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The set of all elements of an analytic function obtained by all analytic continuations (cf. [[Analytic continuation|Analytic continuation]]) of an initial analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237201.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237202.png" /> given initially on a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237203.png" /> of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237204.png" />.
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A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237205.png" /> consisting of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237206.png" /> and a single-valued analytic, or holomorphic, function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237207.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237208.png" /> is called an element of an analytic function or an analytic element, or simply just an element. It is always possible when specifying an analytic function to use a Weierstrass or regular element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c0237209.png" />, which consists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372010.png" /> of a power series
+
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 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
The set of all elements of an analytic function obtained by all analytic continuations (cf. [[Analytic continuation|Analytic continuation]]) of an initial analytic function  $  f = f( z) $
 +
of the complex variable  $  z $
 +
given initially on a certain domain  $  D $
 +
of the extended complex plane  $  \overline{\mathbf C}\; $.
  
and a disc of convergence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372012.png" />, with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372013.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372014.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372015.png" />, a Weierstrass element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372016.png" /> consists of a series
+
A pair  $  ( D, f) $
 +
consisting of a domain  $  D \subset  \overline{\mathbf C}\; $
 +
and a single-valued analytic, or holomorphic, function  $  f $
 +
defined on  $  D $
 +
is called an element of an analytic function or an analytic element, or simply just an element. It is always possible when specifying an analytic function to use a Weierstrass or regular element  $  ( U( a, R), f _ {a} ) $,
 +
which consists for  $  a \neq \infty $
 +
of a power series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
f _ {a}  = f _ {a} ( z)  = \sum_{k=0} ^  \infty  c _ {k} ( z - a) ^ {k}
 +
$$
  
and a domain of convergence of this series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372019.png" />.
+
and a disc of convergence,  $  U( a, R) = \{ {z \in \overline{\mathbf C}\; } : {| z- a | < R } \} $,
 +
with centre  $  a $
 +
and radius  $  R > 0 $.
 +
In the case  $  a = \infty $,
 +
a Weierstrass element  $  ( U( \infty , R), f _  \infty  ) $
 +
consists of a series
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372020.png" /> be the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372021.png" /> to which an initial element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372022.png" /> can be analytically continued over at least one path connecting the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372025.png" />. One must bear in mind the possibility of the situation where for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372026.png" /> analytic continuation is possible along a certain class of paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372027.png" /> but is impossible along any other class of paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372028.png" /> (see [[Singular point|Singular point]]). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372029.png" /> is a domain in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372030.png" />. The complete analytic function (in the sense of Weierstrass) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372031.png" /> generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372032.png" /> is the name given to the set of all Weierstrass elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372034.png" />, obtained by this kind of analytic continuation along all possible paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372035.png" />. The domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372036.png" /> is called the (Weierstrass) domain of existence for the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372037.png" />. The use of an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372038.png" /> instead of a Weierstrass element leads to the same complete analytic function. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372040.png" /> are often called the branches of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372041.png" /> (cf. [[Branch of an analytic function|Branch of an analytic function]]). Any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372042.png" /> of the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372043.png" /> taken as the initial one under analytic continuation leads to the same complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372044.png" />. Each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372045.png" /> of the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372046.png" /> can be obtained from any other element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372047.png" /> by analytic continuation along some path connecting the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372050.png" />.
+
$$ \tag{2 }
 +
f _  \infty  = f _  \infty  ( z) = \sum _{k=0} ^  \infty  c _ {k} z  ^ {-k}
 +
$$
  
