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I posted this on math.SE, but didn't get much response. It might fit better on this site.

Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the entire complex plane. So, just by knowing how the function behaves at a teenie-weenie open disc in $\mathbb{C}$, the behavior at a much larger scale is uniquely defined.

Does this extraordinarily beautiful property have any direct applications in physics? For example, you know how some kind of field looks in a tiny area and you can somehow assume it is described by a holomorphic function and by analytic continuation you "extrapolate" the field to a much bigger domain.

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isn't this also the basic idea behind zeta function regularization (replacing a divergent series by the finite value obtained from analytically continuing the series from a convergent parameter domain)? –  Christoph Dec 5 '13 at 9:20
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Also, one example is to do path integrals in imaginary time, then aggressively exploit the the fact that physical quantities are holomorphic functions of momenta to "Wick rotate" the results back to real time. –  Stan Liou Dec 5 '13 at 9:24
    
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Interestingly, "most" holomorphic functions cannot be continued beyond their specified domain - they have natural boundaries there. See, for example, this MathOverflow thread. –  Emilio Pisanty Feb 28 at 10:05

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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 quantum theory but they are not exclusive. Still more general function spaces are needed to describe other systems in quantum mechanics.

The Hilbert spaces describing the states of many very important systems appearing in almost all areas of physics have a holomorphic realization as (reproducing kernel) Hilbert spaces of holomorphic functions (or sections). This is true for the harmonic oscillator, the electron spin, the hydrogen atom and even in very advanced applications like the Hilbert spaces corresponding to Chern-Simons theories.

The intuition behind the major role played by holomorphic functions in quantum theory is the following:

Imagine $\mathbb{R}^2$ to be the phase space of a particle moving on a line $(x, p) \in \mathbb{R}^2$, where $x$ is the position and $p$ is the momentum. since $\mathbb{R}^2 \equiv \mathbb{C}$ is a complex manifold, we can use the "shorthand”:

$$z = x+ ip$$

Consider the Hilbert space of functions on $ \mathbb{C}$ corresponding to the Gaussian inner product:

$(\psi, \phi) = \int \overline{\psi(z)} \phi(z) e^{-\bar{z}z} d\mathrm{Re}z d\mathrm{Im}z$

Taking the whole Hilbert space would allow construction of wave functions arbitrarily concentrated around any point $(x_0, p_0)$ in phase space, this is in contradiction with the Heisenberg uncertainty principle. On the other hand restricting the Hilbert space to holomorphic functions

$$ \frac{\partial \psi}{\partial \bar{z}} = 0$$

will restrict the "width" of the functions due to the square integrability condition.

This space (of holomorphic functions) is the Hilbert space of the harmonic oscillator. Another way to look at the holomorphicity restriction is to notice that without this restriction the full Hilbret space corresponds to an infinite number of copies of the harmonic oscillator, each closed under the action position and momentum operators.

The meaning of the restriction in this respect is that restricted Hilbert space carries an irreducible representation of the observable algebra.

This is a basic property required by Dirac in his axiomatic formulation of quantum theory. An irreducible representation which describes a single system is the right choice because we started classically from a single system.

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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 an analytic manifold. We meet such manifolds throughout physics. One place is in general relativity, gauge theories and other fields where differential geometry is heavily used, although, strictly speaking, much of the differential geometry that is used in something like GR can be done with much weaker assumptions than $C^\omega$ (quick word on notation: a $C^n$ function is one $n$-times differentiable in each of its arguments, a $C^\infty$ function one infinitely so (often called "smooth" function) and a $C^\omega$ is an analytic function, i.e. a $C^\infty$ function whose Taylor series also converges). Moreover in general manifolds, one doesn't get a notion of a unique "analytic continuation" (although an analogue of this happens for Lie groups, see below); general manifolds can be spliced together in nonunique ways.

Another mathematical tool used in physics is the notion of a Lie group. Lie groups describe the continuous symmetries of our World - rotations, Lorentz boosts, translations, abstract evolutions in state spaces and the symmetries into Lagrangian formulations of particle interactions either as a result of an experimentally observed continuous symmetry, or one deliberately encoded into a theory to beget, through Noether's theorem, a conserved quantity in line with experimental observation of particle interactions.

A Lie group is a group whose group product can be represented by a continuous (take heed: I did NOT say analytic) function of co-ordinates labelling the group. Lie groups are always analytic ($C^\omega$) manifolds, even if we only assume continuous ($C^0$) behaviour. Since you so admire the rigidity (the word mathematicians use to describe something that takes much less "information" to uniquely specify than one would intuitively believe), let this last statement sink in: we only have to assume continuous behaviour and analyticity automatically follows!. This amazing result, the solution to the so-called Hilbert's Fifth Problem, was the result of the work of Montgomery, Gleason and Zippin applied by Hidehiko Yamabe in 1953. One can go even further for compact semisimple Lie groups (ones that are compact and can be thought of as direct products of simple Lie groups, i.e. ones that have no normal Lie subgroups): for such groups, we don't even have to assume continuity, for there is no other abstract group structure even possible for such a Lie group so even the topology emerges from the algebraic structure alone and every group automorphism as an abstract group preserves the Lie group structure as well (van der Waerden, B. L., "Stetigkeitssätze für halbeinfache Liesche Gruppen", Mathematische Zeitschrift 36 pp780 - 786). The analogue of analytic continuation that holds in Lie groups is the unique definition of the identity-connected component given a specification of the group for any neighbourhood of the identity (or of any other point in the identity connected component). So the bits of the group joined to the identity are thus uniquely specified. Actually it is uniquely specified given the commutation relationships in its Lie algebra as well as the discrete information in its fundamental group. One can broaden a Lie group nonuniquely, however, to beyond its identity connected component. For more information, see my answer here.

In closing, returning to holomorphic functions of a complex variable: in this case, the notions of holomorphicity (independence of $F(z) = \int_a^z f(u){\rm d}u$ of path), analyticity and differentiability are all logically equivalent (any one can be derived from any other), but this is not so for more general sets. Indeed, for holomorphic complex functions, you don't even need the behaviour on a teeny tiny disk: such a function is uniquely specified by its values on any countably infinite set with a limit point.

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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 response function to causality, in particular, leading to the Kramers-Kronig relations.

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