Transcript Document

Complex Variables
Mohammed Nasser
Acknowledgement:
Steve Cunningham
Open Disks or Neighborhoods


Definition. The set of all points z which satisfy
the inequality |z – z0|<, where  is a positive
real number is called an open disk or
neighborhood of z0 .
Remark. The unit disk, i.e., the neighborhood
|z|< 1, is of particular significance.
1
Interior Point

Definition. A point z 0  S is called an interior point
of S if and only if there exists at least one
neighborhood of z0 which is completely contained
in S.
z0
S
Open Set. Closed Set.



Definition. If every point of a set S is an interior point of
S, we say that S is an open set.
Definition. S is closed iff Sc is open. Theorem: S`  S,
i.e., S contains all of its limit points
S is closed set.
Sets may be neither open nor closed.
Open
Closed
Neither
Connected

An open set S is said to be connected if
every pair of points z1 and z2 in S can be
joined by a polygonal line that lies entirely in
S. Roughly speaking, this means that S
consists of a “single piece”, although it may
contain holes.
S
z1
z2
Domain, Region, Closure,
Bounded, Compact



An open, connected set is called a domain. A region
is a domain together with some, none, or all of its
boundary points. The closure of a set S denoted S,
is the set of S together with all of its boundary. Thus
S  S  B(S.)
A set of points S is bounded if there exists a positive
real number r such that |z|<r for every z  S.
A region which is both closed and bounded is said to
be compact.
Review: Real Functions of Real
Variables

Definition. Let   . A function f is a rule which
assigns to each element a   one and only one
element b  ,   . We write f:  , or in the
specific case b = f(a), and call b “the image of a
under f.”
We call  “the domain of definition of f ” or simply
“the domain of f ”. We call  “the range of f.”
We call the set of all the images of , denoted f (),
the image of the function f . We alternately call f a
mapping from  to .
Real Function

In effect, a function of a real variable maps
from one real line to another.
f


Complex Function

Definition. Complex function of a complex
variable. Let   C. A function f defined on 
is a rule which assigns to each z   a
complex number w. The number w is called
a value of f at z and is denoted by f(z), i.e.,
w = f(z).
The set  is called the domain of definition of
f. Although the domain of definition is often a
domain, it need not be.
Remark


Properties of a real-valued function of a real variable
are often exhibited by the graph of the function. But
when w = f(z), where z and w are complex, no such
convenient graphical representation is available
because each of the numbers z and w is located in
a plane rather than a line.
We can display some information about the function
by indicating pairs of corresponding points z = (x,y)
and w = (u,v). To do this, it is usually easiest to draw
the z and w planes separately.
Graph of Complex Function
y
w = f(z)
v
x
z-plane
domain of
definition
u
range
w-plane
Arithmetic Operations in Polar Form

The representation of z by its real and imaginary
parts is useful for addition and subtraction.

For multiplication and division, representation by
the polar form has apparent geometric meaning.
Suppose we have 2 complex numbers, z1 and z2 given by :
z 1  x 1  iy 1  r1e
i1
z 2  x 2  iy 2  r2e

 x
i2
 
  i y
z 1  z 2  x 1  iy 1  x 2  iy 2

1
 x2
z 1z 2  r1e
i1
 r1r2e
magnitudes multiply!
1
 y2
r e 
i2
2
i (1 ( 2 ))
phases add!


Easier with normal
form than polar form
Easier with polar form
than normal form
For a complex number z2 ≠ 0,
z1
z2

r1e
r2e
i1
i2

magnitudes divide!
z1
r1

z2
r2
r1
r2
e
i (1 ( 2 ))

r1
r2
e
i (1 2 )
phases subtract!
z  1  ( 2 )  1   2
Example 1
Describe the range of the function f(z) = x2 + 2i, defined on (the domain is) the
unit disk |z| 1.
Solution: We have u(x,y) = x2 and v(x,y) = 2. Thus as z varies over the closed
unit disk, u varies between 0 and 1, and v is constant (=2).
Therefore w = f(z) = u(x,y) + iv(x,y) = x2 +2i is a line segment from w = 2i to w
= 1 + 2i.
y
domain
f(z)
x
v
range
u
Example 2
Describe the function f(z) = z3 for z in the semidisk given by |z| 2, Im z
 0.
Solution: We know that the points in the sector of the semidisk from
Arg z = 0 to Arg z = 2/3, when cubed cover the entire disk |w| 8
because
  3 
2i
2e
  8e
The cubes of the remaining points of z also fall into this disk,
overlapping it in the upper half-plane as depicted on the next screen.
i 2
3
w = z3
y
v
8
2
-2
2
x
-8
8
-8
u
If Z is in x+iy form
z3=z2..z=(x2-y2+i2xy)(x+iy)=(x3-xy2+i2x2y+ix2y-iy3 -2xy2)
=(x3-3xy2) +i(3x2y-y3)
If u(x,y)= (x3-3xy2) and v(x,y)=(3x2y-y3), we can write
z3=f(z)=u(x,y) +iv(x,y)
Example 3
f(z)=z2, g(z)=|z| and
h(z)= z
i)
D={(x,x)|x is a real number}
ii)
D={|z|<4| z is a complex number}
Draw the mappings
Sequence