It may happen that the initial element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372051.png" /> cannot be analytically continued to any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372052.png" />. In that case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372053.png" /> is the natural domain of existence or domain of holomorphy of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372054.png" />, while the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372055.png" /> is the natural boundary of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372056.png" />. For example, for the Weierstrass element
+
and a domain of convergence of this series  $  U( \infty , R) = \{ {z \in \overline{\mathbf C}\; } : {| z | > R } \} $,  
 +
$  R \geq  0 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372057.png" /></td> </tr></table>
+
Let  $  E _ {f} $
 +
be the set of all points  $  \zeta \in \overline{\mathbf C}\; $
 +
to which an initial element  $  ( U( a, R), f _ {a} ) $
 +
can be analytically continued over at least one path connecting the points  $  a $
 +
and  $  \zeta $
 +
in  $  \overline{\mathbf C}\; $.
 +
One must bear in mind the possibility of the situation where for a point  $  \zeta \in E _ {f} $
 +
analytic continuation is possible along a certain class of paths  $  L _ {1} $
 +
but is impossible along any other class of paths  $  L _ {2} $(
 +
see [[Singular point|Singular point]]). The set  $  E _ {f} $
 +
is a domain in the plane  $  \overline{\mathbf C}\; $.
 +
The complete analytic function (in the sense of Weierstrass)  $  f _ {W} $
 +
generated by the element  $  ( U( a, R), f _ {a} ) $
 +
is the name given to the set of all Weierstrass elements  $  ( U( \zeta , R), f _  \zeta  ) $,
 +
$  \zeta \in E _ {f} $,
 +
obtained by this kind of analytic continuation along all possible paths  $  L \subset  \overline{\mathbf C}\; $.
 +
The domain  $  E _ {f} $
 +
is called the (Weierstrass) domain of existence for the complete analytic function  $  f _ {W} $.
 +
The use of an arbitrary element  $  ( D, f) $
 +
instead of a Weierstrass element leads to the same complete analytic function. The elements  $  ( D, f) $
 +
of  $  f _ {W} $
 +
are often called the branches of the analytic function  $  f _ {W} $(
 +
cf. [[Branch of an analytic function|Branch of an analytic function]]). Any element  $  ( D, f) $
 +
of the complete analytic function  $  f _ {W} $
 +
taken as the initial one under analytic continuation leads to the same complete analytic function  $  f _ {W} $.  
 +
Each element  $  ( U( \zeta , R), f _  \zeta  ) $
 +
of the complete analytic function  $  f _ {W} $
 +
can be obtained from any other element  $  ( U( a, R), f _ {a} ) $
 +
by analytic continuation along some path connecting the points  $  a $
 +
and  $  \zeta $
 +
in  $  \overline{\mathbf C}\; $.
  
the natural boundary is the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372058.png" />, the boundary of the disc of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372059.png" />, since this element cannot be analytically continued to any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372060.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372061.png" />. No matter what the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372062.png" />, one can construct an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372063.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372064.png" /> is the natural domain of existence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372065.png" /> and where the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372066.png" /> is the natural boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372067.png" /> (this follows, for example, from the [[Mittag-Leffler theorem|Mittag-Leffler theorem]]).
+
It may happen that the initial element  $  ( D, f) $
 +
cannot be analytically continued to any point $  \zeta \notin D $.  
 +
In that case, $  D = E _ {f} $
 +
is the natural domain of existence or domain of holomorphy of the function  $  f $,
 +
while the boundary $  \Gamma = \partial  D $
 +
is the natural boundary of the function  $  f $.  
 +
For example, for the Weierstrass element
  
The complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372068.png" /> in its domain of existence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372069.png" /> is, in general, not a function of points in the usual sense of the word. A situation frequently encountered in the theory of analytic functions is that the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372070.png" /> is a multi-valued function: For each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372071.png" /> there exists, in general, an infinite set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372072.png" /> with centre at this point. However, this set is at most countable (the theorem of Poincaré–Volterra). On the whole, the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372073.png" /> can be regarded as a single-valued analytic function only on the corresponding Riemann surface, which is a multi-sheeted covering surface over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372074.png" />. For example, the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372075.png" /> is multi-valued in its domain of existence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372076.png" />; at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372077.png" /> it takes the countable set of values
+
$$
 +
\left ( U( 0, 1), f _ {0} ( z) = \sum _{k=0} ^  \infty  z  ^ {k!} \right )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372078.png" /></td> </tr></table>
+
the natural boundary is the circle  $  \Gamma = \{ {z \in \overline{\mathbf C}\; } : {| z | = 1 } \} $,
 +
the boundary of the disc of convergence  $  U( 0, 1) $,
 +
since this element cannot be analytically continued to any point  $  \zeta $
 +
such that  $  | \zeta | \geq  1 $.
 +
No matter what the domain  $  D \subset  \overline{\mathbf C}\; $,
 +
one can construct an analytic function  $  f _ {D} ( z) $
 +
for which  $  D $
 +
is the natural domain of existence for  $  f _ {D} ( z) $
 +
and where the boundary  $  \Gamma = \partial  D $
 +
is the natural boundary of  $  f _ {D} ( z) $(
 +
this follows, for example, from the [[Mittag-Leffler theorem|Mittag-Leffler theorem]]).
  
and each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372079.png" /> corresponds to a countable set of elements
+
The complete analytic function  $  f _ {W} $
 +
in its domain of existence  $  E _ {f} $
 +
is, in general, not a function of points in the usual sense of the word. A situation frequently encountered in the theory of analytic functions is that the complete analytic function  $  f _ {W} $
 +
is a multi-valued function: For each point $  \zeta \in E _ {f} $
 +
there exists, in general, an infinite set of elements  $  ( U( \zeta , R), f _  \zeta  ) $
 +
with centre at this point. However, this set is at most countable (the theorem of Poincaré–Volterra). On the whole, the complete analytic function  $  f _ {W} $
 +
can be regarded as a single-valued analytic function only on the corresponding Riemann surface, which is a multi-sheeted covering surface over  $  \overline{\mathbf C}\; $.
 +
For example, the complete analytic function  $  f( z) = \mathop{\rm ln}  z = \mathop{\rm ln}  | z | + i  \mathop{\rm arg}  z $
 +
is multi-valued in its domain of existence  $  E _ {f} = \{ {z \in \overline{\mathbf C}\; } : {0 < | z | < \infty } \} $;
 +
at each point  $  \zeta \in E _ {f} $
 +
it takes the countable set of values
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372080.png" /></td> </tr></table>
+
$$
 +
f _  \zeta  ( \zeta ; s)  =   \mathop{\rm ln}  | \zeta | + i  \mathop{\rm Arg}  \zeta + 2 \pi si,\ \
 +
s= 0, \pm  1 \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372081.png" /></td> </tr></table>
+
and each point  $  \zeta \in E _ {f} $
 +
corresponds to a countable set of elements
  
with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372082.png" />. Usually, one employs a single-valued branch of this complete analytic function, namely the principal value of the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372083.png" />. This is a holomorphic function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372084.png" />, and can be "continuously extended to -∞, 0" , i.e. for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372085.png" /> the limit
+
$$
 +
( U( \zeta , | \zeta | ), f _  \zeta  ( z; s)) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372086.png" /></td> </tr></table>
+
$$
 +
= f _  \zeta  ( \zeta ; s) + \sum _{k=1} ^  \infty 
 +
\frac{(- 1)  ^ {k-1} }{k \zeta  ^ {k} }
 +
( z - \zeta )  ^ {k}
 +
$$
  
exists. (Likewise, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372087.png" /> exists; these limit values do not coincide (their difference is constant, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372088.png" />).)
+
with centre  $  \zeta $.  
 +
Usually, one employs a single-valued branch of this complete analytic function, namely the principal value of the logarithm  $  \mathop{\rm Ln}  z = \mathop{\rm ln}  | z | + i  \mathop{\rm Arg}  z $.
 +
This is a holomorphic function in the domain  $  D = \{ {z \in \overline{\mathbf C}\; } : {0 < | z | < \infty,  - \pi <  \mathop{\rm arg}  z < \pi } \} $,
 +
and can be "continuously extended to -∞, 0" , i.e. for  $  \zeta \in ( - \infty , 0 ) $
 +
the limit
 +
 
