Complex Analysis 03018P

1. COMPLEX NUMBERS

complex conjugate, real part, imaginary part, absolute value (modulus), polar coordinate representation, argument, stereographic projection, nth root, exponential form, Euler's Formula, de Moivre's Formula, complex current, impedance, discrete LTI system, transfer function, frequency response, amplitude response, phase response, placing zeros and poles, Discrete Fourier Transform

2. COMPLEX FUNCTIONS

polynomials, rational function, exponential function, logarithmic function, trigonometric functions, complex powers, inverse trigonometric functions, hyperbolic and area functions, mapping properties, limits, continuity

3. COMPLEX DERIVATIVE, ANALYTICITY

derivative, analytic function, conformal mapping, Cauchy-Riemann equations, harmonic functions, conjugate harmonic function

4. COMPLEX SERIES

convergence, absolute convergence, uniform convergence, radius of convergence, power series, Taylor series

5. COMPLEX INTEGRATION

contour integrals, definition, integral function, Cauchy's Theorem, Cauchy's Integral Formula, Mean Value Theorem, Taylor's Theorem, Laurent's series, classification of singularities, poles, calculation of residues, Residue Theorem, evaluation of definite integrals, Principle of the Argument, Rouche's Theorem

6. MÖBIUS TRANSFORMATION (BILINEAR TRANSFORMATION)

mapping properties, applications to signal analysis

7. DISCRETE LTI SYSTEM, STABILITY

causal discrete LTI system, input, output, convolution, transfer function, impulse response, inverse Z-transformation (by complex integration, by Laurent's series), stability, Schur's polynomials, Hurwitz' polynomials