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2. Exponentials

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The case of the exponential function is specially simple and gives some clues about the generalization of the derivatives. Following (1.2),

 (2.1)

the above limit exists for any complex number a. However, it should be noted that in the substitution of the binomial formula a natural number has been considered. We shall deal with this problem later to get our first generalization of the derivative. Applying this to the imaginary unit,

 (2.2)

and

 (2.3)

solving this system we have the next definition for the sine and cosine derivatives,

 (2.4)

and

 (2.5)

We could expect these relations for the sine and cosine derivatives to be maintained in the generalization of the derivative.

Applying the above method we also can calculate the following,

 (2.6)

and

 (2.7)

Thus,

 (2.8)

and

 (2.9)

Indeed, the above result of the exponential can be applied to any function that can be expanded in exponentials

 (2.10)

Expanding the function in Fourier series,

 (2.11)

This method can be useful for calculating fractional derivatives of trigonometric functions.