 +
$$
 +
\lim\limits _ {\begin{array}{c}
 +
z \rightarrow \zeta \\
 +
  \mathop{\rm Im} z > 0
 +
\end{array}
 +
} \
 +
\mathop{\rm Ln}  z  =  \mathop{\rm Ln} _ {+}  z
 +
$$
 +
 
 +
exists. (Likewise,  $  \lim\limits _ {z \rightarrow \zeta ,  \mathop{\rm Im}  z < 0 }  \mathop{\rm Ln}  z =  \mathop{\rm Ln} _ {-z} $
 +
exists; these limit values do not coincide (their difference is constant, equal to $  2 \pi i $).)
  
 
Inversion of the Weierstrass elements (1) and (2) (see [[Inversion of a series|Inversion of a series]]) gives rise to elements of more general nature, correspondingly defined by Puisieux series:
 
Inversion of the Weierstrass elements (1) and (2) (see [[Inversion of a series|Inversion of a series]]) gives rise to elements of more general nature, correspondingly defined by Puisieux series:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
f _ {a}  = \sum _ {k= \mu } ^  \infty  c _ {k} ( z- a) ^ {k/ \nu } ,\ \
 +
f _  \infty  = \sum _ {k= \mu } ^  \infty  c _ {k} z ^ {- k/ \nu } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372090.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372091.png" /> is a natural number, and the discs of convergence of these series are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372093.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372095.png" />, the series (3) coincide with (1) and (2), which define regular elements; a difference from these is that the elements defined by the series (3) are called singular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372096.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372097.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c02372099.png" />, the series of (3) define correspondingly unbranched and (algebraic) branched elements.
+
where $  \mu $
 +
is an integer and $  \nu $
 +
is a natural number, and the discs of convergence of these series are $  U( a, R) $
 +
and $  U( \infty , R) $.  
 +
In particular, for $  \mu \geq  0 $
 +
and $  \nu = 1 $,  
 +
the series (3) coincide with (1) and (2), which define regular elements; a difference from these is that the elements defined by the series (3) are called singular for $  \mu < 0 $
 +
or $  \nu > 1 $.  
 +
For $  \nu = 1 $
 +
and $  \nu > 1 $,  
 +
the series of (3) define correspondingly unbranched and (algebraic) branched elements.
  
If under continuation of the initial element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720100.png" /> one allows for the occurrence of special elements, with series of the form (3), which in general are multi-valued (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720101.png" />) and have singularities of pole-type (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720102.png" />), one gets the Riemann domain of existence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720103.png" /> (which is larger than the Weierstrass domain of existence), and the correspondingly larger set of elements which are defined by series of the form (3) is called the [[Analytic image|analytic image]]. An analytic image differs from a complete analytic function by the addition of all singular elements obtained under extension of a given regular element. When a corresponding topology has been introduced, the analytic image becomes the Riemann surface for the given function.
+
If under continuation of the initial element $  ( U( a, R), f _ {a} ) $
 +
one allows for the occurrence of special elements, with series of the form (3), which in general are multi-valued (for $  \nu > 1 $)  
 +
and have singularities of pole-type (for $  \mu < 0 $),  
 +
one gets the Riemann domain of existence $  E _ {R} $(
 +
which is larger than the Weierstrass domain of existence), and the correspondingly larger set of elements which are defined by series of the form (3) is called the [[Analytic image|analytic image]]. An analytic image differs from a complete analytic function by the addition of all singular elements obtained under extension of a given regular element. When a corresponding topology has been introduced, the analytic image becomes the Riemann surface for the given function.
  
