Math413

Course Description form

Faculties and departments requiring this course (if any): Faculty of

     Science/Department of Mathematics.

Objectives: 

1-    Know the basic need to define complex number system and its relation with real number system.

2- Recognize the importance and usefulness of complex  

     analysis.

3- Extend some concepts studied in Math 311 and 312 and  

     calculus course

4- Understand the concept of analytic functions, and their    

    relation with Cauchy-Riemann differential equations.

5- Understand the concept of complex integration.

6- Be able to apply theorems and facts on analytic functions to   

     complex integration.

7- Understand the concept of Laurent's series.

8- Understand and apply theorems and facts on Residues.

Contents:

1. Complex Numbers

a.    Basic algebraic properties                       

b.    Moduli and complex conjugates               

c.    Exponential form                                 

d.    The Polar form                                           

e.    Roots of complex numbers                     

2. Complex Valued Functions

a.    Limit of complex valued functions      

b.    Continuity                                  

c.    Derivatives                                          

d.    Cauchy-Riemann equations                     

e.    Sufficient conditions for  differentiability                      

f.     Analytic functions                                 

g.    Harmonic functions                      

3. Elementary Functions

a.    The Exponential function                        

b.    The Logarithmic function              

c.    Complex exponents                               

d.    Trigonometric functions                         

e.    Hyperbolic functions                          

4. Complex Integration

a.    Contours                                             

b.    Contour Integrals                       

c.    Antiderivatives                                     

d.              Cauchy-Goursat Theorem             

e.    Cauchy Integral Formula                    

f.      Liouville's Theorem                          

5. Series and Residues

a.   Taylor's and Laurent's series                  

b.   Classification of singularities

c.   Cauchy's residue theorem                     

d.   Zeros and poles                                   

e.   Evaluation of improper integrals          

Course Outcomes:

A-   Knowledge:

     (Specific facts and knowledge of concepts, theories, formula etc.)

     Students will know the most applicable methods and results which are very useful in sciences and engineering. For example, the students will know the methods which will be derived from the Cauchy’s Theorem, Integral Formula and Residue Theorem and apply these methods to evaluate improper integrals. Classroom discussions, Chapter  review Quiz and  Chapter  review exercises (i.e. problem sheets) are key  teaching strategies  to develop a solid knowledge of  COMPLEX ANALYSIS.

B-Cognitive Skills:

     (Thinking, problem solving )

     In effective strategy instruction, the teacher serves as a mediator by helping to activate prior knowledge, represent information, select learning strategies, construct meaning, monitor understanding, assess the use of a strategy, organize and relate ideas, summarize, extend learning and predict consequences. By reflecting on what they did during this learning process, students can learn the thinking skills that were successful in learning the subject matter. In addition, by learning thinking skills in meaningful contexts, students are able to recognize the usefulness of these skills for practical purposes.

C- Interpersonal skills and responsibilities:

    (group participation, leadership, personal responsibility, ethic and moral behavior, capacity for self directed learning)

     Healthy interpersonal skills reduce stress, reduce conflict, improve communication, enhance intimacy, increase and promote understanding. The task of the teacher is to provide the student with a level of support & guidance. Teams can be formed to figure out solutions to problems and teacher can empower them to carry out the solutions.

D- Analysis and communication:

)communication, mathematical and IT skills)

     Communication skills are the most important when we talk about effective teaching. The tone, volume, rhythm and emotions of the communicator play a vital role while dealing with students.

    We may improve interpersonal skills with students by using technical skills too i.e. ability to work with latest teaching aids like computers, multimedia or other technical equipments. We have to develop a problem sheet which consists of such a questions to encourage the students to use computers, multimedia or other technical equipments.

Assessment methods for the above elements;

     Effective understanding of content subject matter is more likely to occur when students are required to explain and  elaborate; the burden of explanation is often the push needed to make them evaluate, integrate, and elaborate knowledge in new ways. By interacting with  teachers, students can master the subject more thoroughly.

    To ask your listeners to paraphrase what they think you have said. This concept helps the teacher to keep the attention of the student and keep them participating in discussion.

    Ask open-ended questions that begin with "How...," "What...," "When...," "Where...," and "Why."

    The students will be expected to demonstrate their learning through quizzes, homework assignments, group discussions and examinations on the subject.

Supplementary references:

   A first Course in Complex Analysis with Applications (2003)

   Author: Dennis G. Zill and Patrick D. Shanahan

   Publisher: Jones and Bartlett Publishers, London

Other Information Resources

   A First Course in Complex Analysis (Version 1.24) By

   Matthias Beck, Gerald Marchesi, and Dennis Pixton

   http://www.math.binghamton.edu/dennis/complex.pdf

   http://math.sfsu.edu/beck/complex.html.

   For more information about Cauchy, see

   ttp://www-groups.dcs.st-and.ac.uk/_history/Biographies/Cauchy.html.

   For more information about Riemann, see

   http://www-groups.dcs.st-and.ac.uk/_history/Biographies/Riemann.html.

   For more information about Edouard Jean-Baptiste Goursat (1858{1936), see

   http://www-groups.dcs.st-nd.ac.uk/_history/Biographies/Goursat.html.

   For more information about Pierre Alphonse Laurent (1813{1854), see

   http://www-groups.dcs.st-and.ac.uk/_history/Biographies/Laurent Pierre.html.