Math 132 - Complex Numbers and Functions - New …infohost.nmt.edu/~aitbayev/math132/Math132ComplexNumbers.doc · Web viewA complex number whose imaginary part is 0 is of the form

Math 132 - Complex Numbers and Functions

Many engineering problems can be treated and solved by using complex numbers and complex functions. We will look at complex numbers, complex functions and complex differentiation. Part I Complex Numbers In algebra we discovered that many equations are not satisfied by any real numbers. Examples are:

or We must introduce the concept of complex numbers. Definition: A complex number is an ordered pair of real numbers x and y. We call x the real part of z and y the imaginary part, and we write

, .

Two complex numbers are equal where and :

if and only if and

Addition and Subtraction of Complex Numbers: We define for two complex numbers, the sum and difference of and :

and .

Multiplication of two complex numbers is defined as follows:

We need to represent complex numbers in a manner that will make addition and multiplication easier to do.

1

Example 1: and

Example 2: Let and then

and

and .


Complex numbers represented as A complex number whose imaginary part is 0 is of the form and we have

and and

which looks like real addition, subtraction and multiplication. So we identify with the real number and therefore we can consider the real numbers as a subset of the complex numbers.

We let the letter and we call i a purely imaginary number.

Now consider and so we can consider the complex number = the real number . We also get

And so we have:

Now we can write addition and multiplication as follows:

and = .

The Complex PlaneThe geometric representation of complex numbers is to represent the complex number as the point .

y-axis 2 1

1 2 x-axis

2

Example 3: Let and , then

and


So the real number is the point on the horizontal x-axis, the purely imaginary number is on the vertical y-axis. For the complex number , is the real part and y is

the imaginary part.

How do we divide complex numbers? Let’s introduce the conjugate of a complex number then go to division.

Given the complex number , define the conjugate

We can divide by using the following:

Problem Set I Find 1. 2. 3. 4. 5. 6. 7. 8. 9. Let and and , find10. 11. 12.

13. 14. 15.

Graph the following: 16. 17. 18. and its conjugate.

19. Find the solutions of .

20. Find the solution of .

Complex Numbers in Polar Form

It is possible to express complex numbers in polar form. If the point is

represented by polar coordinates , then we can write , and

3

Example 4. Locate 2-3i on the graph above.

Example 5.


. r is the modulus or absolute value of z, , and is

the argument of z, . The values of r and determine z uniquely, but the converse

is not true. The modulus r is determined uniquely by z, but is only determined up to a multiple

of 2. There are infinitely many values of which satisfy the equations ,

but any two of them differ by some multiple of 2. Each of these angles is called an argument

of z, but, by convention, one of them is called the principal argument.

Definition If z is a non-zero complex number, then the unique real number , which satisfies

is called the principal argument of z, denoted by .

Note: The distance from the origin to the point is , the modulus of z; the argument of z

is the angle . Geometrically, is the directed angle measured from the positive x-

axis to the line segment from the origin to the point . When , the angle is

undefined.

The polar form of a complex number allows one to multiply and divide complex number

more easily than in the Cartesian form. For instance, if then

, . These formulae follow directly from DeMoivre’s formula.

4

Example 6. For , we get and . The

principal value of is , but would work also.

1

2

1z

1 2z zy

x


Multiplication and Division in Polar Form

Let and then we have

and

We can use =And so: DeMoivre's Theorem:

where n is an positive integer.

5

Example 7: and

Then =

Since

And

= =


Let to get: .

Roots of Complex Numbers:

Consider = (Equation 1)

where . Then , and so or .

However also satisfies Equation 1 and so . And

implies . However implies .

And continuing implies . for k any integer up to n.

We get , k=0, 1, 2, 3, , (n-1).

6

Example 8: Compute

Example 9: Find the square roots of i.

Since , we let

is one square root of . The

second square root of is :

.


7

Example 10. Find the sixth root of

There will be six roots:


Example 11: Compute

Solution: = =

Where . So

=

.

Problem Set II

Write in polar form:

1. 2. 3.

Write in rectangular form:

4. 5. 6.

7. Find 8. Find

9. Find 10. Find

8


Part II Functions, Neighborhoods and Limits We consider the concept of a function of a complex number. For , where x and y are real numbers, we know about limits, continuity and derivatives. Let the complex number

, where and . Here x and y are independent variables and where and . This is an example of a complex valued function, w, of a complex variable z. In general, let

where and

z

Distance in the complex plane. In the complex plane let and then

is the distance between the complex numbers and Note that is the distance from the origin to z. Definition: A neighborhood of the point in the complex plane is the set of points where and Definition: A function is said to have a limit L as approaches ( and written

) iff is defined in a neighborhood , except at and the

values of f are close to L for all z close to . Mathematically: for every , we can find such that for all in the neighborhood , so that if , then

. Notice that this definition of a limit is similar to the definition in calculus. The difference here is that z can approach from any direction in the complex plane. Definition: A function is continuous at if is defined and .

By definition a complex function that is continuous at is defined in a neighborhood of . is continuous in a domain D if it is continuous at each point in the domain D.

9


The derivative of a function at a point z is defined as:

provided that limit exists.

The rules for derivatives are the same for calculus:

c=constant

Definition: A function is defined to be analytic in the domain D if is defined and has a derivative at all points of D. The Exponential Function

The definition of the exponential function is given in terms of the real functions , , and

To show why this is true, consider the power series expansion: =

and substitute for x, to get:

=

= + = .

10

Example 12: Let . To compute we consider


The formula is known as Euler’s formula.

Theorem: is never zero.

Proof: , therefore, cannot be zero.

Theorem: If y is real, then .

Proof: and , which implies .

Theorem: if and only if , where n is an integer.

Proof: If , . Conversely, suppose , then . Since , this implies ; and where k is an integer.

Substituting this into implies . Hence k is even, since . Therefore, .

Theorem: if and only if , where n is an integer.

Proof: if and only if . Then use the previous theorem.

Definition of and

To get the definitions of and we substitute for y in Euler’s Formula to get:

= + = .

Adding and together and dividing by 2, we get:

.

Subtracting from , and dividing by , we get: .

We now define and .

11


Substitute ix for y in to get .

Similarly we get .

Facts: 1. For , to get 2. The derivative of is . 3. 4. For we get . 5. In polar form . 6. It now follows that . 7.

8.

9.

Logarithmic Functions

If then we write , called the natural logarithm of z. Thus the natural logarithm function is the inverse of the exponential function and can be defined by

where

where .

So we have that is multiple-valued, and in fact is infinitely many-valued. If we let then , where called the principal value or principal branch of .

To find the values of , let , where and so which implies .

12


We equate the real and imaginary parts:

(1) (2)

Squaring (1) and (2) and adding, we find or . Thus, from (1) and (2),

and so .

Hence we have

Problem Set III.

Sketch the point sets defined by the following 1. 2. 3.

Show that 4. 5. 6.

Evaluate each of the following7. 8. 9. 10. 11.

Find all possible values of :

12. 13. 14.

15. 16. 17.

18. 19.

20 Show that if and only if where n is any integer.

13

Example 13: Compute

Since , we have

The principle value is (let ).

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Math 132 - Complex Numbers and Functions - New …infohost.nmt.edu/~aitbayev/math132/Math132ComplexNumbers.doc · Web viewA complex number whose imaginary part is 0 is of the form