# User:Maximilian Janisch/latexlist/Algebraic Groups/Singular point

A singular point of an analytic function $f ( z )$ is an obstacle to the analytic continuation of an element of the function $f ( z )$ of a complex variable $8$ along any curve in the $8$-plane.

Let $f ( z )$ be defined by a Weierstrass element $( U ( \zeta , R ) , f _ { \zeta } )$, consisting of a power series

\begin{equation} f _ { \zeta } = f _ { \zeta } ( z ) = \sum _ { k = 0 } ^ { \infty } c _ { k } ( z - \zeta ) ^ { k } \end{equation}

and its disc of convergence

\begin{equation} U ( \zeta , R ) = \{ z \in \overline { C } : | z - \zeta | < R \} \end{equation}

with centre $\zeta \neq \infty$ and radius of convergence $R > 0$. Consider all possible curves $L : [ 0,1 ] \rightarrow \overline { C }$, i.e. continuous mappings $L : z = \phi ( t )$ of the interval $0 \leq t \leq 1$ into the extended complex plane $\overline { C }$, which begin at the centre of this element $5$, $\zeta = \phi ( 0 )$. If the analytic continuation of the given element is possible along any such curve to any point $z \in \overline { C }$, then the complete analytic function $f ( z )$ thus obtained reduces to a constant: $f ( z ) =$. For non-trivial analytic functions $f ( z ) \neq$, the existence of obstacles to the analytic continuation along certain curves $L$ is characteristic.

Let $12$ be a point in the extended plane $\overline { C }$ on a curve $L _ { 1 } : z = \phi _ { 1 } ( t )$, $\alpha = \phi _ { 1 } ( \tau _ { 1 } )$, $0 < \tau _ { 1 } \leq 1$, $\phi _ { 1 } ( 0 ) = \zeta$, and on a curve $L _ { 2 } : z = \phi _ { 2 } ( t )$, $\alpha = \phi _ { 2 } ( \tau _ { 2 } )$, $0 < \tau _ { 2 } \leq 1$, $\phi _ { 2 } ( 0 ) = \zeta$, and let analytic continuation along $I$ and $12$ to all preceding points $z = \phi _ { 1 } ( t )$, $0 \leq t < \tau _ { 1 }$, and $z = \phi _ { 2 } ( t )$, $0 \leq t < \tau _ { 2 }$, be possible. Two such curves $I$ and $12$ are said to be equivalent with respect to the analytic continuation of the given element $( U ( \zeta , R ) , f _ { \zeta } )$ to the point $12$ if there is for any neighbourhood $V ( \alpha )$ of $12$ in $\overline { C }$ a number $\epsilon > 0$ such that the Weierstrass element obtained from $( U ( \zeta , R ) , f _ { \zeta } )$ by analytic continuation along $I$ to any point $z ^ { \prime } = \phi _ { 1 } ( \tau ^ { \prime } )$, $\tau _ { 1 } - \epsilon < \tau ^ { \prime } < \tau _ { 1 }$, can be continued along a certain curve located in $V ( \alpha )$ to an element obtained by continuation along $12$ from $( U ( \zeta , r ) , f _ { \zeta } )$ to any point $z = \phi _ { 2 } ( \tau ^ { \prime \prime } )$, $\tau _ { 2 } - \epsilon < \tau ^ { \prime \prime } < \tau _ { 2 }$.

If analytic continuation to a point $12$ is possible along a curve $L$, then it is also possible along all curves of the equivalence class $\{ L \}$ containing $L$. In this case, the pair $( \alpha , \{ L \} )$ is said to be regular, or proper; it defines a single-valued regular branch of the analytic function $f ( z )$ in a neighbourhood $V ( \alpha )$ of the point.

If analytic continuation along a curve $L : z = \phi ( t )$, $0 \leq t \leq 1$, $\phi ( 0 ) = \zeta$, which passes through $12$, $a = \phi ( \tau )$, $0 < \tau \leq 1$, is possible to all points $\phi ( t )$, $0 \leq t < \tau$, preceding $12$, but is not possible to the point $a = \phi ( \tau )$, then $12$ is a singular point for analytic continuation of the element $( U ( \zeta , R ) , f _ { \zeta } )$ along the curve $L$. In this instance it will also be singular for continuation along all curves of the equivalence class $\{ L \}$ which pass through $12$. The pair $( \alpha , \{ L \} )$, consisting of the point $x \in \overline { C }$ and the equivalence class $\{ L \}$ of curves $L$ which pass through $12$ for each of which $12$ is singular, is called a singular point of the analytic function $f ( z )$ defined by the element $( U ( \zeta , R ) , f _ { \zeta } )$. Two singular points $( \alpha , \{ L \} )$ and $( b , \{ M \} )$ are said to coincide if $a = b$ and if the classes $\{ L \}$ and $\{ M \}$ coincide. The point $12$ of the extended complex plane $\overline { C }$ is then called the projection, or $8$-coordinate, of the singular point $( \alpha , \{ L \} )$; the singular point $( \alpha , \{ L \} )$ is also said to lie above the point $a \in C$. In general, several (even a countable set of) different singular and regular pairs $( \alpha , \{ L \} )$ obtained through analytic continuation of one and the same element $( U ( \zeta , R ) , f _ { \zeta } )$ may lie above one and the same point $x \in \overline { C }$ (cf. Branch point).

