Suppose in the Schwarz-Christoffel transformation (8.9) or (8.10), page 247, one point, say [latex]x_{n}[/latex], is chosen at infinity. Show that the last factor is not present. [latex]{\frac{d w}{d z}}=A(z-x_{1})^{\alpha_{1}/\pi-1}(z-x_{2})^{\alpha_{2}/\pi-1}\cdot\cdot\cdot(z-x_{n})^{\alpha_{n}/\pi-1}[/latex] (8.9) [latex]w=A\int(z-x_{1})^{\alpha_{1}/\pi-1}(z-x_{2})^{\alpha_{2}/\pi-1}\cdot\cdot\cdot(z-x_{n})^{\alpha_{n}/\pi-1}d z+B[/latex] (8.10)

In (8.9), page 247, let [latex]A=K /\left(-x_{n}\right)^{\alpha_{n} / \pi-1}[/latex] where [latex]K[/latex] is a constant. Then, the right side of (9) can be written [latex]K\left(z-x_{1}\right)^{\alpha_{1} / \pi-1}\left(z-x_{2}\right)^{\alpha_{2} / \pi-1} \cdots\left(z-x_{n-1}\right)^{\alpha_{n-1} / \pi-1}\left(\frac{x_{n}-z}{x_{n}}\right)^{\alpha_{n} / \pi-1}[/latex] As [latex]x_{n} \rightarrow \infty[/latex], this last factor approaches 1 ; this is equivalent to removal of the factor.

Question 8.19: Suppose in the Schwarz –Christoffel transformation (8.9) or ...

Schaum’s Outline of Complex Variables

Suppose in the Schwarz-Christoffel transformation (8.9) or (8.10), page 247, one point, say $x_{n}$ , is chosen at infinity. Show that the last factor is not present.

${\frac{d w}{d z}}=A(z-x_{1})^{\alpha_{1}/\pi-1}(z-x_{2})^{\alpha_{2}/\pi-1}\cdot\cdot\cdot(z-x_{n})^{\alpha_{n}/\pi-1}$ (8.9)

$w=A\int(z-x_{1})^{\alpha_{1}/\pi-1}(z-x_{2})^{\alpha_{2}/\pi-1}\cdot\cdot\cdot(z-x_{n})^{\alpha_{n}/\pi-1}d z+B$ (8.10)

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