For the construction of the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720104.png" /> one can use the concept of a germ of an analytic function instead of the concept of an element. It involves localizing the concept of an element, and discarding the radius of convergence, which in this case is not important. Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720106.png" /> such that the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720108.png" /> contain a common point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720109.png" /> are called equivalent at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720110.png" /> if there exists a neighbourhood around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720111.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720112.png" />. This equivalence relation has the usual properties of reflexivity, symmetry and transitivity. An equivalence class of elements at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720113.png" /> is called a germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720114.png" /> of the analytic function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720115.png" />. The germ characterizes the local properties of the function at the given point. Two germs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720117.png" /> are equal if any two representatives of the equivalence classes coincide in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720118.png" />. Similarly, one can define arithmetic operations and differentiation on germs by means of representatives. The complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720119.png" /> is the set of all germs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720121.png" />, of the analytic function obtained from a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720122.png" /> by analytic continuation along all paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720123.png" />. Equality of two complete analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720125.png" /> and operations on complete analytic functions are defined as equality of the germs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720127.png" /> at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720128.png" /> and operations on the germs.
+
For the construction of the complete analytic function $  f _ {W} $
 +
one can use the concept of a germ of an analytic function instead of the concept of an element. It involves localizing the concept of an element, and discarding the radius of convergence, which in this case is not important. Two elements $  ( D, f) $
 +
and $  ( G, h) $
 +
such that the domains $  D $
 +
and $  G $
 +
contain a common point $  a $
 +
are called equivalent at the point $  a $
 +
if there exists a neighbourhood around $  a $
 +
at which $  f \equiv h $.  
 +
This equivalence relation has the usual properties of reflexivity, symmetry and transitivity. An equivalence class of elements at a given point $  a \in \overline{\mathbf C}\; $
 +
is called a germ $  \mathbf f _ {a} $
 +
of the analytic function at the point $  a $.  
 +
The germ characterizes the local properties of the function at the given point. Two germs $  \mathbf f _ {a} $
 +
and $  \mathbf g _ {a} $
 +
are equal if any two representatives of the equivalence classes coincide in some neighbourhood of $  a $.  
 +
Similarly, one can define arithmetic operations and differentiation on germs by means of representatives. The complete analytic function $  f _ {W} $
 +
is the set of all germs $  \mathbf f _  \zeta  $,  
 +
$  \zeta \in E _ {f} $,  
 +
of the analytic function obtained from a given $  \mathbf f _ {a} $
 +
by analytic continuation along all paths in $  \overline{\mathbf C}\; $.  
 +
Equality of two complete analytic functions $  f _ {W} $
 +
and $  g _ {W} $
 +
and operations on complete analytic functions are defined as equality of the germs $  \mathbf f _ {a} $
 +
and $  \mathbf g _ {a} $
 +
at some point $  a \in E _ {f} \cap E _ {g} $
 +
and operations on the germs.
  
The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720129.png" />, the Weierstrass elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720130.png" /> and the germs of analytic functions of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720132.png" />, are defined exactly as above, but by means of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720133.png" /> in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720134.png" /> or by polydiscs of convergence,
+
The elements $  ( D, f) $,  
 +
the Weierstrass elements $  ( U  ^ {n} ( a, R), f _ {a} ) $
 +
and the germs of analytic functions of several complex variables $  z = ( z _ {1} \dots z _ {n} ) $,  
 +
$  n \geq  1 $,  
 +
are defined exactly as above, but by means of domains $  D $
 +
in the complex space $  \mathbf C  ^ {n} $
 +
or by polydiscs of convergence,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720135.png" /></td> </tr></table>
+
$$
 +
U  ^ {n} ( a, R)  = \{ {z \in \mathbf C  ^ {n} } : {| z _ {j} - a _ {j} | < R _ {j} ,\
 +
j= 1 \dots n } \}
 +
;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720136.png" /></td> </tr></table>
+
$$
 +
R _ {j}  >  0, j = 1 \dots n; \  a  = ( a _ {1} \dots a _ {n} );
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720137.png" /></td> </tr></table>
+
$$
 +
= ( R _ {1} \dots R _ {n} ),
 +
$$
  
 
of multiple power series
 
of multiple power series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720138.png" /></td> </tr></table>
+
$$
 +
f _ {a}  = f _ {a} ( z) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023720/c023720139.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {k _ {1} \dots k _ {n} = 0 } ^  \infty  c _ {k _ {1}  \dots k _ {n} } ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} } .
 +
$$
  
 
The concept of a complete analytic function of several complex variables is constructed in complete analogy with the case of one variable.
 
