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Chap 1. Complex Numbers.1. Sums and Products.

Complex numbers can be defined as ordered pairs (x,y) of real numbers that are to be interpreted as points in the complex plane, with rectangular coordinates x and y, just as real numbers x are thought of as points on the real line.

• (x,0) real number

• (0, y) pure imaginary number

• It is customary to denote a complex number (x,y) by z , so that

x: real part of z Re z = x

y: imaginary part of z Im z = y

,z x y

2

1 1 1 2 2 2

1 2 1 1 2 2 1 2 1 2

1 2 1 1 2 2 1 2 1 2 1 2 1 2

Sum and product of , , , are defined as

, , ,

, , ,

z x y z x y

z z x y x y x x y y

z z x y x y x x y y y x x y

2

, ,0 0,

but 0,1 ,0 0,

,0 0,1 ,0

0,1

(0,1)(0,1) 1

z x y x y

y y

z x y

let i

Then z x iy

i

•

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2. Algebraic Properties

1 2 2 1

1 2 2 1

1 2 3 1 2 3

1 2 3 1 2 3

z z z z

z z z z

z z z z z z

z z z z z z

commutative law

associative law

1 2 1 2z z z zz zz distributive law

z+0=z 0=(0,0) additive identity

z 1=z 1=(1,0) multiplicative identity

For each z, there is a -z

such that z+(-z)=0 additive Inverse

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For any nonzero z=(x,y), multiplicative Inverse

There is a such that less obvious than additive inverse1z 1 1zz

2 2 2 2

12 2 2 2

, , 1,0

1

0

, , 0

x y u v

xu yv x yu v

yu xv x y x y

x yz z

x y x y

• Division by a non-zero complex number

111 2 2

2

0z

z z zz

if 1 1 1 2 2 2

11 1 2 1 2 1 2 1 21 2 2 2 2 2

2 2 2 2 2

1 2 1 2 1 2 1 22 2 2 2

2 2 2 2

, , ,

,

z x y z x y

z x x y y x x x yz z

z x y x y

x x y y x x x yi

x y x y

1 1 2 21

2 2 2 2 2

x iy x iyz

z x iy x iy

得到相同結果

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Other Identities

1 1 11 2 1 2 1 2

1 2 1 23 4

3 4 3 4

0, 0

0, 0

z z z z z z

z z z zz z

z z z z

Example.

1 1 1

2 3 1 2 3 1

1 5 5

5 5 5 26

5 5 1

26 26 26 26

i i i i

i i

i i i

ii

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3. Moduli and conjugates

It is natural to associate any nonzero complex number z=x+iy with the directed line segment, or vector, from the origin to the point (x,y) that represent z in complex plane.

In fact, we often refer to z as the point z or the vector z.

(x,y)

x0

x+iy

y

(-2,1) -2+i1z

2z

-Z2Z1-Z2

Z1+Z2

Z1

Z2

-Z2

1 2 1 2 1 2,z z x x y y

1 2 1 2z z z z

1 2

1 2

Although the product of two complex number z and z is itself a complex number

represented by a vector, that vector lies in the same plane as the vectors for z and z .

This product is neither the sc

alar nor the vector product used in vector analysis.

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•The modulus, or absolute value, of a complex number z=x+iy is defined as

2 2z x y length of the vector z.

distance between point z and 0

the distance between two points

is

1 1 1

2 2 2

z x iy

z x iy

2 2

1 2 1 2 1 2

0

z z x x y y

z z R a circle

RZ0

• 2 2 2Re Im

Re Re

Im Im

z z z

z z z

z z z

• complex conjugate of z =x+iy is

z x iy

z x+iy

x-iyz

8

• If

1 1 1 2 2 2

1 2 1 2 1 2 1 1 2 2

1 2

1 2 1 2

1 2 1 2

1 12

2 2

,

, 0

z x iy z x iy

z z x x i y y x iy x iy

z z

z z z z

z z z z

z zz

z z

2

1 2 1 2

2Re 2 Im

Re Im2 2

z z z z z i z

z z z zz z

i

z z z

z z z z

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4. Triangle Inequality

1 2 1 2z z z z

geometrically

1 2z z

2z

1z

1 2Equality holds when , ,0 are colinear.z z

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algebraically,

2

2

1 2 1 2 1 2 1 2 1 2

1 1 1 2 1 2 2 2

2 2

1 1 2 2

2 2

1 1 2 2

2 2

1 1 2

2

1 2

1 2 1 2

2Re

2

2

z z z z z z z z z z

z z z z z z z z

z z z z

z z z z

z z z z

z z

z z z z

Now

1 1 2 2 1 2 2 1 2 2

1 2 1 2

1 2 2 1 1 2 1 2similarly

z z z z z z z z z z

z z z z

z z z z z z z z

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2 2,

1 2 1 2

1 2 1 2

When is replaced by

z z

z z z z

z z z z

Example:

z on unit circle

1 or 0 1z z 33 3

33

2 2 2 3

2 2 1

z z z

z z

2

The triangle inequality can be generalized by mathematical induction to sums …

1 2 1 2... ... 2,3,...n nz z z z z z n

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5. Polar coordinates and Euler’s Formula

Let r, and be polar coordinates of the point (x,y) that corresponds to a non-zero complex number z=x+iy.

since

if z=0, the coordinate is undefined.

the length of the radius vector for z.

has an possible values.

