Analytical continuation

In analysis , the analytic continuation of a function that is defined on a subset of real or complex numbers is understood to mean an analytic function that is defined on a complex area that includes and that corresponds to the original function on the subset . Here, almost exclusively the cases are of interest in which the continuation (and usually also a maximum area) is clearly determined by the given set and the function defined on it . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle f}$

In function theory , especially when investigating functions in several complex variables, the term is used more abstractly. Here analytical continuation means the continuation of a holomorphic function or a holomorphic function nucleus . A distinction is made between the continuation of the germ along a path and the continuation to a function in an area.

It is significant that holomorphic functions - unlike continuous functions or functions that can only be differentiated as often as required - can be reconstructed very well from local data in a very small environment.

Analytical continuation in analysis

→ Main article : Identity theorem for holomorphic functions

The following statements about continuability are important for elementary analysis:

• Let be a real (open or closed) interval. Then a function can be analytically continued if and only if ${\displaystyle M}$ ${\displaystyle f\colon M\rightarrow \mathbb {R} }$

• if for every point of the interval there is an open neighborhood on which the function can be represented by an absolutely convergent power series , or
• if at every point of the interval it is arbitrarily often differentiable and the Taylor series has a non-vanishing radius of convergence for every point of the interval . ${\displaystyle f}$

In both cases, the series mentioned - theoretically only locally, but in many practically important cases with a suitable choice of the development point in a complex area that covers the entire interval - a description of the analytical continuation clearly determined here as a power series. ${\displaystyle M}$

• If the closed hull of an infinite set is connected , for example a real interval, and there is an analytical continuation of to a domain , then a second holomorphic function already agrees with the continuation , ${\displaystyle M\subset \mathbb {C} }$ ${\displaystyle g}$ ${\displaystyle f\colon M\rightarrow \mathbb {C} }$ ${\displaystyle G\supset M}$ ${\displaystyle G}$ ${\displaystyle h}$ ${\displaystyle g}$

• when used with on an infinite subset of that in itself accumulates matches or ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle M}$
• if at some fixed point of the function values ​​and all derivatives of and agree. ${\displaystyle M}$ ${\displaystyle g}$ ${\displaystyle h}$

The statements mentioned here and a few other statements about analytical continuability and the uniqueness of continuation are contained in the following, more abstract formulations of function theory as special cases.

Examples

• Each whole rational function on , so any real function, the function term is a polynomial in is, can be attributed to analytically by the function with the same function term continue. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} [x]}$ ${\displaystyle \mathbb {C} }$
• The fractional rational function can be continued on the area . In the inside of the unit circle the continuation can be represented by the power series , in the outside by the Laurent series . Both continuations can be analytically continued locally beyond their convergence area through power series. They can therefore be put together to form a common analytical continuation on , as always with fractional-rational real functions, this is of course the complex fractional-rational function . ${\displaystyle f(x)={\tfrac {1}{(1-x^{2})}}}$ ${\displaystyle F=\mathbb {C} \setminus \lbrace \pm 1\rbrace }$ ${\displaystyle G=\lbrace z\in \mathbb {C} :|z|<1\rbrace }$ ${\textstyle g(z)=\sum _{k=0}^{\infty }z^{2k}}$ ${\displaystyle H=\lbrace z\in \mathbb {C} :|z|>1\rbrace }$ ${\textstyle h(z)=-\sum _{k=1}^{\infty }z^{-2k}}$ ${\displaystyle g,h}$ ${\displaystyle F}$ ${\displaystyle z\mapsto {\tfrac {1}{(1-z^{2})}}}$
• The real exponential functions , the sine function and the cosine function can be represented as power series with the radius of convergence . Therefore one can analytically extend them to whole functions , which can then be represented by the same power series. ${\displaystyle \rho =\infty }$
• The on defined factorial function as analytic continuation has the gamma function . This continuation, however, only becomes clear through the additional condition that the continuation should be logarithmically convex . → See Bohr-Mollerup theorem . ${\displaystyle M=\mathbb {N} _{0}}$ ${\displaystyle f(n)=n!}$ ${\displaystyle z\mapsto \Gamma (z+1)}$

