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Main
UQU Programs
Mathematics

Complex Analysis

MTH4304
4 hours English

Complex Analysis

, Complex Analysis
MTH3302 Real Analysis 1
4 hours One requisite
MTH2103 Multi-variables Calculus
4 hours One requisite
1 Exponential and Trigonometric functions, Analytic (Holomorphic) functions, Necessary and sufficient condition for analyticity (Cauchy-Riemann equations), Polar form of Cauchy-Riemann equations, Harmonic functions, Harmonic Conjugate
2 Complex Integration, Cauchy’s theorem, Cauchy’s Integral formula, higher order derivative of analytic function, Morera’s Theorem, Poisson’s integral formula for a circle, Cauchy’s Inequality, Liouville’s theorem
3 Expansion of analytic function as power series: Taylor andLaurent theorem, Zerosof an analytic function, Singularities and types, Meromorphic functions, principle of Argument, Roche’s theorem, fundamental theorem of algebra, Maximum Modulus Principle, Schwarz Lemma
4 Residue, Cauchy’s Residue theorem, evaluation of definite integrals, their properties and classification, Definitions and examples of conformal transformation, bilinear transformation, their properties and classification
5 Others: Revision and activities, Exam, quizzes
1.1 Mapped to: K1

Demonstrate understanding of domains, regions, and contours in the complex plane.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exam.
1.2 Mapped to: K1

Define and determine the domain of definition, range, limits and limits involving infinity, continuity, differentiability, and boundedness of complex functions.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
1.3 Mapped to: K2

Define the exponential, logarithm, trigonometric, hyperbolic, and power functions of complex variables, and contrast their properties with those of their counterparts in the real case.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
2.1 Mapped to: S6

Perform algebraic operations on complex numbers and compute their powers and roots.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
2.2 Mapped to: S6, S7

Determine the analyticity of a complex function and recognize various properties of analytic functions.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
2.3 Mapped to: S1, S2, S6, S7

Evaluate contour integrals directly by the antiderivatives and derive bounds for the modulus of contour integrals

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
2.4 Mapped to: S1, S2, S5, S7

Prove and apply Cauchy’s integral theorem and Cauchy’s integral for formulas for the analytic functions and their derivatives.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
2.5 Mapped to: S1, S2, S7

Classify singularities and poles of a complex function and apply the residue theorem to evaluate real and complex integrals.

Teaching Strategy Lectures
Assessment Methods Homework, Periodic and final Exams.
3.1 Mapped to: V1, V2

Develop self-learning skills and solve problems both individually and in collaboration with the classmates

Teaching Strategy Lectures
Assessment Methods Homework and assignments.
Assessment
Weight
Midterm Exam
30%
Homework and assignments
20%
Final Exam
50%
Total
100%