Thread: Analytic Continuation

What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.

Originally Posted by Rudolf

What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.

Uh? Take for example the series . This series converges

and defines an analytic function within the unit disk, but we can continue its definition as to the whole

plane except at the point .

There are other much more important examples, say the zeta Riemann function (see http://numbers.computation.free.fr/C...neralities.pdf , for ex.). Google it.

Tonio

Originally Posted by Rudolf

What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.

Have you ever seen a function that wasn't defined at a point and then redefined to make it continuous in Calc II?

I think so:
x/sin x, at the point x = 0.