|Complex numbers, Cartesian and polar representation of complex numbers, powers and roots of complex numbers. Limits and continuity of a complex function. Analytic functions, Cauchy-Riemann equations, harmonic functions. Exponential, trigonometric, hyperbolic functions and logarithmic functions. Complex integration, contour integrals, Cauchy's theorem, Cauchy's formula. Bounds on analytic functions. Series representation of analytic functions, Taylor and Laurent series, power series, Zeros and singularities. Residue theory. Applications to real and improper integrals.