The theory of functional calculus deals with the idea of 'inserting operators into functions'. The holomorphic (or Dunford-Riesz) functional calculus is based on a class of functions can be represented by the Cauchy-integral formula. A topological extension means that the primary calculus is extended to a wider class of functions and the definition of the operator is obtained as the limit of a sequence of operators defined by the primary calculus. Sectorial and half-plane type operators are operators that have some spectral conditions. In the case of sectorial operators, the topologically extended holomorphic functional calculus serves several benefits. The Hirsch calculus for non-negative operators can be incorporated into this holomorphic approach. The resolvent of the logarithm can be defined without the requirement of the injectivity of the operator. Furthermore, this approach can be used in order to obtain convergence rates for the Euler's approximations for bounded semigroups. In the case of half-plane type operators, the main benefit, the topologically extended holomorphic functional calculus serves, is that the Hille-Phillips calculus for generators of bounded strongly continuous semigroups can be incorporated into this holomorphic approach.