SSH3503 COMPLEX VARIABLES

Topic outline

• General
  

  General

  Lecturer : ALI HASSAN BIN MOHAMED MURID Semester : II Semester 2 2011/2012
  Synopsis : This course introduces calculus of functions of a single complex variable. Topics to be covered are functions of a complex variable, complex differentiation, complex integration, complex series including Taylor and Laurent series, the theory of residues with applications to the evaluation of complex and real integrals, and conformal mapping.
  This work, SSH3503 COMPLEX VARIABLES by ALI HASSAN BIN MOHAMED MURID is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License
• Topic 1
  

  Topic 1

  This chapter provides a brief history of complex numbers, arithmetic operations on complex numbers, including conjugate, absolute value, and inequalities of complex numbers.

  • Chapter 1: Complex Numbers File
• Topic 2
  

  Topic 2

  This chapter describes geometric properties of complex numbers, polar form, powers and roots, and sets of complex numbers.

  • Chapter 2: Geometry of Complex Numbers File
• Topic 3
  

  Topic 3

  This chapter discusses functions of a complex variable, limits and continuity, derivatives, analytic functions, and harmonic functions.

  • Chapter 3: Functions of Complex Variable File
• Topic 4
  

  Topic 4

  This chapter extends the definitions of the elementary functions from real variables to complex variables. The elementary functions include the complex exponential function, complex trigonometric functions, complex hyperbolic functions, and the complex logarithmic function.

  • Chapter 4: Elementary Complex Functions File
• Topic 5
  

  Topic 5

  This chapter introduces complex integral as a generalization of a real integral studied in calculus. The topics include line integrals, the fundamental theorem, Cauchy-Goursat theorem, path independence, and Cauchy's integral formula.

  • Chapter 5: Complex Integration File
• Topic 6
  

  Topic 6

  In calculus, several convergence tests have been established for series of real numbers or functions of real variables, such as the divergence test, ratio test, and root test. The notions of power series and Taylor series for representing real functions are also discussed in calculus. In this chapter we shall extend these ideas to series of complex numbers or functions of complex variables. The complex Taylor series for representing analytic functions has generalization to Laurent series which has no analogue in calculus.

  • Chapter 6: Complex Series File
• Topic 7
  

  Topic 7

  In this chapter, based on Laurent series, we shall develop a powerful technique for evaluating a larger class of complex integrals. The topics include residue theory and its applications for evaluating definite real integrals and improper real integrals.

  • Chapter 7: Residue Theory File
• Topic 8
  

  Topic 8

  This chapter describes a class of analytic functions with the property that angles between curves are preserved in magnitude as well as in direction, known as conformal mapping. Some classes of conformal mappings that frequently arise in applications are the Moebius transformations, the Schwarz-Christoffel mapping, and the Riemann map.

  • Chapter 8: Conformal Mapping File
• Topic 9
  

  Topic 9

  Some useful references for Complex Variables.

  • References File
• Topic 10
  

  Topic 10
• Topic 11
  

  Topic 11
• Topic 12
  

  Topic 12
• Topic 13
  

  Topic 13
• Topic 14
  

  Topic 14
• Topic 15
  

  Topic 15