### Complex Calculus (MATH 305)

2019 Spring
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
6.00 / 6.00 ECTS (for students admitted in the 2013-14 Academic Year or following years)
Turgay Bayraktar tbayraktar@sabanciuniv.edu,
English
MATH102
Formal lecture
Interactive,Communicative

### CONTENT

Analytic functions, Cauchy's theorem and the Cauchy integral formula. Taylor series. Singularities of analytic functions, Laurent series and the calculus of residues. Infinite products. Conformal mappings.

### OBJECTIVE

To give an introduction to main methods of Complex Analysis which are needed for successful activity in many fields (engineering, economics, etc.) and for developing of mathematical thinking, as well

### LEARNING OUTCOME

Upon completion of this course, students should be able to:
Operate with complex numbers;

Differentiate and integrate complex valued functions;
Distinguish analyticity from differentiability by real variables (Cauchy-Riemann equation);
Understand how analytic and harmonic functions are connected;
Formulate Cauchy Theorem and Cauchy Formula and apply them consciously for integration;
Identify Taylor and Laurent expansions and distinguish isolated singularities;
Apply Cauchy Residue Theorem to calculations of definite integrals;

### PROGRAMME OUTCOMES

1. Understand the world, their country, their society, as well as themselves and have awareness of ethical problems, social rights, values and responsibility to the self and to others. 1

2. Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice. 4

3. Think critically, follow innovations and developments in science and technology, demonstrate personal and organizational entrepreneurship and engage in life-long learning in various subjects. 2

4. Communicate effectively in Turkish and English by oral, written, graphical and technological means. 2

5. Take individual and team responsibility, function effectively and respectively as an individual and a member or a leader of a team; and have the skills to work effectively in multi-disciplinary teams. 2

1. Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems. 4

2. Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. 4

3. Develop, choose and use modern techniques and tools that are needed for analysis and solution of complex problems faced in engineering applications; possess knowledge of standards used in engineering applications; use information technologies effectively. 3

4. Ability to design a complex system, process, instrument or a product under realistic constraints and conditions, with the goal of fulfilling specified needs; apply modern design techniques for this purpose. 3

5. Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. 2

6. Knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. 2

7. Knowledge of impact of engineering solutions in a global, economic, environmental, health and societal context; knowledge of contemporary issues; awareness on legal outcomes of engineering solutions; understanding of professional and ethical responsibility. 2

1. Use mathematics (including derivative and integral calculations, probability and statistics, differential equations, linear algebra, complex variables and discrete mathematics), basic sciences, computer and programming, and electronics engineering knowledge to (a) Design and analyze complex electronic circuits, instruments, software and electronics systems with hardware/software. or (b) Design and analyze communication networks and systems, signal processing algorithms or software 5

1. Applying fundamental and advanced knowledge of natural sciences as well as engineering principles to develop and design new materials and establish the relation between internal structure and physical properties using experimental, computational and theoretical tools. 4

2. Merging the existing knowledge on physical properties, design limits and fabrication methods in materials selection for a particular application or to resolve material performance related problems. 2

3. Predicting and understanding the behavior of a material under use in a specific environment knowing the internal structure or vice versa. 2

1. Formulate and analyze problems in complex manufacturing and service systems by comprehending and applying the basic tools of industrial engineering such as modeling and optimization, stochastics, statistics. 3

2. Design and develop appropriate analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy. 3

3. Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 3

1. Familiarity with concepts in statistics and optimization, knowledge in basic differential and integral calculus, linear algebra, differential equations, complex variables, multi-variable calculus, as well as physics and computer science, and ability to use this knowledge in modeling, design and analysis of complex dynamical systems containing hardware and software components. 5

2. Ability to work in design, implementation and integration of engineering applications, such as electronic, mechanical, electromechanical, control and computer systems that contain software and hardware components, including sensors, actuators and controllers. 2

1. Design, implement, test, and evaluate a computer system, component, or algorithm to meet desired needs and to solve a computational problem. 3

2. Demonstrate knowledge of discrete mathematics and data structures. 2

3. Demonstrate knowledge of probability and statistics, including applications appropriate to computer science and engineering. 3

### ASSESSMENT METHODS and CRITERIA

 Percentage (%) Midterm 22.5 Exam 67.5 Participation 10