Definition. A sequence of complex numbers,
denoted
k
zn1, is a function f, such that f: N  C, i.e, it is a
function whose domain is the set of natural numbers
between 1 and k, and whose range is a subset of
the complex numbers. If k = , then the sequence is
called infinite and is denoted by zn, or more
1
often, zn . (The notation f(n) is equivalent.)

Having defined sequences and a means for
measuring the distance between points, we proceed
to define the limit of a sequence.
Meaning of Zn
Zn
Z0
|zn-z0|=rn
Where rn=
, xn
x0,,, yn
Z0
0
(x n  x 0 )2  (y n  y 0 )2
y0
I proved it in previous classes
Geometric Meaning of Zn
Z0
zn tends to z0 in any linear or curvilinear way.
Limit of a Sequence

Definition. A sequence of complex numberszn is
1
said to have the limit z0 , or to converge to z0 , if for
any  > 0, there exists an integer N such that |zn –
z0| <  for all n > N. We denote this by
lim zn  z0 or
n

zn  z0 as n  .
Geometrically, this amounts to the fact that z0 is the
only point of zn such that any neighborhood about it,
no matter how small, contains an infinite number of
points zn .
Theorems and Exercises
Theorem. Show that zn=xn+iyn
only if xn x0, yn y0 .
z0=x0+iy0 if and
Ex. Plot the first ten elements of the following sequences
and find their limits if they exist:
i)
1/n +i 1/n
ii)
1/n2 +i 1/n2
iii)
n +i 1/n
iv)
(1-1/n )n +i (1+1/n)n
Limit of a Function


We say that the complex number w0 is the limit of
the function f(z) as z approaches z0 if f(z) stays
close to w0 whenever z is sufficiently near z0 .
Formally, we state:
Definition. Limit of a Complex Sequence. Let f(z) be
a function defined in some neighborhood of z0
except with the possible exception of the point z0 is
the number w0 if for any real number  > 0 there
exists a positive real number  > 0 such that |f(z) –
w0|<  whenever 0<|z - z0|< .
Limits: Interpretation
We can interpret this to mean that if we observe points z within a radius  of z0,
we can find a corresponding disk about w0 such that all the points in the disk
about z0 are mapped into it. That is, any neighborhood of w0 contains all the
values assumed by f in some full neighborhood of z0, except possibly f(z0).
v
y
w = f(z)
 z
0
z-plane

w0
x
w-plane
u
Complex Functions : Limit and
Continuity
f: Ω1
Ω2
Ω1 and Ω2 are domain and codomain respectively.
Let z0 be a limit point of Ω1 , w0 belongs to Ω2 .
Let us take any B any nbd of w0 in Ω2 and take inverse of B, f-1{B}.
f-1{B} contains a nbd of z0 in Ω1..
In the case of Continuity the only difference is w0
=f(z0)),
Properties of Limits
as z  z0, lim f(z)  A, then A is unique
If as z  z0, lim f(z)  A and lim g(z)  B,
 If
then
lim [ f(z)  g(z) ] = A  B
 lim f(z)g(z) = AB, and
 lim f(z)/g(z) = A/B. if B  0.

Continuity

Definition. Let f(z) be a function such that f: C
C. We call f(z) continuous at z0 iff:

F is defined in a neighborhood of z0,
The limit exists, and

zz0


lim f ( z)  f ( z0 )
A function f is said to be continuous on a set
S if it is continuous at each point of S. If a
function is not continuous at a point, then it is
said to be singular at the point.
Note on Continuity

One can show that f(z) approaches a limit
precisely when its real and imaginary parts
approach limits, and the continuity of f(z) is
equivalent to the continuity of its real and
imaginary parts.
Properties of Continuous
Functions


If f(z) and g(z) are continuous at z0, then so
are f(z)  g(z) and f(z)g(z). The quotient
f(z)/g(z) is also continuous at z0 provided that
g(z0)  0.
Also, continuous functions map compact sets
into compact sets.
Exercises

i.
ii.
iii.
iv.
v.
vi.
Find domain and range of the following
functions and check their continuity:
f1(z)=z
f2(z)=|z|
f3(z)=z2
f4(z)=
f5(z)=1/(z-2)
f6(z)=ez/log(z)/z1./2/cos(z)
z
f: <S1,d1>
s in S1.
For all sn
<S2,d2> is continuous at
s
f(s )

n
f(s)
it is true in a general metric space but not
in general topological space.
Test for Continuity of Functions