Authors: Jonathan W. Tooker
We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.
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