Final answer:
The question from complex analysis concerns proving the product of two analytic functions is analytic and verifying the product rule for differentiation, which results in the derivative f'(z)g(z) + f(z)g'(z) when f(z) and g(z) are analytic.
Step-by-step explanation:
In mathematics, particularly in complex analysis, the question is related to the proof that the product of two analytic functions is also analytic and a demonstration of the product rule for differentiation. To prove that f(z)g(z) is analytic when both f(z) and g(z) are analytic, one can utilize the fact that analytic functions can be expressed as power series. When functions are represented as power series, the product of two power series is well-defined and will also be a power series, hence showing analyticity. Now, to demonstrate the differentiation rule, we apply the standard differentiation to the product resulting in the derivative of f(z)g(z) being f'(z)g(z) + f(z)g'(z).