The characterization of analytic functions defined on open sets is simple: we know that the sum of every power series is holomorphic in its open disk of convergence and conversely that every holomorphic function defined on an open set is locally the sum of a power series. Hence, we have the:

The definition of analytic function is not limited to open sets; we may also extend the definition of holomorphic function to sets that may not be open:

This definition is appropriate because – at least on “well-behaved” sets – the classes of analytic and holomorphic functions are identical, as they are on open sets. For example, convex sets are well-behaved:

Conversely, let \(A\) be a convex subset of \(\mathbb{C}\) and let \(f: A \to \mathbb{C}\) be an analytic function. There is a positive function \(a \in A \mapsto r_a\) and a family \(a \in A \mapsto f_a\) of holomorphic functions defined on \(D(a, r_a)\) such that \(f\) and \(f_a\) are equal on \(A \cap D(a, r_a).\) Let \(\phi\) be the complex-valued function defined on the union \(\Omega\) of all disks \(D(a, r_a)\) by \[ \phi(z) = f_a(z) \; \mbox{ if } \; a \in A \; \mbox{ and } \; z \in D(a, r_a). \] This definition is non-ambiguous: if \(a \in A,\) \(b \in A\) and \(z\) belongs to \(D(a, r_a)\) and \(D(b, r_b),\) then by convexity of \(A,\) the domain of \(f\) contains the set \(L = [a,b] \cap D(a, r_a) \cap D(b, r_b).\) By the isolated zeros theorem, the functions \(f_a\) and \(f_b,\) equal on \(L,\) are also equal on \(D(a, r_a) \cap D(b, r_b),\) therefore \(f_a(z) = f_b(z).\) The function \(\phi\) is by construction an extension of \(f\) which is holomorphic. ‌\(\blacksquare\)

Conversely, assume that \(f: A \to \mathbb{C}\) is analytic. Let \(r_z\) be a collection of positive radii indexed by \(z \in A\) and \(f_z: D(z,r_z) \to \mathbb{C}\) the corresponding collection of holomorphic functions such that \[ \forall \, z \in A, \; \forall \, w \in A \cap D(z,r_z), \; f_z(w) = f(w). \] There is another collection with the same property, but associated with a constant and positive radius \(r.\) It is easy to build this collection – by a simple restriction of the \(f_z\) – if we can first extend the functions of the original collection to make sure that \(z \in A \mapsto r_z\) has a positive lower bound. To do so, we may consider for any \(z \in A\) the set of holomorphic functions defined on open disks centered on \(z\) with the same values as \(f\) on the points that also belong to \(A.\) This set is not empty, as it contains \(f_z.\) Additionally, if two functions belong to this set, as \(z\) is not isolated in \(A,\) they are equal on the intersection of their domain of definition. Hence, we may select as a new \(f_z\) the function in the set whose domain of definition has the largest radius. Now, if we assume that the infimum of the \(r_z\) is \(0,\) there is a sequence \(z_n\) in \(A\) such that \(r_{z_n} \to 0\) and thus a subsequence \(w_n\) that converges to some \(z \in A.\) But for any \(w \in D(z,r_z),\) the radius of the largest disk at \(w\) satisfies \[r_{w} + |w-z| \geq r_z,\] thus we have a contradiction eventually when \(|w_n - z| < r_z / 2\) and \(r_{w_n} < r_z /2.\)