If the radius of convergence of the initial series (1) $R < \infty$, then on the boundary circle $\Gamma = \{ z \in \overline { C } : | z - \zeta | = R \}$ of the disc of convergence $U ( \zeta , R )$ there lies at least one singular point $12$ of the element $( U ( \zeta , R ) , f _ { \zeta } )$, i.e. there is a singular point of the analytic function $f ( z )$ for continuation along the curves $z = \phi ( t )$, $0 \leq t \leq 1$, of the class $\{ L \}$ such that $z = \phi ( t ) \in U ( \zeta , R )$ when $0 \leq t < 1$, $\alpha = \phi ( 1 )$. In other words, a singular point of the element $( U ( \zeta , R ) , f _ { \zeta } )$ is a point $\alpha \in \Gamma$ such that direct analytic continuation of the element $( U ( \zeta , R ) , f _ { \zeta } )$ from the disc $U ( \zeta , R )$ to any neighbourhood $V ( \alpha )$ is impossible. In this situation, and generally in all cases where the lack of an obvious description of the class of curves $\{ L \}$ cannot give rise to ambiguity, one usually restricts to the $8$-coordinate $12$ of the singular point. The study of the position of singular points of an analytic function, in dependence on the properties of the sequence of coefficients $\{ c _ { k } \} _ { k = 0 } ^ { \infty }$ of the initial element $( U ( \zeta , R ) , f _ { \zeta } )$, is one of the main directions of research in function theory (see Hadamard theorem on multiplication; Star of a function element, as well as , , ). It is well-known, for example, that the singular points of the series

\begin{equation} f _ { 0 } ( z ) = \sum _ { k = 0 } ^ { \infty } b ^ { k } z ^ { d ^ { k } } \end{equation}

where $b \in \overline { C }$, $| b | < 1$, and $d > 2$ is a natural number, fill the whole boundary $\Gamma = \{ z \in \overline { C } : | z | = 1 \}$ of its disc of convergence $U ( 0,1 )$, although the sum of this series is continuous everywhere in the closed disc $\overline { U } ( 0,1 ) = \{ z \in \overline { C } : | z | \leq 1 \}$. Here, $I$ is the natural boundary of the analytic function $f _ { 0 } ( z )$; analytic continuation of $f _ { 0 } ( z )$ across the boundary of the disc $U ( 0,1 )$ is impossible.

Suppose that in a sufficiently small neighbourhood $V ( \alpha ) = \{ z \in \overline { C } : | z - \alpha | < R \}$ of a point $a \neq \infty$ (or $V ( \infty ) = \{ z \in \overline { C } : | z | > R \}$), analytic continuation along the curves of a specific class $\{ L \}$ is possible to all points other than $12$, for all elements obtained, i.e. along all curves situated in the deleted neighbourhood $V ^ { \prime } ( \alpha ) = \{ z \in \overline { C } : 0 < | z - \alpha | < R \}$ (respectively, $V ^ { \prime } ( \infty ) = \{ z \in C : | z - \alpha | > R \}$); the singular point $( \alpha , \{ L \} )$ is then called an isolated singular point. If analytic continuation of the elements obtained along the curves of the class $\{ L \}$ along all possible closed curves situated in $V ^ { \prime } ( \alpha )$ does not alter these elements, then the isolated singular point $( \alpha , \{ L \} )$ is called a single-valued singular point. This type of singular point can be a pole or an essential singular point: If an infinite limit $\operatorname { lim } f ( z ) = \infty$ exists when $8$ moves towards $12$ along the curves of the class $\{ L \}$, then the single-valued singular point $( \alpha , \{ L \} )$ is called a pole (of a function); if no finite or infinite limit $\operatorname { lim } f ( z )$ exists when $8$ moves towards $12$ along the curves of the class $\{ L \}$, then $( \alpha , \{ L \} )$ is called an essential singular point; the case of a finite limit corresponds to a regular point $( \alpha , \{ L \} )$. If analytic continuation of the elements obtained along the curves of the class $\{ L \}$ along closed curves surrounding $12$ in $V ^ { \prime } ( \alpha )$ alters these elements, then the isolated singular point $( \alpha , \{ L \} )$ is called a branch point or a many-valued singular point. The class of branch points is in turn subdivided into algebraic branch points and transcendental branch points (including logarithmic branch points, cf. Algebraic branch point; Logarithmic branch point; Transcendental branch point). If after a finite number $m > 2$ of single loops around $12$ in the same direction within $V ^ { \prime } ( \alpha )$, the elements obtained along the curves of the class $\{ L \}$ take their original form, then $( \alpha , \{ L \} )$ is an algebraic branch point and the number $m - 1$ is called its order. Conversely, when the loops around $12$ give more and more new elements, $( \alpha , \{ L \} )$ is a transcendental branch point.