Germ

In order to give a precise definition of an analytical continuation in the sense of function theory , the terms stem and function seed must first be explained: Be a complex manifold and a point. In addition, let two environments of and two holomorphic functions . The two functions are called equivalent in the point if there is a neighborhood of with . The set of all these equivalence classes is referred to as a stalk , the equivalence classes as (functional) germs. The projection of a function onto its nucleus in the point is also noted. ${\displaystyle X}$ ${\displaystyle a\in X}$ ${\displaystyle U,V}$ ${\displaystyle a}$ ${\displaystyle f\colon U\rightarrow \mathbb {C} ,\;g\colon V\rightarrow \mathbb {C} }$ ${\displaystyle a}$ ${\displaystyle W\subseteq U\cap V}$ ${\displaystyle a}$ ${\displaystyle f|W=g|W}$ ${\displaystyle {\mathcal {O}}_{a}}$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle \rho _{a}(f)}$

The germ of a function clearly describes the behavior of in the “immediate” vicinity of . That is more than the mere functional value , because the derivations etc. can also be read from the germ, since they result from every little environment of . ${\displaystyle \rho _{a}(f)}$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle f(a)}$ ${\displaystyle f'(a),f''(a)}$ ${\displaystyle a}$

The stalk naturally bears the structure of an algebra. It is isomorphic to the -algebra of the convergent power series , since the local behavior of a holomorphic function is uniquely determined by its power series expansion. ${\displaystyle {\mathcal {O}}_{a}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a}$

Continuation along a path

Let be a connected complex manifold, two points and as well as two function seeds. is called analytical continuation of along the path with if the following applies: There exist points with open neighborhoods and holomorphic functions such that ${\displaystyle X}$ ${\displaystyle a,b\in X}$ ${\displaystyle \varphi \in {\mathcal {O}}_{a}}$ ${\displaystyle \psi \in {\mathcal {O}}_{b}}$ ${\displaystyle \psi }$ ${\displaystyle \varphi }$ ${\displaystyle c\colon [0,1]\rightarrow X}$ ${\displaystyle c(0)=a,c(1)=b}$ ${\displaystyle x_{0},x_{1},\dots ,x_{n}\in c([0,1])}$ ${\displaystyle U_{0},U_{1},\dots ,U_{n}\subseteq X}$ ${\displaystyle f_{k}\colon U_{k}\rightarrow \mathbb {C} ,\;k=0,1,\dots ,n}$

• ${\displaystyle x_{0}=a,\;x_{n}=b}$
• ${\displaystyle c([0,1])\subset \bigcup _{k=0}^{n}U_{k}}$
• ${\displaystyle \rho _{a}(f_{0})=\varphi ,\;\rho _{b}(f_{n})=\psi }$
• ${\displaystyle f_{k-1}|_{U_{k-1}\cap U_{k}}=f_{k}|_{U_{k-1}\cap U_{k}}}$ For ${\displaystyle k=1,2,\dots ,n}$

In other words, there is a finite series of open neighborhoods that cover the curve. Holomorphic functions are defined on each of these environments, which correspond in the areas where the environments overlap. Often one chooses open circles as sets , because these appear as areas of convergence of series expansion; in this case one speaks of a circular chain . ${\displaystyle U_{k}}$

This continuation generally depends on the choice of path (but not on the intermediate points and the surroundings ). Also, there is generally no in an environment of very holomorphic function with and . ${\displaystyle x_{k}}$ ${\displaystyle U_{k}}$ ${\displaystyle U}$ ${\displaystyle c([0,1])}$ ${\displaystyle f}$ ${\displaystyle \rho _{a}(f)=\varphi }$ ${\displaystyle \rho _{b}(f)=\psi }$

definition

Let be a connected complex manifold, a point and a function seed. The quadruple is called an analytic continuation of if: ${\displaystyle X}$ ${\displaystyle a\in X}$ ${\displaystyle \varphi \in {\mathcal {O}}_{a}}$ ${\displaystyle (Y,p,f,b)}$ ${\displaystyle \varphi }$

• ${\displaystyle Y}$ is a connected complex manifold.
• ${\displaystyle p\colon Y\rightarrow X}$is a holomorphic mapping and a local homeomorphism .
• ${\displaystyle f\colon Y\rightarrow \mathbb {C} }$ is a holomorphic function.
• ${\displaystyle b\in Y}$so that and , where the projection of on the equivalence class of its nucleus in denotes. ${\displaystyle p(b)=a}$ ${\displaystyle \rho _{b}(f)=\varphi \circ p}$ ${\displaystyle \rho _{b}(f)}$ ${\displaystyle f}$ ${\displaystyle b}$