The concept of a complete analytic function of several complex variables is constructed in complete analogy with the case of one variable.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich,   "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G. Springer,   "Introduction to Riemann surfaces" , Chelsea, reprint (1981)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B.A. Fuks,   "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) {{MR|0122987}} {{MR|1530201}} {{MR|0092855}} {{ZBL|0501.30039}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian) {{MR|0155003}} {{ZBL|}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
Line 72: Line 242:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Weyl,   "Die Idee der Riemannschen Fläche" , Teubner (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Weyl,   "The concept of a Riemann surface" , Addison-Wesley (1955) (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.M. Farkas,   I. Kra,   "Riemann surfaces" , Springer (1980) pp. Sect. III.6</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Foster,   "Riemannsche Flächen" , Springer (1977)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Weyl, "Die Idee der Riemannschen Fläche" , Teubner (1955) {{MR|0069903}} {{ZBL|0068.06001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Weyl, "The concept of a Riemann surface" , Addison-Wesley (1955) (Translated from German) {{MR|1440406}} {{MR|0166351}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 {{MR|0583745}} {{ZBL|0475.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Foster, "Riemannsche Flächen" , Springer (1977)</TD></TR></table>

Latest revision as of 20:02, 15 January 2024


The set of all elements of an analytic function obtained by all analytic continuations (cf. Analytic continuation) of an initial analytic function $ f = f( z) $ of the complex variable $ z $ given initially on a certain domain $ D $ of the extended complex plane $ \overline{\mathbf C}\; $.

A pair $ ( D, f) $ consisting of a domain $ D \subset \overline{\mathbf C}\; $ and a single-valued analytic, or holomorphic, function $ f $ defined on $ D $ is called an element of an analytic function or an analytic element, or simply just an element. It is always possible when specifying an analytic function to use a Weierstrass or regular element $ ( U( a, R), f _ {a} ) $, which consists for $ a \neq \infty $ of a power series

$$ \tag{1 } f _ {a} = f _ {a} ( z) = \sum_{k=0} ^ \infty c _ {k} ( z - a) ^ {k} $$

and a disc of convergence, $ U( a, R) = \{ {z \in \overline{\mathbf C}\; } : {| z- a | < R } \} $, with centre $ a $ and radius $ R > 0 $. In the case $ a = \infty $, a Weierstrass element $ ( U( \infty , R), f _ \infty ) $ consists of a series

$$ \tag{2 } f _ \infty = f _ \infty ( z) = \sum _{k=0} ^ \infty c _ {k} z ^ {-k} $$

and a domain of convergence of this series $ U( \infty , R) = \{ {z \in \overline{\mathbf C}\; } : {| z | > R } \} $, $ R \geq 0 $.

Let $ E _ {f} $ be the set of all points $ \zeta \in \overline{\mathbf C}\; $ to which an initial element $ ( U( a, R), f _ {a} ) $ can be analytically continued over at least one path connecting the points $ a $ and $ \zeta $ in $ \overline{\mathbf C}\; $. One must bear in mind the possibility of the situation where for a point $ \zeta \in E _ {f} $ analytic continuation is possible along a certain class of paths $ L _ {1} $ but is impossible along any other class of paths $ L _ {2} $( see Singular point). The set $ E _ {f} $ is a domain in the plane $ \overline{\mathbf C}\; $. The complete analytic function (in the sense of Weierstrass) $ f _ {W} $ generated by the element $ ( U( a, R), f _ {a} ) $ is the name given to the set of all Weierstrass elements $ ( U( \zeta , R), f _ \zeta ) $, $ \zeta \in E _ {f} $, obtained by this kind of analytic continuation along all possible paths $ L \subset \overline{\mathbf C}\; $. The domain $ E _ {f} $ is called the (Weierstrass) domain of existence for the complete analytic function $ f _ {W} $. The use of an arbitrary element $ ( D, f) $ instead of a Weierstrass element leads to the same complete analytic function. The elements $ ( D, f) $ of $ f _ {W} $ are often called the branches of the analytic function $ f _ {W} $( cf. Branch of an analytic function). Any element $ ( D, f) $ of the complete analytic function $ f _ {W} $ taken as the initial one under analytic continuation leads to the same complete analytic function $ f _ {W} $. Each element $ ( U( \zeta , R), f _ \zeta ) $ of the complete analytic function $ f _ {W} $ can be obtained from any other element $ ( U( a, R), f _ {a} ) $ by analytic continuation along some path connecting the points $ a $ and $ \zeta $ in $ \overline{\mathbf C}\; $.

