Ciro Santilli
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Identity theorem | 🗖 nosplit | ↑ parent "Analytic continuation" | 89, 2, 120

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Essentially, defining an holomorphic function on any open subset, no matter how small, also uniquely defines it everywhere.
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This is basically why it makes sense to talk about analytic continuation at all.
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One way to think about this is because the Taylor series matches the exact value of an holomorphic function no matter how large the difference from the starting point.
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Therefore a holomorphic function basically only contains as much information as a countable sequence of numbers.
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