Study On Complex Integration And Some Of Its Application

Project and Seminar Material for Physics

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

Complex Number:

A complex numbercan 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.

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 .

Differentiable Function:

Function f is said to be differentiable at when

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.

Entire Function:

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

Line Integral:

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

Contour:

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

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.

Contour Integral:

We write = . Let and take the limit k

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Summary of the Study


The study investigated Complex Integration and application. It aimed applying Complex Integration to Forces and Moment due to Fluid Flow. The findings in the study showed that integrals can be evaluated using Blasius Theorem. The application of the theorems cut across computer science, electrical and electronic engineering.


Conclusions


Complex integrals will be zero unless the indefinite integrals ∫W(z)2dz and ∫W(z)2zdz are functions that can have different values at the start and the end of the loop

Only one type of function has this property logez = loger + iθsince may increase by 2π in traveling around the loop. In general loge(z-zi) increases by 2πiwhen passing around any loop enclosing zi

Only functions of the form (z-zi)-1integrate to loge(z-zi).

Also, function of a complex variable is equivalent to two functions of a real variable and standard interpretation of a function of a real variable as being a curve on anxy plane no longer holds.

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg (z) the phase


Recommendations


  1. Functions of a complex variable find a very elegant application in the mathematical treatment of two-dimensional fluid flow.
  2. The complex number field should be considered relevant in the mathematical formulation of quantum mechanics
  3. Complex numbers can be used in signal analysis and other fields for a convenient description for periodically varying signals.
  4. In engineering, it can be help in understanding the behaviour of circuits which contain reactance (produced by capacitors or inductors) when applied to a.c. signals and provides a way to think about oscillations. This is useful when we want to apply concepts like the conservation of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.
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