2.2 Power Series 101
Multiplication and Division of Powers Series*
Suppose that the two power series f (z) = 23’ anz" and 9(2) = 23’ b,z" both have
radius of convergence at least R for some positive number R; the product f (z)g(z)
is then deﬁned and analy
2.2 Power Series 99
= 1115-2 i (3)0: — 26)"!‘61'
F2
1:0
< mus-2 (3)02 — 26)"-i5i
= |h|6‘2[(R — 25) + 5]" = “ms-2m — a)".
This leads to the estimate
f(2 + h)—f(z)
h — 9(2)
s Ihla-z i2 lan|(R — a)".
This last quantity goes to zero as h —> 0 (recall that
2.2 Power Series 97
f(z) = i 4"('“"z".
n=0
That is,
The two power series
n=0
and
co 1 2n+1
f2(z) = "20 42n+1 2
have radii of convergence R 1 = 1% and R2 = 4, respectively, and
1
1m=ﬁm+ﬁm if m<1
Hence, the radius of convergence of f is at least %. But it
2.1.1 Flows, Fields, and Analytic Functions 87
Flows
Imagine a thin layer of incompressible liquid ﬂowing smoothly across a domain D
in the complex plane (Fig. 2.1). At each point z e D, we can ﬁnd the direction and
speed of the liquid, which we assume do
2.2 Power Series 95
|z — zol > R implies that Z an(z — 20)" diverges.
0
That is, R is the largest number with the property that whenever |z — zol < R, the
series 2a,(z — 20)" converges. When |z — zol = R, the series 23’ a,(z — 20)" may
converge or may div
2.1.1 Flows, Fields, and Analytic Functions 91
is a parametrization of the path, the tangent to the path is just dx/dt + i(dy/dt), so
— +1— = E = x(t) — iy(t).
Hence dx/dt = x, d y/dt = — y, and consequently x(t) = cle', y(t) = cze" for con-
stants c1 and
2.2 Power Series 93
2.2 Power Series
This section is devoted to examining functions of a very special nature: power series.
These functions represent a fusion of the ideas on infinite series introduced in
Section 4 of Chapter 1 with the concept of an anal
2.141 Flows, Fields, and Analytic Functions 89
termed fluxless if the net ﬂux of f across any smooth closed curve y in D is zero; that
is, if I, f ‘ n ds = 0 for any smooth closed curve y in D.
The reader will have noticed by now that there is a differenc
2.1 Analytic and Harmonic Functions; the Cauchy—Riemann Equations 85
For each function f listed in Exercises 8 to 11, ﬁnd an analytic function F with
Fl
8.
ll.
12.
13.
14.
15.
16.
17.
18.
19.
20.
f.
f(z)=z—2
f (z) = cosh (22)
Let f and g be analytic on a
2.1 Analytic and Harmonic Functions; the Cauchy—Riemann Equations 83
where E3 and E4 depend on 6 and v and approach zero as 6 and v approach
zero. Since the Cauchy—Riemann equations hold, replace 60/0y(x0, yo) with
(6u/0x)(xo, yo) and (6u/0y)(xo, yo) with
2.1 Analytic and Harmonic Functions; the Cauchy—Riemann Equations 81
out) that both it and u have continuous partial derivatives of ﬁrst and second order,
then
62a 62a 6 6a 6 6a
6x 6y 6x 6x 6y 6y
_ a 92 ,2 _av
_6x 6y 6y 6x
620 62v
=aa—y‘m=°
Hence, if u =
2
Basic Properties of
Analytic Functions
2.1 Analytic and Harmonic Functions; the Cauchy—Riemann Equations
Analytic functions and their close relatives, harmonic functions, are the stuff of
which the subject of complex variables is built. This section int
2.1 Analytic and Harmonic Functions; the Cauchy—Riemann Equations 79
To obtain this inequality, we used the triangle inequality, as well as the simple facts
that
However, each of the four quantities within absolute value signs approaches zero
as o and 1 i
1.6 Line Integrals and Green's Theorem 75
Iu(z)| S C for all 2. Let vR be the circle ]z| = R. Show that
, u(z)
l m ——
Rl—vq) N (Z — Zo)2
dz=0,
for each 20. (Hint: Use (3).)
16. Let 20 be outside a piecewise smooth simple closed curve y. Extend Example 13
Why are mitosis and meiosis both important to a living organism? When would an organism need to undergo each process of mitosis and meiosis? What would happen if meiosis did not occur?
Mitosis: Organisms have a certain number of chromosomes in their cel