(e) log (-3).
(f) log8.
(g) log ( 4i).
(h) log (- ./3 -i).5.2 • THE COMPLEX LOGARITHM 169- Use the properties of arg (z) in Section 1.4 to est ablish
(a) Equation (5-18).
(b) Equation (5-19).- Find all the values of 3 for which each equation holds.
(a) Log (z) = 1 -if.
(b) Log (z - 1) = i~.
(c) exp (z) = -ie.
(d) exp(z + 1) = i.- Refer to Theorem 5.2.
(a) Explain why -7r < Arg (z 1 ) + Arg (z2) $ 7r implies that Arg (z 1 z2) =
Arg (z1) + Arg (z2).
(b) Prove the "only if" part.- Pick an appropriate branch of the logarithm (see Equation (5-20)), find ~';,and
state where the formula is valid for
(a) w =log" (z - 1 -i).
(b) w = log 0 (l+i./3-z).
(c) w = z log" (z).
(d) w = log 0 (iz).(e) w = log 0 (z^2 -z+ 2).
(f) w = log 0 (z^2 + z + 1).
- Show that f (z) = ~2~;;.:J is analytic everywhere except at the points -1, -2,
and on the ray {(x,y): x $ -5,y = 0}. - Show that the following are harmonic functions in the right half-plane { z : Rez > 0}.
(a) u (:z:, y) =In (x^2 + y^2 ).
(b) v(x,y) =Arc tao(~).- Show that z" =exp [nlog,. (z)], where n is an integer and log., is any branch of
the logarithm. - Construct a branch off (z) =log (z + 4) that is analytic at the point z = - 5 a.nd
takes on the value 77ri there.