Analytic Series

Mathematicians are interested in trying to make sense of this kind of puzzle, which often comes up in physics too. As a teaser, let us prove that Let . Then it is easy to see that if we multiply it by 2 and add 1 you get the same series again. In other words we can simply write which implies that . This is a counter-intuitive result that is, nevertheless, correct. While this simple derivation was somewhat lacking in rigor, we can be mathematically more precise by means of “analytic continuation.” We recall the expansion of which is related to geometric series The right-hand side of the equation only makes sense for . But the left-hand side makes sense even if . We can use the left-hand side to make sense of what we mean by the right-hand side even if . This is called an analytic continuation of what the right-hand side represents for .

As another example, let us talk about the sum of all natural numbers The fact that this is finite and negative is again counter-intuitive. We can define an analytic function, which is called Riemann’s zeta function, as follows: This function is defined in the complex plane when has real part , but it can be analytically continued in the complex plane to a unique function, just as we did with the geometric series above. It turns out that once we do this, it satisfies . If we naively substitute in the defining expression, then one would be forced to conclude that Again, this kind of calculation crops up in physics (specifically in finding the number of dimensions that bosonic strings live in), which comes from the equation where is the number of dimensions. This is where the 26 dimensions of early versions of string theory come from. ¹ 

In physics, singularities pop up here and there, which we are not well-equipped to deal with. But we can go past the singularity and find that there are still well-defined points beyond it. This is why ideas from complex analysis are helpful for theoretical physics. When we encounter this kind of infinite series, we try to proceed analytically, which means we try to come up with functions, such as the zeta function, that make sense even in regions where one would not expect them to. Sometimes the way to proceed analytically may not be unique, but if several methods gives us the same answer, then we can be more confident that we can use it in a physical theory.

Puzzle