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Chapter (1) Complex numbers

In this chapter, we survey the algebraic and geometric structure of the complex number system. We assume various corresponding properties of real numbers to be known.

1. SUMS AND PRODUCTS Definitions: (1) 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. (2) Complex numbers of the form (x, 0) correspond to points on the x axis and are called pure real numbers. The x axis is, then, referred to as the real axis. (3) Complex numbers of the form (0, y) correspond to points on the y axis and are called pure imaginary numbers. The y axis is, then, referred to as the imaginary axis. (4) The real numbers x and y are, moreover, known as the real and imaginary parts of z, respectively; and we write Re z = x, Im z = y.

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Properties of complex numbers: (1)Two complex numbers z1 = ( x 1, y1) and z2 = (x2, y2) are equal whenever they have the same real parts and the same imaginary parts. Thus the statement zl = z2 means that zl and z2 correspond to the same point in the complex, or z- plane. (2) The sum zl + z2 and the product zlz2 of two complex numbers zl= (xl, yl) and z2 = (x2, y2) are defined as follows: (x1 ,y1)+(x2,y2)=(x1+x2, y1+y2) (x1 ,y1) (x2,y2) = (x1x2-y1y, x1y2+x2y1) Notes: 1) The operations defined above become the usual operations of addition and multiplication when restricted to the real numbers: (x1,0)+(x2,0)=(x1+x2,0) (x1,0)(x2,0) = (x1x2,0) 2) Any complex number z = (x, y) can be written z = (x, 0) + (0, y), and it is easy to see that (0, l)(y, 0) = (0,y). Hence

z = (x,0)+(0,1)(y,0)

3) let i denote the imaginary number (0, 1), it is clear that

z = x+ i y

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Also, with the convention z2 = z z, z3 = z z2, etc,, we find that i2 = (0, l)(0, 1) = (-1, 0), or i2 = -1 So, (x1 + iy 1) +(x2 + iy2) = (x 1 + x2) +i(y1 + y2), (x1 + iy 1) (x2 + iy2) = (x1 x2 - y1 y2)+i(y1x2+x1y2).

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2. BASIC ALGEBRAIC PROPERTIES

Various properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the more basic of these algebraic properties and verify some of them.

1) The commutative laws: z1+z2=z2+z1, z1z2=z2z1

2) The associative laws: (z1+z2)+z3=z1+(z2+z3) and (z1z2) z1= z1(z2z3)

3) The distributive law: z1(z2+z3)=z1z2+z1z3

These follow easily from the definitions in Sec. 1 of addition and multiplication of complex numbers and the fact that real numbers obey these laws. 4) The additive identity 0 = (0,0) and the multiplicative

identity 1 = (1,0) for real numbers carry over to the entire complex number system. That is, z+0=z and z . 1 = z

for every complex number z. Furthermore, 0 and 1 are the only complex numbers with such properties.

5) There is associated with each complex number

z = (x,y) an additive inverse - z=(-x,-y) = -x - iy satisfying the equation z + (-z) = 0.

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Moreover, there is only one additive invers for any given z.

Additive inverses are used to define subtraction: z1-z2=z1+(-z2) = (x1-x2,y1-y2)= (x1-x2)+i(y1-y2). 6) Multiplicative inverse: for any nonzero complex

number z = (x, y),there is a number z-1 such that zz-1=1. Examples: H.W. 1,2,3,4,8,9,10

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3. FURTHER PROPERTIES :

In this section, we mention a number of other algebraic properties of addition and multiplication of complex numbers that follow from the ones already described in sec2.

Properties :

1) if zl z2 = 0, either z1 = 0 or z2 = 0; or possibly both zl and z2 equal zero.

Another way to state this result is that if two complex numbers zl and z2 are nonzero, then so is their product z lz2.

2) Division by a nonzero complex number is defined as follows:

a) 2

1

zz

= z1z2-1 , z2 0≠

b) 3

2

3

1132

131

1321

3

21 )(zz

zzzzzzzzz

zzz

+=+=+=+ −−− , z3≠ 0

using the relation .0,0,)( 211

21

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21 ≠≠= −−− zzzzzz the following identity hold.

d) 21

1zz

= ).0,(),1)(1()( 2121

12

11

121 ≠== −−− zz

zzzzzz

e) .0,(, 434

2

3

1

43

21 ≠

= zz

zz

zz

zzzz

Examples:

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4. MODULI 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 represents z (Sec. 1) in the complex plane. In fact, we often refer to z as the point z or the vector z. According to the definition of the sum of two

complex numbers zl = x1+iy1 and z2=x2+y2 , the number z1 + z2 corresponds to the point (x1+x2,y1,y2) .

