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Math Help - Analytic Continuation

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by Rudolf View Post
    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{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{\frac{1}{1-z}} to the whole

    plane except at the point 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
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    Quote Originally Posted by Rudolf View Post
    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?
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  4. #4
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    I think so:
    x/sin x, at the point x = 0.
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