Study On Complex Integration And Some Of Its Application

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:

Investigations of theorems in Complex Integrals to include Cauchy theorem, Gauss, mean value theorem, Liouville’s theorem, Maximum Modulus theorem and Blasius Theorems
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. 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. 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. Differentiable Function:

Function f is said to be differentiable at when exists.

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. Entire Function:

If the function is analytic everywhere in the complex plane, it is entire.

6. Line Integral:

Line integral of scalar fields over a curve C do not dependent on the chose parametrization r of C.

7. Contour:

Is defined as a curve consisting of a finite number of smooth curves joined end to end.

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. Contour Integral:

We write = . Let and take the limit k

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