^{2/3}\left(s-1\right)^{2/3}}}d s.}Values of f can be approximated using a CAS. For example, using the NIntegrate command in Mathematica we find that
f(i)\approx0.4244+0.3323i\quad\mathrm{and}\quad f(1+i)\approx0.5756+0.3323i.