In the integral of Exercise 8. let the numbers Z i ( j =

Chapter 0, Problem 11.15

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In the integral of Exercise 8. let the numbers Z i ( j = I. 2 . .... 11) be the 11th roots of unity. \\!rite u.> = cxp(2,'7i/11) and Z 1 = I. Z 2 = w ..... Z11 = w11 - 1 (sec Sec. 10). Let each of the numbers k i (j = I. 2 . .... 11) have the value 2/ 11. The integral in Exerci sc 8 SEC. 131 r-u:ID r-1.ow '" .\ CllA!\!\EL T11Roue111 A SuT 407 then becomes z u~ = A' f ./o dS ----+8. (Sn_ 1)2:'11 Show lhal \Vhen A' = I and B = 0. this transformation maps the interior of the unit circle IZJ = I onto the interior of a regular polygon ofn sides and that the center oflhc polygon is the point rr = 0. Sugge.\'T ion: The image of each of the points Z i (j = I. 2 . .... 11) is a vertex of some polygon with an exterior angle of 2,7/11 al that vertex. \\I rile "'1 = l 1 c/S . 0 (.\11 _ I )2:'11 where the path of the integration is along the positive real axis from Z = 0 lo Z = I and the principal value of chc 11ch rool of (S11 - I ) 2 is to be taken. Then show lhal chc images of the points Z2 = w . .... Z,, = w11 - 1 arc che points cJ)tr 1 w" - 1 u 1. respective) y. Thus verify thal the polygon is regular and is centered al tr = 0.