Let \(V\) be an open neighbourhood of \(A\) where the projection onto \(A\) is unique. We may assume that \(r\) is smaller than the distance between \(A\) and \(\mathbb{C} \setminus V.\) For any \(z \in \Omega = A + D(0,r),\) denote \(\pi(z)\) the projection of \(z\) onto \(A\) and define \(g(z) = f_{\pi(z)}(z);\) the function \(g: \Omega \mapsto \mathbb{C}\) obviously extends \(f.\) Additionally, it is holomorphic, because it satisfies \[ \forall \, z \in \Omega, \; \exists \, \epsilon > 0, \; \forall \, w \in D(z, \epsilon), \; g(w) = f_{\pi(z)}(w). \] Indeed, if \(|w - z| < \epsilon = r - |z - \pi(z)|,\) then \[ |w - \pi(z)| \leq |w - z| + |z - \pi(z)| < r, \] so \(w\) that belongs to \(D(\pi(w), r)\) by construction also belongs to \(D(\pi(z), r).\) By construction, the functions \(f_{\pi(z)}\) and \(f_{\pi(w)}\) have the same values on every point of \(A\) that belongs to both of their domains of definition. Naturally, \(\pi(z)\) belongs to the domain of \(f_{\pi(z)};\) since \[ |\pi(w) - \pi(z)| \leq |w - z| < r \] it also belongs to the domain of \(f_{\pi(w)}.\) As none of the points of \(A\) is isolated, by the isolated zeros theorem, the functions \(f_{\pi(w)}\) and \(f_{\pi(z)}\) are identical on \(D(\pi(z), r) \cap D(\pi(w),r),\) which leads to \(g(w) = f_{\pi(w)}(w) = f_{\pi(z)}(w).\) ‌\(\blacksquare\)

The characterization of analytic functions on convex sets provides:

We also have:

Conversely, if \(f\) is real analytic, let \(g\) be one of its holomorphic extensions. At every \(x\) in \(I,\) \[ f'(x) = \lim_{y \in I \to x} \frac{f(y) - f(x)}{y - x} = \lim _{y \in I \to x} \frac{g(y) - g(x)}{y - x} = g'(x); \] the real derivative of \(f\) exists and is equal to the complex derivative of \(g,\) hence \(g'\) is a holomorphic extension of \(f'.\) By induction, at every order \(n,\) \(f^{(n)}\) exists and accepts \(g^{(n)}\) as a holomorphic extension. Consequently, the local expansion of \(g\) as a Taylor series provides a local expansion of \(f\) as a (real) Taylor series. ‌\(\blacksquare\)

We define the continuation for open and connected sets only so that:

Like analytic continuations to sets, analytic continuations along paths are also unique whenever they exist:

Actually, an analytic continuation along a path carries more information than a simple value from the initial point to the terminal point of the path: it defines a new holomorphic function in the some open neighbourhood of the terminal point. To formalize this, we introduce a new definition.

Let \(z \in \Omega\) and let \(\phi_z\) be the analytic continuation of \(f\) along \([c \to z].\) By construction, there is a \(\epsilon \in \left]0,1\right]\) such that \(\phi_z(t) = f((1-t)c + tz)\) for any \(0 \leq t < \epsilon.\) If we still denote \(\phi_z\) the holomorphic extension of \(\phi_z\) to some open neighbourghood \(\{w \in \mathbb{C} \;|\; d(w,[0,1])< r\}\) of \([0,1],\) then for any complex number \(w\) in some non-empty open disk centered at \(0,\) we have \[\phi_z(w) = f((1-w)c + wz).\] Now, for any \(z \in \Omega \setminus D(z_0,r)\) and any complex number \(h\) such that \(|h| < r |z-c|,\) the function \[ \psi_{z,h}: t \in [0,1] \mapsto \phi_z(w) \; \mbox{ with } \; w = t \frac{z+h-c}{z-c} \] is defined, analytic and as \(w\) satisfies \[ (1 - t) c + t (z+h) = (1-w) c + wz, \] for small values of \(t,\) we have \[ \psi_{z,h}(t) = \phi_z(w) = f((1-w)c + wz) = f((1-t)c + t(z+h)), \] therefore \(\psi_{z,h}\) is the analytic continuation of \(f_c\) along \([c \to z+h]\): \[\psi_{z,h}=\phi_{z+h}.\] This leads for small values of \(h\) to the equality \[ \phi_{z+h}(1) = \psi_{z,h}(1) = \phi_z\left(\frac{z+h-c}{z-c}\right). \] The function \(z \in \Omega \mapsto \phi_z(1)\) is an extension of \(f\) to \(\Omega;\) by the above equality it is also holomorphic, thus it is the analytic extension of \(f\) to \(\Omega.\) ‌\(\blacksquare\)