Product of Analytic Functions is Again Analytic (Non-Standard Proof)

[TaylorM0192]

Hello,

I'm trying to prove that the product of two analytic functions is analytic. Specifically, given their power series representations about some point (let's just say they're both analytic at z = 0, and have the same radii of convergence for convenience), prove that the Cauchy product of these power series (a) converges and (b) converges to the desired limit (i.e. the product of the original limits of partial sums of the power series).

The theorem is trivial if we appeal to Merten's theorem (or Abel's of course), as both power series are absolutely convergent on the interior of their disks of convergence, and therefore for all z in the disk, the Cauchy product not only converges, but converges to the product of limits.

I want to prove this another way, and the hint is to apply the discrete Fubini theorem, which states that (under appropriate hypotheses) you may interchange a double sum without affecting the limit. In particular, if you have a double sequence {a_ij} and sum(|a_j|) converges for each i = 1, 2, ... to some sequence {b_i} and sum(b_i) converges, then the interchange is valid (a sufficient condition which I proved in an earlier exercise is that if each term of a_ij is positive, then the interchange is always valid, and this follows from rearrangements not affecting the convergence of absolutely converging series: that is, the interchanges each converge to the same value, or diverge together to infinity).

I did a little scratch work and wasn't really able to come up with anything substantive toward a solution. Basically my idea was to form the Cauchy product of the series and then find suitable candidates for the a_j and b_i series...but I'm having difficulty in even seeing the connection of this theorem to the problem.

If anyone could provide a more illuminating hint, I'd appreciate it.

Thanks ~

 

[jvicens]

Hello,

Thank you for your question. The product of two analytic functions is indeed analytic, and there are a few different ways to prove this. One approach is to use the Cauchy integral formula, which states that the coefficient of z^n in the power series expansion of a function f(z) is given by the contour integral of f(z)/z^(n+1) around a small circle centered at the origin. Using this, we can show that the Cauchy product of two power series is also a power series, and thus the product of the two functions is analytic.

To prove this using the discrete Fubini theorem, we can consider the Cauchy product as a double sum of the coefficients of the two power series. As you mentioned, we can interchange the sums as long as the terms are positive. In this case, we can show that the terms are indeed positive by noting that the coefficients of the power series are non-negative, and thus the Cauchy product is also non-negative. From there, we can use the convergence of the individual power series to show that the Cauchy product also converges, and thus the product of the two functions is analytic.

I hope this helps guide you towards a solution. If you have any further questions, please don't hesitate to ask. Good luck with your proof!