I was going over proofs of the Basel Problem given on wikipedia :https://en.wikipedia.org/wiki/Basel_problem
I was interested in something mentioned in the following proof
$$ \frac{sin(\pi x)}{\pi x} = x\prod_{n=1}^{\infty} \left(1-\frac{x^2}{n^2 \pi^2}\right) $$
It then says : $\color{red}{\textrm{The product is analytic, so taking the natural logarithm of both sides and differentiating yields}} $
$$ \frac{\pi cos(\pi x)}{sin(\pi x} - \frac{1}{x} = -\sum_{n=1}^{\infty} \frac{2x}{n^2-x^2}$$
- Will omit rest of proof due to it not being relevant to the question
So my question is : Since they do mention that the product is analytic and thus they can take the logarithm and differentiate , are there any cases when one is not allowed to take the logarithm of an infinite product and differentiate as shown above (given that we have a convergent product to begin with) ?
Apologies for the simple question ,
Thank you kindly for your help and time.
Ignoring first few factors we see that the remaining factors are positive and so it is legitimate to take natural logarithm for those factors. But the question then is how do we differntiate an infinite sum? This is where analyticity comes in. A power series can be differentiated term by term inside the circle of convergnce and that is exactly what you have to do in this case.