# Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain 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 becomes divergent.

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 singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

## Initial discussion

Analytic continuation of natural logarithm (imaginary part)

Suppose f is an analytic function defined on a non-empty 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

${\displaystyle \displaystyle F(z)=f(z)\qquad \forall z\in U,}$

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 the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U

F1(z) = F2(z) = f(z),

then

F1 = F2

on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions.

## 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.

## Worked example

Analytic continuation from U (centered at 1) to V (centered at a=(3+i)/2)

Begin with a particular analytic function ${\displaystyle f}$. In this case, it's given by a power series centered at ${\displaystyle z=1}$:

${\displaystyle f(z)=\sum _{k=0}^{\infty }(-1)^{k}(z-1)^{k}}$.

By the Cauchy–Hadamard theorem, its radius of convergence is 1. That is, ${\displaystyle f}$ is defined and analytic on the open set ${\displaystyle U=\{|z-1|<1\}}$ which has boundary ${\displaystyle \partial U=\{|z-1|=1\}}$. Indeed, the series diverges at ${\displaystyle z=0\in \partial U}$.

Pretend we don't know that ${\displaystyle f(z)=1/z}$, and focus on recentering the power series at a different point ${\displaystyle a\in U}$:

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}(z-a)^{k}}$.

We'll calculate the ${\displaystyle a_{k}}$'s and determine whether this new power series converges in an open set ${\displaystyle V}$ which is not contained in ${\displaystyle U}$. If so, we will have analytically continued ${\displaystyle f}$ to the region ${\displaystyle U\cup V}$ which is strictly larger than ${\displaystyle U}$.

The distance from ${\displaystyle a}$ to ${\displaystyle \partial U}$ is ${\displaystyle \rho =1-|a-1|>0}$. Take ${\displaystyle 0; let ${\displaystyle D}$ be the disk of radius ${\displaystyle r}$ around ${\displaystyle a}$; and let ${\displaystyle \partial D}$ be its boundary. Then ${\displaystyle D\cup \partial D\subset U}$. Using Cauchy's differentiation formula to calculate the new coefficients,

${\displaystyle a_{k}={\frac {f^{(k)}(a)}{k!}}={\frac {1}{2\pi i}}\int _{\partial D}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}={\frac {1}{2\pi i}}\int _{\partial D}{\frac {\sum _{n=0}^{\infty }(-1)^{n}(\zeta -1)^{n}d\zeta }{(\zeta -a)^{k+1}}}}$
${\displaystyle ={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }(-1)^{n}\int _{\partial D}{\frac {(\zeta -1)^{n}d\zeta }{(\zeta -a)^{k+1}}}={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }(-1)^{n}\int _{0}^{2\pi }{\frac {(a+re^{i\theta }-1)^{n}rie^{i\theta }d\theta }{(re^{i\theta })^{k+1}}}}$
${\displaystyle ={\frac {1}{2\pi }}\sum _{n=0}^{\infty }(-1)^{n}\int _{0}^{2\pi }{\frac {(a-1+re^{i\theta })^{n}d\theta }{(re^{i\theta })^{k}}}}$
${\displaystyle ={\frac {1}{2\pi }}\sum _{n=0}^{\infty }(-1)^{n}\int _{0}^{2\pi }{\frac {\sum _{m=0}^{n}{\binom {n}{m}}(a-1)^{n-m}(re^{i\theta })^{m}d\theta }{(re^{i\theta })^{k}}}={\text{(boring details)}}}$
${\displaystyle =(-1)^{k}a^{-k-1}.}$

That is,

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}(z-a)^{k}=\sum _{k=0}^{\infty }(-1)^{k}a^{-k-1}(z-a)^{k}={\frac {1}{a}}\sum _{k=0}^{\infty }{\Big (}1-{\frac {z}{a}}{\Big )}^{k},}$

which has radius of convergence ${\displaystyle |a|}$, and ${\displaystyle V=\{|z-a|<|a|\}}$. If we choose ${\displaystyle a\in U}$ with ${\displaystyle |a|>1}$, then ${\displaystyle V}$ is not a subset of ${\displaystyle U}$ and is actually larger in area than ${\displaystyle U}$. The plot shows the result for ${\displaystyle a=(3+i)/2}$.

We can continue the process: select ${\displaystyle b\in U\cup V}$, recenter the power series at ${\displaystyle b}$, and determine where the new power series converges. If the region contains points not in ${\displaystyle U\cup V}$, then we will have analytically continued ${\displaystyle f}$ even farther. This particular ${\displaystyle f}$ can be analytically continued to the punctured complex plane ${\displaystyle \mathbb {C} \setminus \{0\}}$.

## Formal definition of a germ

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

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

be a power series converging in the disk Dr(z0), r > 0, defined by

${\displaystyle D_{r}(z_{0})=\{z\in \mathbf {C} :|z-z_{0}|.

Note that without loss of generality, here and below, we will always assume that a maximal such r was chosen, even if that r 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

${\displaystyle g=(z_{0},\alpha _{0},\alpha _{1},\alpha _{2},\ldots )}$

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.

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 ${\displaystyle {\mathcal {G}}}$.

## The topology of the set of germs

Let g and h be germs. If |h0g0| < r where r is the radius of convergence of g and if the power series defined by g and h specify identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write gh. 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 ${\displaystyle \cong }$.

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

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

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

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

## Examples of analytic continuation

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

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

${\displaystyle g=\left(1,0,1,-{\frac {1}{2}},{\frac {1}{3}},-{\frac {1}{4}},{\frac {1}{5}},-{\frac {1}{6}},\ldots \right)}$

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.

## Natural boundary

Suppose that a power series has radius of convergence r and defines an analytic function f inside that disc. Consider points on the circle of convergence. A point for which there is a neighbourhood on which f has an analytic extension is regular, otherwise singular. The circle is a natural boundary if all its points are singular.

More generally, we may apply the definition to any open connected domain on which f is analytic, and classify the points of the boundary of the domain as regular or singular: the domain boundary is then a natural boundary if all points are singular, in which case the domain is a domain of holomorphy.

## Monodromy theorem

The monodromy theorem gives a sufficient condition for the existence of a direct analytic continuation (i.e., an extension of an analytic function to an analytic function on a bigger set).

Suppose D is an open set in C, and f an analytic function on D. If G is a simply connected domain containing D, such that f has an analytic continuation along every path in G, starting from some fixed point a in D, then f has a direct analytic continuation to G.

In the above language this means that if G is a simply connected domain, and S is a sheaf whose set of base points contains G, then there exists an analytic function f on G whose germs belong to S.

For a power series

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{n_{k}}}$

with

${\displaystyle \liminf _{k\to \infty }{\frac {n_{k+1}}{n_{k}}}>1}$

the circle of convergence is a natural boundary. Such a power series is called lacunary. This theorem has been substantially generalized by Eugen Fabry (see Fabry's gap theorem) and George Pólya.

## Pólya's theorem

Let

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

be a power series, then there exist εk ∈ {−1, 1} such that

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

has the convergence disc of f around z0 as a natural boundary.

The proof of this theorem makes use of Hadamard's gap theorem.