algebraically closed field analytic continuation analytic function analytic number theory applied mathematics carl friedrich gauss complex dynamics complex plane derivative directional derivative exponential function fractal harmonic function holomorphic function holomorphic sheaf hue imaginary number laurent series line integral list of complex analysis topics mandelbrot set mathematics meromorphic function methods of contour integration natural logarithm number theory open set partial derivative partial differential equation physics polynomial power engineering quantum field theory real analysis real number residue theorem riemann mapping theorem riemann surface riemann zeta function several complex variables string theory subset taylor series tristan needham vector bundle vector calculus