2.4 • BRANCHES OF F UNCTIONS 79- Let f (z) = z~ = r~ (cos~+ isin n, where z = re'^8 , r > 0, and -?r < 8 ::5 7r. Use
the polar form of z and show that
(a) f (z) _, i as z--> - 1 along the upper semicircle r = 1, 0 < 8 ~ 7r.
(b) f ( z) _, -i as z -+ -1 along the lower semicircle r = 1, -?r < 8 < 0.
Let f (z) = z~~;v• when z f 0, and let f (0) = 1. Show that f (z) is not continuous
at zo = 0.
Let f (z) = xeV + iy^2 e-•. Show that f (z) is continuous for all values of z.
Use the definition of the limit to show that .t:-3+4.i lim z^2 = - 7 + 24i.
Let f (z) = R;'ljl when z # 0, and let f (0) = 1. Is f (z) continuous at the origin?
Let f (z) = ~ when z f 0, and let f (0) = 0. Is f (z) continuous at the
origin?
J l i Ar.ii:(z)
Let f (z) = z2 = lzl 2 e • , where z # 0. Show that f (z) is discontinuous at
ea.ch point along the negative x-axis.
Let f (z) = In lzl +iArg z, where -?r < Arg z ::5 7r. Show that f ( z) is discontinuous
at zo = 0 and at each point along the negative x-axis.
Let lg(z)I <Mand lim f(z) = 0. Show that lim f(z)g(z) = 0. Note:
z-.to ~-ro
Theorem 2.2 is of no use here because you don't know whether lim g (z) exists.
::-to
Give an e, 15 argument.
20. Let {),.z = z -zo. Show that lim f (z) = wo iff lim f (zo + f),.z) = wo.
z-zo Az- o- Let f (z) be continuous for all values of z.
(a) Show that g (z) = f (z) is continuous for a.II z.
(b) Show that g (z) = f (z) is continuous for all z.- Verify the identities
(a) (2-18).
(b} (2-19).
(c) (2-19).- Verify the results of Theorem 2.4.
- Show that the principal branch of the argument, Arg z , is discontinuous at 0 and
all points along t he negative real axis.
2.4 Branches of Funct ions
In Section 2.2 we defined the principal square root function and investigated some
of its properties. We left unanswered some questions concerning the choices
of square roots. We now look at these questions because they are similar to
situations involving other elementary functions.