1. Establish the following:
(a) lim _n_ = 1
n-oo n+ 1
(b) lim –;- = 0
n-oo n + 1
() 1· 2nP + 5n + 1 _ 2
c 1m – ,
n–+oo nP + 3n + 1 p>l
zn
(d)-lim -,=0, ZEC.
n–+oo n.
2. For which values of z is the sequence {zn}~=1 bounded? For which
values of z does the sequence converge to O?
3. Show that {nnzn} converges only for z = O.
4. Show that J~oo Nk(:~ k)! = 1, k ~ O.
40 II Analytic Functions
5. Show that the sequence
1 1 1
bn = 1 + “2 + 3″ + … + -;;; – log n, n ~ 1,
is decreasing, while the sequence an = bn -lin is increasing. Show
that the sequences both converge to the same limit ‘Y. Show that
~ < ‘Y < ~. Remark. The limit of the sequence is called Euler’s
constant. It is not known whether Euler’s constant is a rational
number or an irrational number.
6. For a complex number a, we define the binomial coefficient “a
choose n” by
(~) = 1, ( a) = a(a-1) … (a-n+1),
n n! n~1.
Show the following.
(a) The sequence (~) is bounded if and only if Rea ~ -1.
(b) (~) –+ 0 if and only if Rea> -1.
(c) If a# 0,1,2, … , then (n: 1) / (~) –+-1.
(d) If Rea ~ -1, a# -1, then I(n: 1) I> I(~) Ifor all n~ O.
(e) If Re a> -1 and ais not an integer, then I(n ~ 1) I < I(~) I
for n large.
7. Define Xo = 0, and define by induction Xn+l = x; + ~ for n~ O.
Show that Xn –+ ~. Hint. Show that the sequence is bounded and
monotone, and that any limit satisfies x = x2 + ~.
8. Show that if Sn –+ S, then ISn – sn-ll –+ O.
9. Plot each sequence and determine its lim inf and lim sup.
1
(a) Sn = 1 + – + (-It (c) Sn = sin(7fnI4)
n
(b) Sn = (-nt (d) Sn = xn (x E lR fixed)
10. At what points are the following functions continuous? Justify your
answer. (a) z, (b) z/lzl, (c) z2/1zl, (d) z2/1z13.
11. At what points does the function Arg z have a limit? Where is
Arg z continuous? Justify your answer.
12. Let h(z) be the restriction of the function Arg z to the lower half-
plane {Imz < O}. At what points does h(z) have a limit? What is
the limit?
Exercises 41
13. For which complex values of Q does the principal value of ZCl have
a limit as z tends to O? Justify your answer.
14. Let h(t) be a continuous complex-valued function on the unit inter-
val [0,1), and consider
H(z) = r1 h(t) dt.
Jo t – z
Where is H(z) defined? Where is H(z) continuous? Justify your
answer. Hint. Use the fact that if If(t) – g(t)1 < c for 0::; t ::; 1,
then f01 If(t) – g(t)ldt < c.
15. Which of the following sets are open subsets of C? Which are closed?
Sketch the sets. (a) The punctured plane C\{O}, (b) the exterior
of the open unit disk in the plane, {I z I ~ I}, (c) the exterior of the
closed unit disk in the plane, {Izl > I}, (d) the plane with the open
unit interval removed, C\(O,l), (e) the plane with the closed unit
interval removed, C\[O, 1], (f) the semidisk {Izl < 1, Im(z) ~ O}, (g)
the complex plane C.
16. Show that the slit plane C\( -00,0) is star-shaped but not convex.
Show that the slit plane C\[-l, 1] is not star-shaped. Show that a
punctured disk is not star-shaped.
17. Show that a set is convex if and only if it is star-shaped with respect
to each of its points.
18. Show that the following are equivalent for an open subset U of the
complex plane.
(a) Any two points of U can be joined by a path consisting of
straight line segments parallel to the coordinate axes.
(b) Any continuously differentiable function hex, y) on U such that
‘V’h = 0 is constant.
(c) If V and Ware disjoint open subsets of U such that U = VU W,
then either U = V or U = W. Remark. In the context of
topological spaces, this latter property is taken as the definition
of connectedness.
19. Give a proof of the fundamental theorem of algebra along the fol-
lowing lines. Show that if p(z) is a nonconstant polynomial, then
Ip(z) I attains its minimum at some point Zo E C. Assume that the
minimum is attained at Zo = 0, and that p(z) = 1 + az m + … ,
where m ~ 1 and a i- o. Contradict the minimality by showing
that IP(ce i9o )1 < 1 for an appropriate choice of Bo
