487 ريض (التحليل المركب 1)

Complex numbers: Algebr aic (arithmetic) properties. Complex (and extended) plane (Cartesian and Polar forms of complex

numbers). Powers and roots of complex numbers. Pla ner sets (aspects of connectedness).
Functions of a complex variable: Limits and continuity. Differentiability and holomorphy. Cauchy-Riemann theory. Harmonic functions.
Elementary functions: Exponential, Trigonometric and hyperbolic functions. Logarithmic functions and Branch concepts. The inverses of such functions.
Complex integration: Contour (line) integrals. Cauchy’s theorem. Cauchy’s integral formula and its applications (such as: Maximum modulus principle, Mean value property (analytic and harmonic), Cauchy’s estimate, Liouville’s theorem, Fundamental theorem of algebra…).
Series representation for analytic functions: Sequences and infinite series. Taylor series. Power series and analyticity. Laurent series.
Residue theory: Zeros and singularities of complex variable functions. The residue theorem. Applications to trigonometric integrals. Application to improper integrals. Application to series summations.