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HU Credits:
4
Degree/Cycle:
1st degree (Bachelor)
Responsible Department:
mathematics
Semester:
2nd Semester
Teaching Languages:
Hebrew
Campus:
E. Safra
Course/Module Coordinator:
Prof. Evgeny Strahov
Coordinator Office Hours:
N/A
Teaching Staff:
Prof Evgeny Strahov
Mr.
Course/Module description:
The course is an introduction to the theory of complex valued functions of one complex variable. A basic knowledge of the calculus of real variables is assumed.
Course/Module aims:
The course is an introduction to the theory of complex valued functions of one complex variable.
Learning outcomes - On successful completion of this module, students should be able to:
1. To be able to perform basic operations with complex numbers and elementary functions.
2. To know fundamentals of the theory of analytic functions.
3. To know applications of the theory of analytic functions to complex integration.
4. To be familiar with basic conformal mappings.
Attendance requirements(%):
0
Teaching arrangement and method of instruction:
Lectures + exercises
Course/Module Content:
1. Complex numbers, elementary complex functions and their properties.
Stereographic projections. Introduction to the concept of analytic function.
The Cauchy-Riemann conditions as necessary and sufficient conditions for the existence of a complex derivative.
2. Elementary conformal mappings and their geometric properties. The bilinear transformations. The general form of a bilinear transformation which carries (a) the upper half plane to itself; (b) the upper half plane to a disc; (c) a disc to itself; (d) three given points to another three given points.
3. Complex integration. The Cauchy theorem. The Cauchy integral formula and its consequences. The theorem on the expansion of an
analytic function into a power series. The Cauchy-type integrals.
The uniqueness theorem for analytic functions. Analytic continuation.
The Liouville theorem. The Morera theorem and its applications.
4. The Laurent series. Singularities of complex functions. The behavior of complex functions near isolated singularities, and near zeros.
5. The Cauchy residue theorem and its applications to the evaluation of integrals. The argument principle. The Rouche theorem and its applications.
Required Reading:
none
Additional Reading Material:
1) M. Ablowitz, A. Fokas. Complex variables. Introduction and applications.
2) L. Ahlfors. Complex analysis.
3) E. Stein, R. Shakarchi. Complex Analysis. (Princeton Lectures in Analysis).
useful reference: Fundamentals of complex analysis with applications to engineering and science by
Saff and Snider
Course/Module evaluation:
End of year written/oral examination 90 %
Presentation 0 %
Participation in Tutorials 0 %
Project work 0 %
Assignments 10 %
Reports 0 %
Research project 0 %
Quizzes 0 %
Other 0 %
Additional information:
Prerequisites - Inf. Calculus I
Recommended - Inf. Calculus II
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