Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined makes no good sense.

The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

Initial discussion

Suppose f is an analytic function defined on an open subset U of the complex plane C. If V is a larger open subset of C, containing U, and F is an analytic function defined on V such that

F(z) = f(z) for all z in U,

then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.

Analytic continuations are unique in the following sense: if V is connected and F₁ and F₂ are two analytic continuations of f defined on V, then

F₁ = F₂

everywhere. That is because the difference is an analytic function which vanishes on the intersection of their domains, a non-empty open set, and an analytic function which vanishes on a non-empty open set must vanish everywhere on its domain (assuming the domain is connected) and hence must be identically zero.

For example, if a power series with radius of convergence r about a point a of C is given, one can consider analytic continuations of the power series, i.e. analytic functions F which are defined on larger sets than the open disc of radius r at a, in symbols

{z : |z − a| < r},

and agree with the given power series on that set. The number r is maximal in the following sense: there always exists a complex number z with

|z − a| = r

such that no analytic continuation of the series can be defined at z. Therefore there is a limitation to analytic continuation to bigger discs with the same centre a. On the other hand there may well be analytic continuations to some larger sets. That depends on the radius of convergence when you expand about points b other than a; if that is greater than

r − |b − a|

then we win the right to use that expansion on an open disc, part of which lies outside the original disc of definition. If not, there is a natural boundary on the bounding circle.

Applications

A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, and then extending it by analytic continuation. In practice, this continuation is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. Examples are the Riemann zeta function and the gamma function.

The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces.

The power series defined above is generalized by the idea of a germ. The general theory of analytic continuation and its generalization is known as sheaf theory.

Formal definition of a germ

Let

$f(z)=\sum _{k=0}^{\infty }\alpha _{k}(z-z_{0})^{k}$

be a power series converging in the disc D_r(z₀) := {z in C : |z - z₀| < r} for r > 0. (Note, without loss of generality, here and in the sequel, we will always assume that a maximal such r was chosen, even if it is ∞.) Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector

g = (z₀, α₀, α₁, α₂, ...)

is a germ of f. The base g₀ of g is z₀, the stem of g is (α₀, α₁, α₂, ...) and the top g₁ of g is α₀. The top of g is the value of f at z₀, the bottom of g.

Any vector g = (z₀, α₀, α₁, ...) is a germ if it represents a power series of an analytic function around z₀ with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs ${\mathcal {G}}$.

The topology of the set of germs

Let g and h be germs. If |h₀ - g₀| < r where r is the radius of convergence of g and if the power series that g and h represent define identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write g ≥ h. This compatibility condition is neither transitive, symmetric nor antisymmetric. If we extend the relation by transitivity, we obtain a symmetric relation, which is therefore also an equivalence relation on germs (but not an ordering). This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted $\cong$.

We can define a topology on ${\mathcal {G}}$. Let r > 0, and let

$U_{r}(g)=\{h\in {\mathcal {G}}:g\geq h,|g_{0}-h_{0}|<r\}.$

The sets U_r(g), for all r > 0 and g ∈ ${\mathcal {G}}$ define a basis of open sets for the topology on ${\mathcal {G}}$.

A connected component of ${\mathcal {G}}$ (i.e., an equivalence class) is called a sheaf. We also note that the map φ_g(h) = h₀ from U_r(g) to C where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas for ${\mathcal {G}}$, hence ${\mathcal {G}}$ is a Riemann surface. ${\mathcal {G}}$ is sometimes called the universal analytic function.

Examples of analytic continuation

$L(z)=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}(z-1)^{k}$

is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ

g = (1, 0, 1, −1, 1, −1, 1, −1, ...)

This germ has a radius of convergence of 1, and so there is a sheaf S corresponding to it. This is the sheaf of the logarithm function.

The uniqueness theorem for analytic functions also extends to sheaves of analytic functions: if the sheaf of an analytic function contains the zero germ (i.e., the sheaf is uniformly zero in some neighborhood) then the entire sheaf is zero. Armed with this result, we can see that if we take any germ g of the sheaf S of the logarithm function, as described above, and turn it into a power series f(z) then this function will have the property that exp(f(z))=z. If we had decided to use a version of the inverse function theorem for analytic functions, we could construct a wide variety of inverses for the exponential map, but we would discover that they are all represented by some germ in S. In that sense, S is the "one true inverse" of the exponential map.

In older literature, sheaves of analytic functions were called multi-valued functions. See sheaf for the general concept.

Hadamard's gap theorem

For a power series

$f(z)=\sum _{k=0}^{\infty }\alpha _{k}(z-z_{0})^{k}$

with coefficients mostly zero in the precise sense that they vanish outside a sequence of exponents k(i) with

k(i + 1)/k(i) > 1 + δ

for some fixed δ > 0, the circle centre z₀ and with radius the radius of convergence is a natural boundary. (See for example E. C. Titchmarsh, Theory of Functions.)