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In your question you ask about analyticity of a "field", but in fact such properties are important as well for other physically important functions such as Green's functions / correlation functions. The most concrete application of analytic continuation / analytic properties of complex functions that comes to my mind is the story relating analyticity of a ...

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The role of holomorphic functions (and their generalizations in the form of holomorphic sections of vector bundles) in physics is invaluable. Please see for example the following review by B.C. Hall, discussing holomorphic methods in mathematical physics, especially in quantum mechanics. It should be emphasized that these theories cover important parts of ...

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The property of holomorphic functions that begets the behaviour you (and I) so admire is analyticity, i.e. the ability equate a function $f:X\to Y$, over some open subset $U\subset X$, with its Taylor series expanded about some point $x_0\in X$. As such, this notion can be broadened to $\mathbb{R}^N$, $\mathbb{C}^N$ and is one of the defining properties of ...

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