For example, for the function

\begin{equation} f ( z ) = \frac { 1 } { ( 1 + z ^ { 1 / 2 } ) ( 1 + z ^ { 1 / 6 } ) } \end{equation}

the points $a = 0$, $\infty$ (for all curves) are algebraic branch points of order 5. As a point function, $f ( z )$ can be represented as a single-valued function only on the corresponding Riemann surface $5$, consisting of 6 sheets over $\overline { C }$ joined in a specific way above the points $0 , \infty$. Moreover, three proper branches of $f ( z )$ lie above the point $a = 1$, which are single-valued on the three corresponding sheets of $5$; on one sheet of $5$ there is a pole of the second order, and on two sheets of $5$ there are poles of the first order. In general, the introduction of the concept of a Riemann surface is particularly convenient and fruitful when studying the character of a singular point.

If the radius of convergence of the initial series (1) $R = \infty$, then it represents an entire function $f ( z )$, i.e. a function holomorphic in the entire finite plane $m$. When $f ( z ) \neq$, this function has a single isolated singular point $\alpha = \infty$ of single-valued character; if $\alpha = \infty$ is a pole, then $f ( z )$ is an entire rational function, or a polynomial; if $\alpha = \infty$ is an essential singular point, then $f ( z )$ is a transcendental entire function.

A meromorphic function $f ( z )$ in the finite plane $m$ is obtained when analytic continuation of the series (1) leads to a single-valued analytic function $f ( z )$ in $m$ all singular points of which are poles. If $\alpha = \infty$ is a pole or a regular point, then the total number of poles of $f ( z )$ in the extended plane $\overline { C }$ is finite and $f ( z )$ is a rational function. For a transcendental meromorphic function $f ( z )$ in $m$, the point at infinity $\alpha = \infty$ can be a limit point of the poles — this is the simplest example of a non-isolated singular point of a single-valued analytic function. A meromorphic function in an arbitrary domain $D \subset \overline { C }$ is defined in the same way.

Generally speaking, the projections of non-isolated singular points can form different sets of points in the extended complex plane $\overline { C }$. In particular, whatever the domain $D \subset \overline { C }$, an analytic function $f _ { D } ( z )$ exists in $\Omega$ for which $\Omega$ is its natural domain of existence, and the boundary $\Gamma = \partial D$ is its natural boundary; thus, analytic continuation of the function $f _ { D } ( z )$ across the boundary of $\Omega$ is impossible. Here, the natural boundary $I$ consists of accessible and inaccessible points (see Limit elements). If a point $\alpha \in \Gamma$ is accessible along the curves of a class $\{ L \}$ (there may be several of these classes), all situated in $\Omega$ except for the end point $12$, then only singular points of the function $f _ { D } ( z )$ can lie above $12$, since if this were not the case, analytic continuation of $f _ { D } ( z )$ across the boundary of $\Omega$ through a part of $I$ in a neighbourhood of $12$ would be possible; the accessible points form a dense set on $I$.

The role of the defining element of an analytic function $f ( z )$ of several complex variables $z = ( z 1 , \dots , z _ { r } )$, $n > 1$, can be played by, for example, a Weierstrass element $( U ^ { n } ( \zeta , R ) , f _ { \zeta } )$ in the form of a multiple power series

\begin{equation} f _ { \zeta } = f _ { \zeta } ( z ) = \end{equation}

\begin{equation} \sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } } \end{equation}

and the polydisc of convergence of this series

\begin{equation} U ^ { n } ( \zeta , r ) = \{ z \in C ^ { n } : | z _ { v } - \zeta _ { v } | < R _ { v } , v = 1 , \ldots , n \} \end{equation}

with centre $\zeta = ( \zeta _ { 1 } , \ldots , \zeta _ { n } ) \in C ^ { n }$ and radius of convergence $R = \{ R _ { 1 } > 0 , \ldots , R _ { n } > 0 \}$. By taking in the process of analytic continuation of the element (2) along all possible curves $L : [ 0,1 ] \rightarrow C ^ { n }$, mappings of the interval $0 \leq t \leq 1$ into the complex space $C ^ { \prime \prime }$ as basis, a general definition of the singular points $( \alpha , \{ L \} )$, $\alpha \in C ^ { \prime \prime }$, of the function $f ( z )$ is obtained, which is formally completely analogous to the one mentioned above for the case $n = 1$.