Each value of is called an argument of z.

and the set of all such values is arg z

The principal value of arg z, Arg z, is that unique , s.t.

cos , sin ,

cos sin

x r y r

z r i

r z

arg Arg 2 0, 1, 2,...z z n n

using Euler’s formula

then

cos sini

i

e i

z re

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• Two non-zero complex numbers

are equal iff

1 21 1 2 2

i iz re z r e

1 2 1 2 and 2 , 0, 1, 2,...r r n n

0 0Re Re

0 2

i iz z R z z z z R

6. Product and Quotients in Exponential From

1 2

1 2

1 1 2 2

1 2 1 2

cos sin cos sin

cos sin

i i

i

e e i i

i

i e

If

1 2

1 2

1 1 2 2

1 2 1 2

,i i

i

z re z r e

z z r r e

(1)

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1 21 2

2 2

1 1 1

2 2 2

1 1 1

0, 1, 2,...

i ii

i i

i

n n in

z r e e re

z r e e r

z ez r

z r e n

1, 1, 2,...

cos sin cos sin

ni in

n

if r e e n

i n i n

Moivre’s formula

Ex. Find 7

3 i

76 6

77

7 663 2 2 2

64 3

2i i iii e e e e

i

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argument of product

1 2 1 2arg arg argz z z z

From Expression (1) is a value of

Z1Z2

Z2

Z11 2

1(7)

If we know two of these, can find the third.

A. If

B. If

1 1

2 2

arg

arg

z

z

1 2

1 2 1 2arg 2z z n

1 1 1arg 2z n

If we choose 2 2 1arg 2z n n

(7) is satisfied.

1 2arg z z

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C. Similarly for

1 2 1 2

2 2 2

arg 2

arg 2

z z n

z n

Then choose 1 1 2arg 2z n n

Z1Z2

Z2

Z11 2

1

1

2 1 2arg arg argZZ z z Finally

Ex:

1

2

1 2

Arg arg

1

z i

z

z z i

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7. Roots of Complex Numbers

Suppose z is nth root of a nonzero number .

0

0

0

0 0 2 0, 1, 2,...

n

in jn

n

z z

or r e r e

r r and n k k

00

0

2

20, 1, 2,...

nk

r rn

kk

n n

00

2exp 0, 1, 2,...n

kz r i k

n n

are the nth root of

These roots are on the circle

and are equally spaced every

0nz r

2

n

0z

0z

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All of the distinct roots are obtained when

k = 0,1,2,…,n-1

Let 00

2exp 0,1,2,..., 1n

k

kC r i k n

n n

and denote the set of nth roots of

(і) if is a positive real number then denotes the entire set of roots.

(іі) if in (1) is the principal value of arg

1

0nz

1

0nr

0

0

00 0 expnC r i

n

is referred to as the principal root.

denote these distinct roots

0z

0z 0r

0z

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Ex. nth roots of unity

1

1 1exp 0 2 0, 1, 2,..., 1

0 21 1exp

2exp 0,1,2,..., 1

2 : 1

3 :

nn

i k k n

ki

n n

ki k n

n

n

n

n = 4 ：

1

1

20

1 2 1

2Let exp

2then exp

and 1 1, , ,...,n

n

kn

nn n n

in

ki

n

Ex.2. Find 1

38i

8 8 8exp 2 0, 1, 2,...2

22exp 0,1,2

6 3k

i i i k k

kC i k

0

1 0 3

22 0 3

2exp 36

2

3

C i i

C i C

C i C

c1

c0c22

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• Regions in the complex Plane

closeness of points to one another

• -neighborhood or neighborhood

of a given point

0z z

• Deleted neighborhood

• Interior point

A point is said to be an interior point of a set S whenever there is

some neighborhood of that contains only points of S.

• Exterior point

when there exists a neighborhood of containing no points of S

00 z z

Z

Z0

SZ0

0z

0z

0z

0z

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• Boundary point all of whose neighborhoods contain points is S and points not in S Boundary = { all boundary points }

Ex. is the boundary of

and

1z

1

1

z

z

• A set is open if it contains none of its boundary points

• A set is closed if it contains all of its boundary points.

• The closure of a set S is the closed set consisting of all points in S together with

the boundary of S

is open

is closed and closure of

1

1

z

z

1, 1z z

0 1z - neither open nor closed.

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- The set of all complex number is both open and closed since it has no boundary points.

• An open set S is connected if each pair of points and in it can be joined by a polygonal line that lies entirely in S.

• An open set that is connected is called a domain. (any neighborhood is a domain )

• A domain together with some, none, or all of its boundary points is a region.

• A set S is bounded if every point of S lies inside some circle ; otherwise it is unbounded.

Z1

Z2 21open, connected

1 2z

Z R

1z 2z

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• A point is said to be an accumulation point of a set S if each deleted

neighborhood of contains at least one point of S.

- If a set S is closed, then it contain s each of its accumulation points.

pf: If an accumulation point were not in S, it would be a boundary point of S;

(can not be exterior points)

but this contradicts the fact that a closed set contains all of its boundary points.

Ex: For the set

the origin is the only accumulation point.

1,2,...n

iz n

n

0z

0z

0z