STUDY ON COMPLEX INTEGRATION AND SOME OF ITS APPLICATION

CHAPTER ONE

1.0             INTRODUCTION

1.1             Background of the Study

The fundamental theorem, often called Blasius’s Theorem, is valid for both simply and multiple connected regions. It was first proved by use of Green’s theorem with the added restriction the  be contour. However, Cauchy gave a proof which removed this restriction. For this reason, the theorem is sometimes called the Cauchy – Gousart theorem. The Cauchy Integral theorem is an important statement about the integrals for holomorphic functions in the complex plane.

1.2             Statement of the Problem

The projects studies applications of Complex Integral. However, the problem is to investigate Complex Integral Theorems and their applications.

1.3             Aim and Objectives of the Study

The aim of this study is to examine Complex Integration and some of its applications

The objectives are to:

  1. Investigations of theorems in Complex Integrals to include Cauchy theorem, Gauss, mean value theorem, Liouville’s theorem, Maximum Modulus theorem and Blasius Theorems
  2. Examinations of some application of Complex Integral

 

 

1.4             Significance of the Study

This study reviews a number of relevant theorems in complex analysis. And ultimately, it reviews Complex Integrals and its application in Engineering

1.5             Definition of Terms

  1. 1.     Complex Number: A complex number can be expressed in the form a + bi, where a andb are real numbers and i is the imaginary unit which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.
  2. 2.     Complex Function: Let S be a set of complex numbers. A functionf defined on S is a rule that assigns to each z in S a complex number w. The number w is called the value of f at z and is denoted by f (z); that is .
  3. 3.    Differentiable Function: Function f is said to be differentiable at when exists.
  4. 4.     Analytic Function: A function of a complex variable that has a derivative at every point within a region of complex plane is said to be analytic (or regular or holomorphic) over that region.
  5. 5.     Entire Function: If the function is analytic everywhere in the complex plane, it is entire.
  6. 6.     Line Integral: line integral of scalar fields over a curve C do not dependent on the chose parametrization r of C.
  7. 7.     Contour: Is defined as a curve consisting of a finite number of smooth curves joined end to end.
  8. 8.     Simple Close Contour: A contour is said to be a simple closed contour if the initial and final values of  are the same and the contour does not cross itself.
  9. 9.     Contour Integral: We write = . Let  and take the limit k

.

The above limit is defined to be the contour integral of  along the contour C.

If the above limit exists, then the function  is said to be integrable along the contour C.

If we write

 

then

 

Writing  and

 

=

Simply Connected Domain: A simply connected domain is a path-connected domain where one can continuously shrink any simple closed curve into a point while remaining in the domain. For two-dimensional regions, a simply connected domain is one without holes in it. For three-dimensional domains, the concept of simply connected is more subtle.

Boundary Value Problem: is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Viscosity: The resistance of a fluid to flow is a fundamental concept to understand current viscometer technologies and liquid characterization.

Arithmetic Operation of Complex Numbers

Various properties 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.

Addition

Complex numbers are added by separately adding the real and imaginary parts of the summands. That is to say:

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtraction

Similarly, subtraction  of complex numbers is defined by

(a - bi) + (c - di) = (a - c) + (b - d)i

Multiplication

The multiplication of two complex numbers is defined by the following formula:

(a - bi) + (c - di) = (ac - bd) + (bc - ad)i

Division

The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. When at least one of c and d is non-zero, we have:

 =  +

Division can be defined in this way because of the following observation:

                    =  =

As shown earlier, c − di is the complex conjugate of the denominator c + di. At least one of the real part c and the imaginary part d of the denominator must be nonzero for division to be defined. This is called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number).

Conjugate of Complex Number

The complex conjugate, or simply the conjugate, of a complex number z = x + iyis defined as the complex number x − iyand is denoted by z; that is,

(1)     = x − iy.

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 (Fig. 1). Note that z = z and |z| = |z| for all z.

 

Figure 1.

Theorems of Complex Conjugate  

(1)     If =  and = , then

= (x1+ x2) –= +.

So the conjugate of the sum is the sum of the conjugates:

(2)     =  +

 =

 +  =  + 

 =

 =

In like manner, it is easy to show that

(3)      =   ( ≠ 0)

 = ,                  =

 = ,                  =

 +  =

            =

 +  =

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

Hence

(4)     Re z = and Imz=

An important identity relating the conjugate of a complex number z = x + iyto its modulus is

 =  +

(5)     z= |z|2,

where each side is equal to x2 + y2. It suggests the method for determining a quotient z1/z2 that begins with expression. That method is, of course, based on multiplying both the numerator and the denominator of z1/z2by , so that the denominator becomes the real number |z2|2.

Absolute

The absolute value of a complex number is defined as its distance in the complex plane from the origin using the Pythagorean theorem. More generally the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers. For any complex number

                   Z = x + iy,

wherex and y are real numbers, the absolute value or modulus of z is denoted |z| and is given by

                   |z| = .

When the imaginary part y is zero this is the same as the absolute value of the real number x.

When a complex number z is expressed in polar form as

                   Z =

With r  0 and  real, its absolute value is

                   |z| = r

 

Properties of Conjugate of Complex Numbers

In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign.

In polar form, the conjugate of is .             

The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proven by writing z and w in the form a + ib.

A significant property of the complex conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the complex number is real.

For any two complex numbers:

 =   +

 

Iff

i.e.                                                       (1)

                                                      (2)

If

i.e.              

then           

               z =