= 1 2i Z C[w,R] f(z) (z w) 2 dz 1 2 max
zC[w,R] f(z) (z w) 2 2R = maxzC[w,R]
| f(z)| R M R . The right-hand side can be
made arbitrarily small, as we are allowed to
choose R as large as we want. This implies that f
0 = 0, and hence, by Theorem 2.14, f is
Definition. The series k1 bk converges
absolutely if k1 |bk | converges. CHAPTER 7.
POWER SERIES 96 Theorem 7.20. If a series
converges absolutely then it converges. This
seems like an obvious statement, but its proof
is, nevertheless, nontrivial. Proof.
showed its face in Chapter 2, but we study
them only now in more detail, since we have
more machinery at our disposal. This machinery
comes from complex-valued functions, which
are, nevertheless, intimately connected to
harmonic functions. 6.1 Definition
accumulation point then the sequence
diverges. 7.9. (a) Show that 1 k! 3 k(k+1) for
any positive integer k. (b) Conclude with
Example 7.9 that for any positive integer n, 1 +
1 2 + 1 6 + + 1 n! 3 . 7.10. Derive the
Archimedean Property from the Monotone
S
the function f restricted to D. It follows
immediately that the coefficients of a power
series are unique: Corollary 8.6. If k0 ck (z
z0) k and k0 dk (z z0) k are two power
series that both converge to the same function
on an open disk centered at z0, th
Example 6.4. Revisiting Example 6.1, we can see
that u(x, y) = xy is harmonic in C also by noticing
that f(z) = 1 2 z 2 = 1 2 x 2 y 2 + ixy is entire
and Im(f) = u. Example 6.5. A second reason
that the function u(x, y) = e x cos(y) from
Example 6.2 is ha
simply connected, so by Theorem 6.6 there is a
function f holomorphic in D[w, R] such that u =
Re f on D[w, R]. Now we apply Corollary 4.25 to
f : f(w) = 1 2 Z 2 0 f w + r eit dt . Theorem 6.9
follows by taking the real part on both sides.
Corollary 4.25
relative maximum or minimum in G. But then
neither does | f(z)|, because ln is monotonic.
We finish our excursion about harmonic
functions with a preview and its consequences.
We say a real valued function u on a region G
has a weak relative maximum at w
we leave the proof of the following to Exercise
7.11. Proposition 7.7. (a) Exponentials beat
polynomials: for any polynomial p(n) (with
complex coefficients) and any c C with |c| >
1, limn p(n) c n = 0 . (b) Factorials beat
exponentials: for any c C, limn
uniform convergence; see, e.g., Exercises 7.20
and 7.21, and the following result, sometimes
called the Weierstra M-test. Proposition 7.28.
Suppose fk : G C for k 1, and | fk(z)| Mk
for all z G, where k1 Mk converges. Then
k1 | fk | and k1 fk converge uni
R already in Example 4.29: Z R dz z 2 + 1 = .
This holds for any R > 1, and so we can take the
limit as R . By Proposition 4.6(d) and the
reverse triangle inequality (Corollary 1.7(b),
Z R dz z 2 + 1 max zR 1 z 2 + 1 R
max zR 1 |z| 2 1 R = R R2 1 which
g
HARMONIC FUNCTIONS 88 (b) Apply Theorem
6.9 to the function u(fa(z) with w = 0 to
deduce u(a) = 1 2i Z C[0,1] u(fa(z) z dz . (6.3)
(c) Recalling, again from Exercise 3.9, that fa(z)
maps the unit circle to itself, apply a change of
variables to (6.3) to p
valid with limx!x0 replaced throughout by
limx!1 (limx!1). How would their proofs
have to be changed? 17. Using the definition
you gave in Exercise 2.1.14, show that (a) lim
x!1 1 1 x 2 D 1 (b) lim x!1 2jxj 1 C x D 2 (c) lim
x!1 sin x does not exist 18. F
For the remainder of this chapter (indeed, this
book) we concentrate on some very special
series of functions. Definition. A power series
centered at z0 is a series of the form k0 ck (z
z0) k where c0, c1, c2, . . . C. Example 7.30.
A slight modification
equal. In Exercises 2.1.282.1.30 consider only
the case where at least one of L1 and L2 is 1.
28. Prove: If limx!x0 f .x/ D L1, limx!x0 g.x/ D
L2, and L1 C L2 is not indeterminate, then lim
x!x0 .f C g/.x/ D L1 C L2: 52 Chapter 2
Differential Calculus of
phrase this precisely, we need the following.
