8. The map z → z is a 4-to-1 function away from the origin, so to define a continuous single-valued inverse function z →

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8 The Map Z Z Is A 4 To 1 Function Away From The Origin So To Define A Continuous Single Valued Inverse Function Z 1 (153.49 KiB) Viewed 32 times

8. The map z → z is a 4-to-1 function away from the origin, so to define a continuous single-valued inverse function z → z¹/4 we must cut the domain and select a branch of range values. In this problem we will examine two such branches. (a) Consider angles 0 € (-3π/2, π/2). What set D of complex numbers z = these angles? reio are described by (b) Define f(z) = = 1/4 ei0/4 where z = reie and 0 belongs to the set above. Check that f agrees with the usual fourth root function x → x¹/4 on the positive real axis. (c) Calculate f(-4) and sketch the branch sheet correspnding to f, i.e., the image set f(D) in the complex plane where D is the domain you found in part (a). (d) Consider a new function g(z) 1/40/4 where 0 € (π/2,5π/2) now. This a different branch of z¹/4. Check this is distinct from the first branch by calculating g(1) and checking f(1) = g(1). = (e) Calculate g(-4) and sketch the branch sheet corresponding to g, i.e., sketch the image set g (D) in the complex plane. (f) What is the relationship between the two branches f and g? Write a formula relating the two.