# 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

##### Complex Number:

A complex numbercan be expressed in the form * a* +

*, where*

*bi**and*

*a**are real numbers and*

*b**is the imaginary unit which satisfies the equation*

*i*

*i*^{2}= −1. In this expression,

*is the*

*a**and*

*real part**is the*

*b**of the complex number.*

*imaginary part*##### Complex Function:

Let * S *be a set of complex numbers. A function

*defined on*

*f**is a rule that assigns to each*

*S**in*

*z**a complex number*

*S**. The number*

*w**is called the*

*w**of*

*value**at*

*f**and is denoted by*

*z**; that is .*

*f (z)*##### Differentiable Function:

Function * f *is said to be

*at when*

*differentiable*##### 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

Complete Material Available

**The Complete Material Will Be Sent to You in Just 2 Steps**

** Quick & Simple…**

**Make Payment (Through Transfer) of ₦5,000 to the Account Below**

Acc No: 1225513212 | |

Samphina Academy | |

Current Account |

**Or CLICK HERE To Pay With Debit Card**

FOR STUDENTS OUTSIDE NIGERIA |

CLICK HERE To Purchase Material ($15) |

**Send the Following Details on WhatsApp ( 08143831497) After Payment**

**Payment Details****TOPIC: Study On Complex Integration And Some Of Its Application**

**The Complete Material Will Be Sent To You On WhatsApp After Receiving Your DetailsT & C Apply**

** Need a Different Topic? ****Perform a Quick Search**

## List of Related Works

Awareness Of Physic Students On The Application And Utilization Of Solar Energy In Nigeria

The Design And Construction Of The Hearing Aid Device

Design And Construction Of A Monitoring System That Measure Light Intensity And Gas Level

Thermal Properties Of Some Selected Materials Used As Ceiling In Building

Exploration Of Local Shielding Materials For Diagnostic X-Ray: A Study Of Properties And Efficacy

Construction And Application Of Heat Sensor

The Phenomenology Of Jets In Astrophysics

Design And Construction Of A Variable Resistance Box

Determination Of Noise Pollution Base From Welding And Fabrication Workshop In Makurdi Metropolis

Design And Construction Of A Pyramid-Shaped Solar

Possible Effects Of Electromagnetic Fields (EMF) On Human Health

Design, Construction And Performance Evaluation Of A Passive Solar Water Heater

Application Of Geoelectrical Resistivity Imaging To Investigate Groundwater Potential

Estimation Of Organ Equivalent And Effective Doses From Diagnostic X-Ray

Variation Of Wind Speed With Attitude In The Dispersion Of Gasseous Pollutants

Effect Of Radioclimatic Variables On Radio Signal Propagation At Sokoto, Nigeria

A Study Into Energy Gap In Super Conductors

Comparative Study Of Some Water Quality Parameters Of Selected Ponds And River Benue

Optical And Structural Properties Of Copper Aluminum Diselenide (CuAlSe2) Compound Thin Films

Database Emission Of Carbon IV Oxide (CO2) In Nigeria

Measurement And Assessment Of Indoor And Outdoor Ambient Radiation Levels At The Take-Off Site

Design And Implementation Of A Query Optimizer

Design And Construction Of Digital Distance Measuring Instrument

Thermal Conductivity Of Selected Aluminium Roofing Sheets In Nigeria

Somatic And Genetic Effects Of Low Sar 2.45 GHz Microwave Radiation On Wistar Rats

Redesign Of A Multiple Heat Transfer Mode Charcoal Oven

Analysis Of Air Temperature Anomaly In Sokoto Metropolis

Improvement On The Designed, Construction And Testing Of Medical Waste Incinerator

Assessment Of Radon-222 In Some Selected Water Sources

Control Rod Calibration Of Nigeria Research Reactor -1 (NIRR-1) Using Positive Period Method

Comparative Study Of Thermal Properties Of Some Common Roofing Materials In Nigeria

The Physics Of Stars And Their Astronomical Identification

## 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)^{2}dz and ∫W(z)^{2}zdz are functions that can have different values at the start and the end of the loop

Only one type of function has this property log_{e}z = log_{e}r + iθsince may increase by 2π in traveling around the loop. In general log_{e}(z-z_{i}) increases by 2πiwhen passing around any loop enclosing z_{i}

Only functions of the form (z-zi)^{-1}integrate to log_{e}(z-z_{i}).

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

- Functions of a complex variable find a very elegant application in the mathematical treatment of two-dimensional fluid flow.
- The complex number field should be considered relevant in the mathematical formulation of quantum mechanics
- Complex numbers can be used in signal analysis and other fields for a convenient description for periodically varying signals.
- 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.