Final answer:
To prove that if f(z) and g(z) are analytic in some domain D ∈ℂ, then f(z) g(z) is analytic in D, we need to show that f(z) g(z) has a derivative at every point in D.
Step-by-step explanation:
To prove that if f(z) and g(z) are analytic in some domain D ∈ℂ, then f(z) g(z) is analytic in D, we need to show that f(z) g(z) has a derivative at every point in D.
Since f(z) and g(z) are analytic, they have derivatives at every point in D. Let's denote these derivatives as f'(z) and g'(z) respectively.
By the product rule, the derivative of f(z) g(z) is given by (f(z) g(z))' = f'(z)g(z) + f(z)g'(z). Since f(z) and g(z) have derivatives at every point in D, their product f(z) g(z) also has a derivative at every point in D. Therefore, f(z) g(z) is analytic in D.