Analytic Continuation

• Nov 30th 2010, 11:30 AM
Rudolf
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.
• Nov 30th 2010, 11:45 AM
tonio
Quote:

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 $\displaystyle \displaystyle{1+z+z^2+...=\sum\limits^\infty_{n=0} z^n}$ . This series converges

and defines an analytic function within the unit disk, but we can continue its definition as $\displaystyle \displaystyle{\frac{1}{1-z}}$ to the whole

plane except at the point $\displaystyle z=1$ .

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
• Nov 30th 2010, 01:53 PM
wonderboy1953
Quote:

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?
• Nov 30th 2010, 02:00 PM
Rudolf
I think so:
x/sin x, at the point x = 0.