# Mathematics

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## ENGINEERING MATHEMATICS E 244

Introductory Complex Analysis, Series and Applications

This module is presented in two sections: Complex analysis, and Series and Applications.  In the first section the basic properties of complex functions and integration in the complex plane are presented. The following topics are covered: Complex numbers and their polar form, mappings of sets in the complex plane, analyticity and differentiability of complex functions, the exponential and  logarithmic functions, as well as trigonometric and  hyperbolic functions and their inverses. Then follows the development of integration in the complex plane: Parametrization of contours, contour integration, path independence and the Cauchy integral formula. A discussion of Taylor and Laurent series is followed by the definition, calculation and applications of residues, as well as conformal mappings.

The first part of the second section, namely Series and Applications, is on infinite sequences and infinite series as well as the theorem of Taylor and the representation of functions by infinite series.  This is followed by the expansion of periodic functions in terms of Fourier series, with applications to the solution of partial differential equations.  Next, Fourier transforms and applications thereof are discussed, and finally Bessel functions.

Module Information

• 47953 244 (15) Engineering Mathematics E 244
• Second year, second semester of the Program in Electrical and Electronic Engineering.
• 4 lectures and one tutorial of 2 hours per week.
• Prerequisite pass modules (PP³50): Engineering Mathematics 145 or Engineering Mathematics 214; Prerequisite module (KP³40): Engineering Mathematics 214.
• Language specification : A & E

Lecturers

• Dr S Mouton (Afrikaans): Office 2022,  Van der Sterr Building (e-mail:  smo@sun.ac.za)
• Dr Z Janelidze (English).: Office 3014, Van der Sterr Building (e-mail:  zurab@sun.ac.za)

Learning Material

Prescribed textbooks:

• [1] J Stewart:  Calculus (5th Edition), Thomson, 2003.
• [2] DG Zill and MR Cullen:  Advanced Engineering Mathematics (3rd Edition), Jones and Bartlett Publishers, 2006.

Module Content

First section:

• [2] Chapter 17: Functions of a complex variable.
• [2] Chapter 18: Integration in the complex plane.
• [2] Chapter 19: Series and residues.
• [2] Chapter 20: Conformal mappings.

Second section:

• [1] Chapter 12: Infinite sequences and infinite series.
• [2] Chapter 12: Sections 12.2 to 12.4: Fourier series.
• [2] Chapter 13: Sections 13.1 to 13.5: Partial differential equations.
• [2] Chapter 15: Sections 15.3 to 15.4: Fourier transforms.
• Notes: Bessel functions.

Learning Opportunities

The course material is dealt with in detail during the lectures. Tutorials provide the opportunity to work on problems and discuss coursework under supervision of the lecturer.

Assessment

• Method: Obtain class mark (KP³40) and pass exam (final mark PP³50).
• The class mark is determined by the class test.  An additional test will take place during the fourth term:  the mark obtained for this test is averaged with that of the formal class test.
• Formula for final mark: PP = 0,4 KP + 0,6 EP.
• To gain admittance to the exam, you need a class mark of at least 40%, with a subminimum of 30% for each section.  To pass the module, you need a performance mark of at least 50%, with a subminimum of 40% for each section.
• The date and time of the class test and the additional test are determined by the Faculty of Engineering (http://www.eng.sun.ac.za), and those of the examination are published in the University’s Exam and Timetables book. For more information on exam regulations see the University Calendar, Parts 1 and 11.
• The class test is returned during the week following the mid-semester break.
• Short tutorial tests will be written at the end of tutorial periods, but they will not be marked.  The solutions will be put on WebCT.

General Information

• Please note the following regarding the tutorials:  (a) Attendance of all tutorials is compulsory, also for repeaters.  (b) No other appointments (academic or otherwise) may be observed during tutorial periods.
• The solutions of tutorial problems and tutorial tests will be made available on WebCT within a week after the tutorial.

Study Hints

• It is important that you understand the basic concepts very well, so that you are able to apply the theory.
• To determine whether you have mastered the course material, you must work a variety of problems from the Exercises in the textbook.  The lecturer will mention a number of these problems during the lectures.  These problems will not be marked, and this remains your own responsibility.
• You are welcome to consult your lecturer if there is anything which you do not understand, or if you get stuck with a problem.  The lecturer will be available right after each lecture, as well as in his/her office (preferably by appointment).
• Revise each section completely as soon as it has been discussed in class.  It is very important that you do not fall behind, since it is very difficult to catch up.

Rationale

The module is presented as part of the Program in Electrical and Electronic Engineering and offers basic training in Mathematics which is necessary for the successful completion of other modules within this Program. The module supports the Program outcome that graduates should be able to use their knowledge of mathematics to solve engineering problems.

Learning Outcomes

After completing this course, students should be able to:

First section:

• Understand the basic notions on complex numbers and complex functions.
• Determine whether, and where, complex functions are analytic and differentiable.
• Work with the exponential, logarithmic and hyperbolic functions.
• Understand the techniques of integration in the complex plane, including the roles played by analyticity of the integrand, path independence and contour integration.
• Apply Cauchy’s integral formula.
• Understand the expansion of a complex function in a Taylor or Laurent series.
• Find residues and apply them to the calculation of several types of definite integrals.

Second section:

• Understand the basic concepts of infinite sequences and infinite series;  determine limits of sequences;  apply convergence tests to determine the convergence or divergence of infinite series (including power series);  know the theorem of Taylor and the Taylor series expansions of the elementary functions.
• Understand the basic concepts about periodic functions and are able to calculate the Fourier expansions of simple periodic functions, including half-range expansions.
• Solve boundary value problems for the basic linear partial differential equations using separation of variables and Fourier expansions of the non-homogeneous boundary values.
• Solve boundary value problems for the basic linear partial differential equations using Fourier transforms.
• Know the definition and properties of Bessel functions.