Definition. The sequence (an) is monotone if it
is either nondecreasing (an+1 an for all n) or
nonincreasing (an+1 an for all n). There are
many equivalent ways of formulating the
completeness property for the
Archimedean Property and the Least Upper
Bound Property can be used in (different)
axiomatic developments of R. 3Archimedes of
Syracuse (287212 BCE) attributes this property
to Eudoxus of Cnidus (408355 BCE). CHAPTER
7. POWER SERIES 93 mention them here
e
R R f = 0 and hence limR R [R,R] f = e .
(d) Conclude, by just considering the real part,
that Z cos(x) x 2 + 1 dx = e . 5.18.
Compute Z cos(x) x 4 + 1 dx . CHAPTER 5.
CONSEQUENCES OF CAUCHYS THEOREM 80
5.19. This exercise outlines how to extend some
of t
on any domain. State why. (a) sin f .x/ D x (b) e
f .x/ D jxj (c) 1 C x 2 C f .x/2 D 0 (d) f .x/f
.x/ 1 D x 2 2. If f .x/ D r .x 3/.x C 2/ x 1 and
g.x/ D x 2 16 x 7 p x 2 9; find Df , Df g, Dfg, and
Df =g. Section 2.1 Functions and Limits 49 3.
Find Df .
1, and (fn) converges uniformly to the zero
function in G. Show that, if (zn) is any sequence
in G, then limn fn(zn) = 0 . (b) Apply (a) to
the function sequence given in Example 7.23,
together with the sequence (zn = e 1 n ), to
prove that the convergenc
POWER SERIES 97 (There is a small technical
detail to be checked here, since we are
effectively ignoring half the partial sums of the
original series; see Exercise 7.17.) Since 1 2k 1
1 2k = 1 2k(2k 1) 1 (2k 1) 2 1 k 2 , k1
(1) k+1 k converges by Corolla
sin z (z 2 + 1 2 ) 2 (h) 1 (z + 4)(z 2 + 1) (i) exp(2z)
(z 1) 2(z 2) 5.4. Compute Z C[0,2] exp z (z
w) 2 dz where w is any fixed complex number
with |w| 6= 2. 5.5. Define f : D[0, 1] C
through f(z) := Z [0,1] dw 1 wz (the integration
path is from 0 to 1
exists then the radius of convergence of k0
ck(z z0) k equals R = if limk ck+1
ck = 0 , limk ck ck+1 otherwise. 7.34.
Find the radius of convergence for each of the
following series. (a) k0 a k 2 z k where a C
(b) k0 k n z k where n Z (c) k0 z k! (d)
k1
by writing f .x/ D e 2x.1 e x /; we find that lim x!
1 f .x/ D lim x!1 e 2x lim x!1 1 lim x!1 e x D 1.1
0/ D 1: 44 Chapter 2 Differential Calculus of
Functions of One Variable Example 2.1.15 Let
g.x/ D 2x2 x C 1 3x2 C 2x 1 : Trying to find limx!1
g.x/ by
(sometimes the nonnegative) integers to the
complex numbers. Its values are usually written
as an (as opposed to a(n) and we commonly
denote the sequence by (an) n=1 , (an)n1 ,
or simply (an). Considering such a sequence as
a function of n, the notion of
(a) Z exp(z) z 3 dz (b) Z exp(z) (z i) 2 dz
(c) Z sin(2z) (z ) 2 dz (d) Z exp(z) cos(z)
(z ) 3 dz 5.2. Prove the formula for f 00 in
Theorem 5.1. Hint: Modify the proof of the
integral formula for f 0 (w) as follows: (a) Write
a difference quotient for f
can be replaced by > in (2.1.20), f is decreasing
on I . In either of these two cases, f is strictly
monotonic on I . Example 2.1.16 The function
f .x/ D ( x; 0 x < 1; 2; 1 x 2; is nondecreasing on
I D 0; 2 (Figure 2.1.4), and f is nonincreasing
on I D 0;
= k1 k Re(z) converges. Viewed as a
function in z, the series (z) is the Riemann zeta
function, an indispensable tool in number
theory and many other areas in mathematics
and physics.4 Another common mistake is to
try to use the converse of Theorem 7.20,
the power series k0 ck(z z0) k has radius of
convergence R > 0 and is a piecewise smooth
path in D[z0, R]. Then Z k0 ck(z z0) k dz =
k0 ck Z (z z0) k dz . In particular, if is
closed then Z k0 ck(z z0) k dz = 0 . Proof.
Let r := maxz |(z) z0| (whose exis
history of complications (eg, malignancies,
opportunistic infections) may not tolerate fulldose therapy or may not tolerate mitomycin
and require dosage adjustment or treatment
without mitomycin. f See Principles of
Chemotherapy (ANAL-A). gSee Principles