What is analytic continuation?

[cragwolf]

I don't understand the concept of analytic continuation, at any level (I have some small amount of experience with introductory undergraduate-level complex analysis from a long time ago, mostly forgotten):

1) Firstly, why would you want to apply analytic continuation on some complex function?

2) Surely if you apply analytic continuation on some complex function, then it's no longer the same function?

3) What is analytic continuation, really?

I came across this concept when I was reading about the Riemann hypothesis in some popular articles and books. They mentioned that the Riemann zeta function is the infinite sum of $$1/n^s$$ over natural numbers, n, but this is only the case when the real part of s is greater than 1; values of the Riemann zeta function for the rest of the complex plane may be found via analytic continuation.

What kind of book would give me an introduction to analytic continuation? A complex analysis book? Any recommendations? Also, what sort of mathematical background do I need to be able to understand one of the standard books devoted to the Riemann zeta function, like "Riemann's Zeta Function" by Harold M. Edwards, or "The Theory of the Riemann Zeta-Function" by E. C. Titchmarsh?

 

[matt grime]

Suppose you have some laurent series (about 0), with radius of convergence R. Then you might want to know of there were some analyitc function that agreed on the disc of radius R but is defined on some larger domain. This is where analytic continuation comes into it.

So you try and find another laurent series around some other point and valid on a disc that overlaps the original disc. Strictly speaking you will have a different function because you have a different domain.

The reason that a laurent series has radius of convergence R is because there is some point on the boundary where the expansion is not valid (the resulting series diverges there) so there is something quite deep going on.

Example:

f(z) = z-z^2/2+z^3/3+...

This a power series for log (1+z) around 0, with radius of convergence 1. There is pole at z=-1. That doesn't mean that we can't continue log(1+z) around the singularity.

Elsewhere there will be different power series with different expanisions with different radii of convergence that glue together (to form a Riemann surface...)

Certain power series have no continuations. I can't remember the exact example but it's soemthing like:

$$f(z)=\sum_{n=1}^{\infty}z^{n!}$$

which is valid for |z|<1, but has a dense set of poles on the unit circle - or at least that is how the counter example is supposed to go, even if that isn't the specific example.

As for the Riemann Zeta function. If you want to understand it, then you need a very good background in analysis and numbre theory. If you just want to know what's going on a little then there are some coffee-table books around.

Try, for various levels, Rudin's Analysis, anything by Beardon, Du Sautoy's music of the primes, and Devlin;s book about the clay institute prizes.

 

[cragwolf]

Thanks matt, I have a lot of learning to do.

 

[mathwonk]

the concept of analytic continuation just means enlarging the domain without giving up the property of being differentiable, i.e. holomorphic or meromorphic.

The power series method Matt described is a technique for doing it.


For example you could ask if it is possible to extend the definition of the sin function from the real line to all complex numbers. That is an easy example of analytic continuation. Namely the same power series that defines the taylor series for sin(x) with real x, also works for complex values of x.