A function that is analytic in domain $D$ is uniquely determined over $D$ by its value in a domain or along a line segment contained in $D$ .
What is the meaning of this theorem and how is it useful , like any potential application of this theorem?
When a Complex function is uniquely determined in a given domain?
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complex-analysis
analytic-functions
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0I think one use is when studying analytic continuations of functions, in particular that they're unique. Take the Riemann-Zeta function: it's continuation from $(1,+\infty)$ to $\mathbb{C}-{1}$ is unique. This is one way to prove that $\sum_{k=1}^{\infty}k=-\frac{1}{12}$ – 2017-01-28
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2@branchedout, no, it is not a way to prove that, as that does not make sense: as you probably know the left hand side of the equality you wrote diverges. – 2017-01-28
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0$\zeta(-1)=-\frac{1}{12}$, then – 2017-01-28
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0Rather, the application to that situation is that there is at most one function on some open set in the plain which contains any open segment of the positive real like which are extends the Riemann zeta function. – 2017-01-28
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1This $-{1 \over 12}$ seems to creep in everywhere. – 2017-01-28