The analytical continuation defined in this way is related to the continuation along a path: If a path has a starting point and an end point , then a path is a starting point and an end point . The function is defined in an environment of by a function nucleus in . ${\displaystyle c\colon [0,1]\rightarrow Y}$ ${\displaystyle c(0)=b}$ ${\displaystyle c(1)=d}$ ${\displaystyle p\circ c\colon [0,1]\rightarrow X}$ ${\displaystyle a=p(b)}$ ${\displaystyle p(d)}$ ${\displaystyle f\colon Y\rightarrow \mathbb {C} }$ ${\displaystyle p(d)}$ ${\displaystyle f\circ p^{-1}}$ ${\displaystyle {\mathcal {O}}_{p(d)}}$

example

${\displaystyle X=\mathbb {C} ,a=1}$and be the seed in that branch of the holomorphic square root with . Analytical continuations of this are, for example: ${\displaystyle \varphi }$ ${\displaystyle 1}$ ${\displaystyle {\sqrt {1}}=1}$

• The function defined by the Taylor series around in the open circular disk . The projection is the natural inclusion image. ${\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)n!^{2}4^{n}}}(x-1)^{n}}$ ${\displaystyle 1}$ ${\displaystyle B(1,1)}$ ${\displaystyle f\colon Y=B(1,1)\rightarrow \mathbb {C} }$ ${\displaystyle p\colon Y\rightarrow \mathbb {C} }$
• The main branch of the square root, defined on the slotted complex plane , again being the natural inclusion mapping. ${\displaystyle Y:=\mathbb {C} \setminus \left\{z\in \mathbb {C} \,:\,\mathrm {Im} \,z=0,\;\mathrm {Re} \,z\leq 0\right\}}$ ${\displaystyle p\colon Y\rightarrow \mathbb {C} }$

All examples have in common that can be understood as a subset of . The last two examples also show that there is no largest area within which the function can be holomorphically continued. The question of the greatest possible continuation leads to the definition of the maximum analytical continuation: ${\displaystyle Y}$ ${\displaystyle X=\mathbb {C} }$ ${\displaystyle \mathbb {C} }$

Maximum analytical continuation

Let be a connected complex manifold, a point and a function seed. An analytic continuation of is called maximal analytic continuation if for every other analytic continuation of : There exists a holomorphic mapping with , and . ${\displaystyle X}$ ${\displaystyle a\in X}$ ${\displaystyle \varphi \in {\mathcal {O}}_{a}}$ ${\displaystyle (Y,p,f,b)}$ ${\displaystyle \varphi }$ ${\displaystyle (Z,q,g,c)}$ ${\displaystyle \varphi }$ ${\displaystyle F\colon Z\rightarrow Y}$ ${\displaystyle f\circ F=g}$ ${\displaystyle F(c)=b}$ ${\displaystyle q=p\circ F}$

Existence and uniqueness

The uniqueness of the maximum analytical continuation up to holomorphic isomorphism follows directly from the definition. The existence can be shown with the help of the sheaf theory : is the connected component of the superposition space of the sheaf of the holomorphic functions , which contains a firmly chosen archetype of the germ . ${\displaystyle Z}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle \varphi }$

example

${\displaystyle X=\mathbb {C} ,\;a=1}$and be the seed of that branch of the holomorphic square root with . The maximum analytical continuation is given by: ${\displaystyle \varphi }$ ${\displaystyle {\sqrt {1}}=1}$ ${\displaystyle (Y,b,p,f)}$

${\displaystyle Y:=\left\{(y_{1},y_{2})\in \mathbb {C} ^{2}\,:\,y_{2}=y_{1}^{2},y_{1}\neq 0\right\}\subset \mathbb {C} ^{2}}$

${\displaystyle b=(1,1)}$

${\displaystyle p(y_{1},y_{2})=y_{2}}$

${\displaystyle f(y_{1},y_{2})=y_{1}}$

For another analytical continuation , the mapping is defined by . ${\displaystyle (Z,q,g,c)}$ ${\displaystyle F\colon Z\rightarrow Y}$ ${\displaystyle F(z):=\left(g(z),q(z)\right)}$

literature

• Heinrich Behnke and Friedrich Sommer: Theory of the analytical functions of a complex variable . Study edition, 3rd edition. Springer, Berlin / Heidelberg / New York 1972, ISBN 3-540-07768-5 .
• Hans Grauert, Klaus Fritzsche: Introduction to the function theory of several variables . Springer-Verlag, Berlin 1974, ISBN 3-540-06672-1 u. ISBN 0-387-06672-1
• Otto Forster: Riemann surfaces . Springer-Verlag 1977 (out of print; English translation available, ISBN 0-387-90617-7 )