Show that the Schwarz-Christoffel transformation f (z) =
Chapter 23, Problem 23.38(choose chapter or problem)
Show that the Schwarz-Christoffel transformation f (z) = 2i z 0 ( + 1) 1/2 ( 1) 1/2 1/2 d maps the upper half-plane onto the rectangle with vertices 0, c, c + ic, and ic where c = (1/2)(1/4)/ (3/4). For this problem, it is necessary to know the integral formula B(x, y) = 1 0 ux1 (1 u) y1 du for the beta function B(x, y), and also to know that, in terms of the gamma function, B(x, y) = (x)(y) (x + y) . Hint: See Section 15.3 ( 39 and 40).
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