Riemann's theorem of liftability

The Riemannian Hebbarkeitssatz (by Bernhard Riemann ) is a basic result of the mathematical part of the area of the function theory . The theorem says that an isolated singularity of a holomorphic function can be removed (“fixed”) if and only if the function is restricted in a neighborhood of the singularity . Such a singularity is called liftable.

sentence

Let it be a domain and let it be a holomorphic function. ${\displaystyle G\subseteq \mathbb {C} }$ ${\displaystyle z_{0}\in G}$ ${\displaystyle f\colon G{\setminus }\{z_{0}\}\to \mathbb {C} }$

There exists a neighborhood of in , so to is limited, there is one on ${\displaystyle U}$ ${\displaystyle z_{0}}$ ${\displaystyle G}$ ${\displaystyle f}$ ${\displaystyle U{\setminus }\{z_{0}\}}$ all holomorphic function with . ${\displaystyle G}$ ${\displaystyle {\tilde {f}}}$ ${\displaystyle {\tilde {f}}|_{G{\setminus }\{z_{0}\}}=f}$

The existence of states that by holomorphic on can continue. As a result, the “gap” in the domain of definition is, so to speak, “canceled”. According to the identity theorem for holomorphic functions , there can only be one such . ${\displaystyle {\tilde {f}}}$ ${\displaystyle f}$ ${\displaystyle z_{0}\mapsto {\tilde {f}}(z_{0})}$ ${\displaystyle z_{0}}$ ${\displaystyle f}$ ${\displaystyle {\tilde {f}}}$

proof

The Riemannian levy theorem can be deduced from the Cauchy estimate of the Laurent series coefficients:

According to the prerequisite, there is a small enough that the dotted environment is still completely in and applies to one and all . Since it is holomorphic, it can be developed into a convergent Laurent series there . In other words: There is (exactly) one sequence of complex numbers such that the following applies to all : ${\displaystyle \varepsilon >0}$ ${\displaystyle {\dot {B}}_{\varepsilon }(z_{0})=\{z\in \mathbb {C} \mid 0<|z-z_{0}|<\varepsilon \}}$ ${\displaystyle G}$ ${\displaystyle |f(z)|\leq M}$ ${\displaystyle M\in \mathbb {R} }$ ${\displaystyle z\in {\dot {B}}_{\varepsilon }(z_{0})}$ ${\displaystyle f}$ ${\displaystyle {\dot {B}}_{\varepsilon }(z_{0})}$ ${\displaystyle (a_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle z\in {\dot {B}}_{\varepsilon }(z_{0})}$

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

The function is of course also limited to each subset of by (in terms of amount), so according to the Cauchy estimate the following applies for and each : ${\displaystyle f}$ ${\displaystyle {\dot {B}}_{\varepsilon }(z_{0})}$ ${\displaystyle M}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle 0<\delta <\varepsilon }$

${\displaystyle |a_{n}|\leq {\frac {M}{\delta ^{n}}}}$

Is , this can be written as, after the border crossing arises . The main part of the Laurent series disappears identically , which is why the singularity of in must be liftable. This elevation then occurs precisely through the value . ${\displaystyle n<0}$ ${\displaystyle |a_{n}|\leq M\cdot \delta ^{|n|}}$ ${\displaystyle \delta \to 0}$ ${\displaystyle |a_{n}|=0}$ ${\displaystyle 0}$ ${\displaystyle f}$ ${\displaystyle z_{0}}$ ${\displaystyle {\tilde {f}}(z_{0})=a_{0}}$

Generalizations

A simple generalization is to abandon the premise of limitation and only require that

${\displaystyle \lim _{z\to z_{0}}(z-z_{0})f(z)=0.}$

The continuability of now follows easily from the above formulation when applied to those in an environment of restricted function . ${\displaystyle f}$ ${\displaystyle z_{0}}$ ${\displaystyle g(z)=(z-z_{0})f(z)}$

reversal

The statement of the deductibility theorem can also be reversed, that is, the following applies:

Has a holomorphic function in a removable singularity, it is in an environment of limited. ${\displaystyle f}$ ${\displaystyle z_{0}}$ ${\displaystyle z_{0}}$

This is a simple consequence of the continuity of the holomorphic continuation at the point . Due to this local restriction, liftable singularities differ fundamentally from poles and essential singularities . ${\displaystyle {\tilde {f}}}$ ${\displaystyle z_{0}}$

Non-existence of a holomorphic root function

In function theory, the levy theorem also serves as an auxiliary proposition in other proofs. For example, it can be used to prove the non-existence of a holomorphic root function.

