Advanced Calculus fi..

620 Advanced Calculus, Fifth Editionwhere A and B are complex constants. The function w = G(z) defines the Schwarz-Christoflel transformation. Every one-to-one conformal mapping of the upper halfplaneonto a simply connected domain bounded by straight lines and line segmentsis representable in the form (8.1 18). This includes every mapping onto the interiorof a polygon.The function G(z) satisfies the boundary conditions:/forarg Gf(z) = h2 for y = 0, xl < x < x2, . . . ,= ef(~)= eu+iu , arg G'(z) = u = Im If (z)l.The quantity arg Gf(z) measures the amount by which directions are rotated in goingfrom the z plane to the w plane (Section 8.20). Hence along the boundary the amountof rotation is piecewise constant; accordingly, each interval x < X I, xl < x < xz, . . .must be mapped on a straight line by w = G(z). The numbers h 1 , h2, . . . give theangles from the direction of the positive real axis to these lines, as suggested inFig. 8.45 for the case of a convex polygonal domain; the numbersare successive exterior angles of the polygon; the (n + 1)th exterior angle is hl +2n - hn+, . It can happen that hn+] = hl + 27r, in which case the polygon has onlyn vertices; otherwise, for a proper convex polygon hl + n < h,+, < hl + 217. Sincethe two cases are as follows:kl + k2 + . a -+ kn = 2 (n vertices),1 < kl + . - . + kn < 2 (n + 1 vertices).Figure 8.45Convex polygonal domain.