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Derivative of a ratio of complex functions

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Show that the derivative of g(z) exists, and hence g(z) is complex analytic, at points where g(z) does not diverge, thereby proving that the points where g(z) diverges are the only singularities.

g(z)= eiz/ z2 +4z+5

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Solution Summary

The solution shows how to prove that the derivative of a quotient of two analytic functions exist, and therefore the quotient is analytic as well (except at some points).

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If the limit (1.1) exists
(1.1)
Then is the derivative of the complex function

Now, assume we have two complex functions and , both differentiable (that is, the limit 1.1 exists for each function)
We define
...

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