There is no holomorphic function that fulfills for everyone . ${\displaystyle \mathbb {C} {\setminus }\{0\}}$ ${\displaystyle f}$ ${\displaystyle f(z)^{2}=z}$ ${\displaystyle z\neq 0}$

Assuming that your amount must then apply. According to this, is is bounded in a neighborhood of and therefore even to be completely holomorphic according to Riemann's theorem of liftability . In particular, it is continuously differentiable in with the derivative . After the identity theorem need and their derivation function to match each with the real root function and its derivative. For positive real arguments , however, the derivative grows as it approaches 0 over all limits, so that a (actual) limit value does not exist: ${\displaystyle |f(z)|={\sqrt {|z|}}}$ ${\displaystyle f}$ ${\displaystyle 0}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle 0}$ ${\displaystyle f'(0)}$ ${\displaystyle f}$ ${\displaystyle f'}$ ${\displaystyle \mathbb {R} ^{+}\subset \mathbb {C} }$ ${\displaystyle x\in \mathbb {R} ^{+}}$

${\displaystyle \lim _{x\searrow 0}f'(x)=\lim _{x\searrow 0}{\frac {1}{2{\sqrt {x}}}}=\infty \neq f'(0)}$

Several variables

In the function theory of several variables, a subset of a domain is called thin if it is locally contained in non-trivial sets of roots , that is, more precisely if there is an open polycircle and a holomorphic function different from 0 for each point , so that . ${\displaystyle X}$ ${\displaystyle G\subset \mathbb {C} ^{n}}$ ${\displaystyle z\in G}$ ${\displaystyle \Delta (z;r)\subset G}$ ${\displaystyle g:\Delta (z;r)\rightarrow \mathbb {C} }$ ${\displaystyle X\cap \Delta (z;r)\subset \{\zeta \in \Delta (z;r)\mid g(\zeta )=0\}}$

If there is also a region, then a function is said to be locally bounded if there is an open polycircle for every point such that . ${\displaystyle G\subset \mathbb {C} ^{n}}$ ${\displaystyle X\subset G}$ ${\displaystyle f:G\setminus X\rightarrow \mathbb {C} }$ ${\displaystyle z\in G}$ ${\displaystyle \Delta (z;r)\subset G}$ ${\displaystyle \sup\{|f(\zeta )|\,\mid \zeta \in \Delta (z;r)\setminus X\}<\infty }$

Riemann's theorem of levability has the following generalization to several dimensions:

• Let it be a thin set of a domain and a holomorphic function that is locally bounded. Then there is a holomorphic function that matches on with . ${\displaystyle X}$ ${\displaystyle G\subset \mathbb {C} ^{n}}$ ${\displaystyle f:G\setminus X\rightarrow \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle {\tilde {f}}:G\rightarrow \mathbb {C} }$ ${\displaystyle G\setminus X}$ ${\displaystyle f}$

For the one-dimensional case , the above classic version of Riemann's theorem of liftability is returned, because in the one-dimensional case, thin sets are discrete because of the identity theorem . In other words, singularities in are always isolated. These situations are always trivial for several variables , because the following applies: ${\displaystyle n=1}$ ${\displaystyle G}$ ${\displaystyle n\geq 2}$

• Every isolated singularity of a holomorphic function with more than one variable can be lifted.

literature

• Klaus Jänich : Function Theory . 6th edition. Springer, 2008.
• Steven G. Krantz: Handbook of Complex Variables . Birkhäuser, 1999, ISBN 978-1-4612-1588-2 .

Web links

• Proof of Riemann's theorem of liftability

Individual evidence

1. ^ Gunning - Rossi : Analytic functions of several complex variables , Prentice-Hall 1965, chap. IC, theorem 3
2. ^ Gunning - Rossi : Analytic functions of several complex variables , Prentice-Hall 1965, chap. IC, Corollary 6th