It also corresponds to a vector with those coordinates as its components The difference zl – z2 = z1+ (-z2) corresponds to the

sum of the vectors for zl and -z2. Although the product of two complex numbers zl and

z2 is itself a complex number represented by a vector, that vector lies in the same plane as the vectors for z1 and z2.

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

22 yx + and is denoted by |z|; that is, .22 yxz +=

Geometrically, the number |z | is the distance between the point (x, y) and the origin, or the length of the vector representing z.

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Note that, while the inequality zl < z2 is meaningless unless both zl and z2 are real, the statement |z1| < |z2| means that the point zl is closer

to the origin than the point z2 is. Example: The distance between two points zl = x1+iy1 and

z2 = x2+iy2 is |z1-z2| = .)()( 2

212

21 yyxx −+−

The complex numbers z corresponding to the points lying on the circle with center z0 and radius R thus satisfy the equation |z – z0| = R, and conversely. Example:

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Proposition: Let z be a complex number. Then 1. |z|2 = (Re z)2 + (Im z)2. 2. Re(z) ≤ |Re(z)| ≤ |z| . 3. Im(z) ≤ |Im(z)| ≤ |z| 4. Triangle inequalities : |z1±z2| ≤ |z1| + |z2|. 5. |z1±z2| ≥ ||z1|-|z2||. Examples:

H.W. (1-5)

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5. COMPLEX CONJUGATES

The complex conjugate, or simply the conjugate, of a complex number z = x + iy is defined as the complex number x - iy and is denoted by z ; that is, iyxz −= .

The number z is represented by the point (x, -y), which is the reflection in the real axis of the point ( x , y) representing z Properties :

1) ||||, zzandzz == for all z.

2) the conjugate of the sum or difference is the sum or difference of the conjugates that is:

3) 2121

2121

,

zzzz

zzzz

=

±=±

4) 2

1

2

1

zz

zz

=

5) The sum zz + of a complex number z = x + iy and its conjugate z = x – iy is the real number 2x, and the difference zz − is the pure imaginary number 2iy. Hence

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6) An important identity relating the conjugate of a complex number z = x + iy to its modulus is:

7) |z1z2| = |z1| |z2|

8) 0 , 22

1

2

1 ≠= zzz

zz

H.W. ( 1-16)

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6. EXPONENTIAL FORM

Let r and θ be polar coordinates of the point (x, y) that corresponds to a nonzero complex number

z = x + iy.

Since x = r cos θ and y = r sinθ , the number z can be

written in polar form as

z =r(cosθ + i sinθ).

Notes:

1) If z = 0, the coordinate θ is undefined; and so it is always understood that z # 0

2) In complex analysis, the real number r is not allowed to be negative and is the length of the radius vector for z; that is, r = |z|.

3) The real number θ represents the angle, measured in radians, that z makes with the positive real axis . 4) As in calculus, θ has an infinite number of

possible values, including negative ones, that differ by integral multiples of 2n.

5) Those values can be determined from the

equation tan θ = y / x , where the quadrant containing the point corresponding to z must be specified.

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Definitions :

1) Each value of θ is called an argument of z, and the set of all such values is denoted by arg z.

2) The principal value of arg z, denoted by Arg z, is that unique value Θ such that -π < Θ ≤ π. Note that 1) arg z = Arg z + 2nπ (n = 0, ±1, ±2, ...).

