Abstract
King’s method applies to solve scalar equations. The local analysis is established under conditions including the fifth derivative. However, the only derivative in this method is the first. Earlier studies apply to equations containing at least five times differentiable functions. Consequently, these articles provide no information that can be used to solve equations involving functions that are less than five times differentiable, although King’s method may converge. That is why the new analysis uses only the operators and their first derivatives which appear in King’s method. The article contains the semi-local analysis for complex plane-valued functions not presented before. Numerical applications complement the theory.
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