# Understanding convergence of an infinite product

I am reading the book Complex Analysis by Gamelin and I am having trouble understanding the following definition:

Let $$p_j$$ be a sequence of non zero complex numbers. We say that $$\prod p_j$$ converges if $$p_j\to 1$$ and $$\sum \log{p_j}$$ converges (Here $$\log$$ denotes de principal branch of the logarithm).

I don't understand why the expression $$\sum \log{p_j}$$ makes sense because the $$p_j$$ can be negative integers, say $$p_1=-1$$, in this case how we define $$\log(-1)$$? (the principal branch of the logarithm is not defined for negative integers)

• Notice that, however, the $p_j$-s will eventually be all in a neighbourhood of $1$. Since the first few terms are inconsequent, you may just consider any tailing sum, without worrying too much to whether $\log(-1)$ has been defined as $i\pi$ or $-i\pi$. – Saucy O'Path Apr 19 at 21:02
• You are right, thanks. – Sebitas Apr 19 at 21:11

## 1 Answer

The principal branch of the logarithm is defined by $$z=re^{i\theta}\mapsto \log(r)+i\theta$$, where we understand $$\log(r)$$ as a function on the positive reals and we force $$\theta\in(-\pi,\pi]$$. Since $$-1=e^{i\pi}$$ we find that $$\log(-1)=i\pi$$ (at least for the principal branch).

Thus, you can make sense of all negative integers, and you can then talk about the convergence of that sum.

• The problem with that is that the logarithm you are defining is not continuous, because $\log(-1+0i)=i\pi$ while $\log(-1-i0)=-i\pi$ – Sebitas Apr 19 at 21:10
• @Sebitas The principal branch of the logarithm isn't supposed to be a continuous function, so all is well. – ItsJustLogsBro Apr 19 at 21:11
• Yes, now I understand a lot of things, thanks. Nice name btw – Sebitas Apr 19 at 21:13