2) when z is a negative real number, Arg z has value

π, not -π . Example : Find Arg (-1-i) and arg(-1-i)

Definition: The symbol eiθ or exp(iθ), is defined by means of Euler's formula as eiθ= cos θ + i sin θ, where θ is to be measured in radians. It enables us to write the polar form in exponential z = r eiθ

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EXAMPLE 2. Write The number -1-i in exponential form. Note: Exponential expression with r = 1 tells us that the numbers eiθ lie on the circle centered at the origin with radius unity. Geometrically it is obvious that eiπ = -1 , eiπ/2 = -I , and e-i4π = 1 Note, too, that the equation z=Reiθ , (0 ≤ θ ≤ 2π) is a parametric representation of the circle |z| = R, centered at the origin with radius R. More generally, the circle |z - zol = R, whose center is zo and whose radius is R, has the parametric representation z = z0 + Reiθ , (0 ≤ θ ≤ 2π) Examples:

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7. PRODUCTS AND QUOTIENTS IN EXPONENTIAL FORM

Properties :

)(

212121 )1 θθ += ierrzz

)(

2

1

2

1 21 )2 θθ −= ierr

zz

θier

z −− =1 )3 1

4) arg z-1 = - arg z .

5) arg( z1z2) = arg z1+ arg z2 .

6) arg(z1/z2) = arg z1 – arg z2 .

Examples:

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7) zn = rn einᶿ , n=0,±1,±2,………..

observe that if r = 1, expression (7) becomes

Called ( de Moivre,s formula)

Examples:

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8. ROOTS OF COMPLEX NUMBERS

Consider now a point z = reiᶿ , lying on a circle centered at the origin with radius r.

Definitions:

1) two nonzero complex numbers z1 = rl eiθ1 and

z2 = r2 eiᶿ2 are equal if and only if rl = r2 and θ1 = θ2 + 2kπ, where k is some integer (k = 0, ± 1, ±2, . . .).

2) An nth root of z0 is a nonzero number z = reiθ such that zn = zo .

Or rn einθ = r0 eiθ0 that is

rn = r0 and nθ = θ0+2kπ , k is integer.

So n rr 0= ,where this radical denotes the unique positive nth root of the positive real number ro , and

Thus the complex numbers

are the nth roots of z0 .

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The distinct roots obtained when k = 0,1,2,…..,n-1. Let ck denote these roots and write

Remarks: 1) The number n r0 is the length of each of the radius

vectors representing the n roots.

2) The first root co has argument θ0/n ; and the two roots when n = 2 lie at the opposite ends of a diameter of the circle |z| = n r0 ,th e second root being -co.

3) When n ≥ 3, the roots lie at the vertices of a regular

polygon of n sides inscribed in that circle.

4) When the value of Θ0 that is used is the principal value of arg zo (-π < Θ0 ≤ π), the number co is referred to as the principal root. Thus when zo is a positive real number ro, its principal root is n r0 .

Section 9: Examples:

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10. REGIONS IN THE COMPLEX PLANE In this section, we are concerned with sets of complex numbers, or points in the z plane, and their closeness to one another. Definitions:

1) An ε neighborhood of z0 | z - z0 | < ε of a given point zo. It consists of all points z lying inside but not on a circle centered at zo and with a specified positive radius ε .

2) A deleted neighborhood 0 < | z - z0|< ε

consisting of all points z in an ε neighborhood of zo except for the point zo itself.

3) A point z0 is said to be an interior point of a set S whenever there is some neighborhood of zo that contains only points of S.

4) A point z0 it is called an exterior point of S when

there exists a neighborhood of it containing no points of S .

5) If zo is neither of these, it is a boundary point of S.

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A boundary point is, therefore, a point all of whose

neighborhoods contain points in S and points not in S.

6) The totality of all boundary points is called the

boundary of S. Example:

7) A set is open if it contains none of its boundary points.

8) A set is closed if it contains all of its boundary points; and the closure of a set S is the closed set consisting of all points in S together with the boundary of S.

9) For a set to be not open, there must be a boundary point that is contained in the set; and if a set is not closed, there exists a boundary point not contained in the set, the point is neither open nor closed .

10) An open set S is connected if each pair of points zl and z2 in it can be joined by a polygonal line, consisting of a finite number of line segments joined end to end, that lies entirely in S.

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11) An open set that is connected is called a domain.

12) A domain together with some, none, or all of its boundary points is referred to as a region.

13) A set S is bounded if every point of S lies inside some circle | z | = R; otherwise, it is unbounded .

14) A point z0 is said to be an accumulation point of a set S if each deleted neighborhood of zo contains at least one point of S .

15) A point zo is not an accumulation point of a set S whenever there exists some deleted neighborhood of zo that does not contain points of S. Examples: