Final answer:
To determine if functions are holomorphic, they must satisfy the conditions of differentiability in the complex plane and for specific cases, can be analyzed via the Cauchy-Riemann equations or convergent power series. If holomorphic, their derivatives are found using standard calculus techniques. However, functions that are non-normalizable, discontinuous, or multiply-valued are generally not holomorphic.
Step-by-step explanation:
The determination of whether functions are holomorphic depends on satisfying the conditions of differentiability in the complex plane. A holomorphic function, also known as an analytic function, must have a derivative at every point in its domain. Without specific functions provided in the question, a general answer cannot be given. However, concepts such as the Cauchy-Riemann equations and convergent power series are usually employed to ascertain holomorphicity. If a function is found to be holomorphic, its derivative can be obtained through standard calculus techniques.
In cases involving power series, if the function can be expressed as a convergent power series, it is analytic and thus holomorphic within its radius of convergence. Non-normalizable functions, discontinuous functions, or functions that are multiply-valued are generally not holomorphic because they fail to meet the necessary conditions.