However, as a result of the overdeterminacy of the Cauchy-Riemann equations when $n > 1$ and the resulting "large power" of analytic continuation, the case $n > 1$ differs radically from the case $n = 1$. In particular, for $n > 1$ there are domains $D \subset C ^ { x }$ which cannot be natural domains of existence of any single-valued analytic or holomorphic function. In other words, on specific sections of the boundary $0 \Omega$ of this domain there are no singular points of any holomorphic function $f ( z )$ defined in $\Omega$, and analytic continuation is possible across them. For example, the Osgood–Brown theorem holds: If a compact set $K$ is situated in a bounded domain $D \subset C ^ { x }$ such that $D \backslash K$ is also a domain, and if a function $f ( z )$ is holomorphic in $D \backslash K$, then it can be holomorphically continued onto the whole domain $\Omega$ (see also Removable set). The natural domains of existence of holomorphic functions are sometimes called domains of holomorphy (cf. Domain of holomorphy), and are characterized by specific geometric properties. Analytic continuation of a holomorphic function $f ( z )$ which is originally defined in a domain $D \subset C ^ { x }$ while retaining its single-valuedness makes it necessary to introduce, generally speaking, many-sheeted domains of holomorphy over $C ^ { \prime \prime }$, or Riemann domains — analogues of Riemann surfaces (cf. Riemannian domain). In this interpretation, the singular points of a holomorphic function $f ( z )$ prove to be points of the boundary $\Gamma = \partial \hat { D }$ of its domain of holomorphy $D$. The Osgood–Brown theorem shows that the connected components of $I$ cannot form compact sets $K$ such that the function $f ( z )$ is holomorphic in $\hat { D } \backslash K$. In particular, for $n > 1$ there do not exist isolated singular points of holomorphic functions.

The simplest types of singular points of analytic functions of several complex variables are provided by meromorphic functions $f ( z )$ in a domain $D \subset C ^ { x }$, $n \geq 1$, which are characterized by the following properties: 1) $f ( z )$ is holomorphic everywhere in $\Omega$ with the exception of a polar set $P$, which consists of singular points; and 2) for any point $\alpha \in P$ there are a neighbourhood $V ( \alpha )$ and a holomorphic function $\psi _ { \alpha } ( z )$ in $V ( \alpha )$ such that the function $\phi _ { a } ( z ) = \psi _ { a x } ( z ) f ( z )$ can be continued holomorphically to $V ( \alpha )$. The singular points $\alpha \in P$ are then divided into poles, at which $\phi _ { \alpha } ( \alpha ) \neq 0$, and points of indeterminacy, at which $\phi _ { \alpha } ( \alpha ) = 0$. In the case of a pole, $\operatorname { lim } f ( z ) = \infty$ when $8$ moves towards $12$, $z \in D \backslash P$; in any neighbourhood of a point of indeterminacy, $f ( z )$ takes all values $w \in C$. For example, the meromorphic function $f ( z ) = z _ { 1 } / z _ { 2 }$ in $c ^ { 2 }$ has the straight line $P = \{ z = ( z _ { 1 } , z _ { 2 } ) \in C ^ { 2 } : z _ { 2 } = 0 \}$ as its polar set; all points of this straight line are poles, with the exception of the point of indeterminacy $( 0,0 )$. A meromorphic function $f ( z )$ in its domain of holomorphy $D$ can be represented globally in $D$ as the quotient of two holomorphic functions, i.e. its polar set $P$ is an analytic set.

A point $\alpha \in C ^ { \prime \prime }$ is called a point of meromorphy of a function $f ( z )$ if $f ( z )$ is meromorphic in a certain neighbourhood of that point; thus, if a singular point is a point of meromorphy, then it is either a pole or a point of indeterminacy. All singular points of the analytic function $f ( z )$ which are not points of meromorphy are sometimes called essential singular points. These include, for example, the branch points of $f ( z )$, i.e. the branching points of its (many-sheeted) domain of holomorphy $D$. The dimension of the set of all singular points of a holomorphic function $f ( z )$ is, in general, equal to $2 n - 1$. Given certain extra restrictions on $f ( z )$ this set proves to be analytic (and, consequently, is of smaller dimension; see ).