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# Complex Variables - PowerPoint PPT Presentation

Complex Variables. Mohammed Nasser Acknowledgement: Steve Cunningham. Open Disks or Neighborhoods. Definition. The set of all points z which satisfy the inequality | z – z 0 |< , where  is a positive real number is called an open disk or neighborhood of z 0 .

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### Complex Variables

Mohammed Nasser

Acknowledgement:

Steve Cunningham

• 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

• Definition. A point is called an interior point of S if and only if there exists at least one neighborhood of z0which is completely contained in S.

z0

S

• 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.

Neither

Open

Closed

• An open set S is said to be connected if every pair of points z1and z2in 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

• 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 , is the set of S together with all of its boundary. Thus .

• 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.

• 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 .

• In effect, a function of a real variable maps from one real line to another.

f

• 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.

• 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.

y

v

w = f(z)

x

u

domain ofdefinition

range

z-plane

w-plane

• 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 :

Easier with normal form than polar form

Easier with polar form than normal form

magnitudes multiply!

For a complex number z2 ≠ 0,

phases subtract!

magnitudes divide!

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) = x2and 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.

v

y

f(z)

range

u

x

domain

Describe the function f(z) = z3for 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

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.

y

v

8

2

u

x

-2

2

-8

8

-8

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)

f(z)=z2, g(z)=|z| and h(z)=

• D={(x,x)|x is a real number}

• D={|z|<4| z is a complex number}

Draw the mappings

• Definition. A sequence of complex numbers, denoted , 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 , or more 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 Z0

Zn Z0

|zn-z0|=rn 0

Where rn=

, xn x0,,, yn y0

I proved it in previous classes

zn tends to z0 in any linear or curvilinear way.

• Definition. A sequence of complex numbers is 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

• Geometrically, this amounts to the fact that z0is the only point of znsuch that any neighborhood about it, no matter how small, contains an infinite number of points zn .

Theorem. Show that zn=xn+iyn z0=x0+iy0 if and only if xn x0, yn y0 .

Ex. Plot the first ten elements of the following sequences and find their limits if they exist:

• 1/n +i 1/n

• 1/n2 +i 1/n2

• n +i 1/n

• (1-1/n )n +i (1+1/n)n

• We say that the complex number w0is the limit of the function f(z) as z approaches z0 if f(z) stays close to w0whenever 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 z0is the number w0if for any real number  > 0 there exists a positive real number  > 0 such that |f(z) – w0|<  whenever 0<|z - z0|< .

We can interpret this to mean that if we observe points z within a radius  of z0, we can find a corresponding disk about w0such that all the points in the disk about z0are 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)

w0

z0

u

x

w-plane

z-plane

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)),

• If as z z0, lim f(z)  A, then A is unique

If as z z0, lim f(z)  A and lim g(z)  B, 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.

• 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

• 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.

• 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.

• 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 z0provided that g(z0)  0.

• Also, continuous functions map compact sets into compact sets.

• 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)

it is true in a general metric space but not in general topological space.

f: <S1,d1> <S2,d2> is continuous at s in S1.

For all sn s f(sn) f(s)