By Jan G. Krzyz
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A simply connected domain whose boundary consists of two circular arcs (not necessarily different, either arc can be replaced by a straight line segment), onto a disk, or half-plane. In particular the mapping w = z" for Ot: suitably chosen carries an angular sector with vertex at the origin into a half-plane. 9. CONFORMAL MAPPING OF CIRCULAR WEDGES 31 function transforms zero angle into a half-plane. It may even happen that a particular circular triangle with two right angles can be mapped onto a circular wedge and hence onto a half-plane.
Show that a function analytic in the extended plane except for a finite number of poles, is rational. 11. (z-a)+ ... + f Any pair of conjugate harmonic functions u,v determines an analytic function u+iv. 1. Find all the functions harmonic 10 C"", (- 00, 0] which are cOllstant one the rays argz = const. 2. Find all the functions harmonic in C"", 0 which are constant on the circles ceO; r). 3. Verify that the functions u = log Izl, v = argz are conjugate harmonic functions in C"", (-00,0] and Logz = log Izl+iArgz, where Argz is the principal value of argument: -7t < Argz < 7t, is analytic in C"",(- 00, 0]. 4. Verify that ~cosy, ~siny are conjugate harmonic functions in C.