It may happen that the initial element $ ( D, f) $ cannot be analytically continued to any point $ \zeta \notin D $. In that case, $ D = E _ {f} $ is the natural domain of existence or domain of holomorphy of the function $ f $, while the boundary $ \Gamma = \partial D $ is the natural boundary of the function $ f $. For example, for the Weierstrass element

$$ \left ( U( 0, 1), f _ {0} ( z) = \sum _{k=0} ^ \infty z ^ {k!} \right ) $$

the natural boundary is the circle $ \Gamma = \{ {z \in \overline{\mathbf C}\; } : {| z | = 1 } \} $, the boundary of the disc of convergence $ U( 0, 1) $, since this element cannot be analytically continued to any point $ \zeta $ such that $ | \zeta | \geq 1 $. No matter what the domain $ D \subset \overline{\mathbf C}\; $, one can construct an analytic function $ f _ {D} ( z) $ for which $ D $ is the natural domain of existence for $ f _ {D} ( z) $ and where the boundary $ \Gamma = \partial D $ is the natural boundary of $ f _ {D} ( z) $( this follows, for example, from the Mittag-Leffler theorem).

The complete analytic function $ f _ {W} $ in its domain of existence $ E _ {f} $ is, in general, not a function of points in the usual sense of the word. A situation frequently encountered in the theory of analytic functions is that the complete analytic function $ f _ {W} $ is a multi-valued function: For each point $ \zeta \in E _ {f} $ there exists, in general, an infinite set of elements $ ( U( \zeta , R), f _ \zeta ) $ with centre at this point. However, this set is at most countable (the theorem of Poincaré–Volterra). On the whole, the complete analytic function $ f _ {W} $ can be regarded as a single-valued analytic function only on the corresponding Riemann surface, which is a multi-sheeted covering surface over $ \overline{\mathbf C}\; $. For example, the complete analytic function $ f( z) = \mathop{\rm ln} z = \mathop{\rm ln} | z | + i \mathop{\rm arg} z $ is multi-valued in its domain of existence $ E _ {f} = \{ {z \in \overline{\mathbf C}\; } : {0 < | z | < \infty } \} $; at each point $ \zeta \in E _ {f} $ it takes the countable set of values

$$ f _ \zeta ( \zeta ; s) = \mathop{\rm ln} | \zeta | + i \mathop{\rm Arg} \zeta + 2 \pi si,\ \ s= 0, \pm 1 \dots $$

and each point $ \zeta \in E _ {f} $ corresponds to a countable set of elements

$$ ( U( \zeta , | \zeta | ), f _ \zeta ( z; s)) = $$

$$ = f _ \zeta ( \zeta ; s) + \sum _{k=1} ^ \infty \frac{(- 1) ^ {k-1} }{k \zeta ^ {k} } ( z - \zeta ) ^ {k} $$

with centre $ \zeta $. Usually, one employs a single-valued branch of this complete analytic function, namely the principal value of the logarithm $ \mathop{\rm Ln} z = \mathop{\rm ln} | z | + i \mathop{\rm Arg} z $. This is a holomorphic function in the domain $ D = \{ {z \in \overline{\mathbf C}\; } : {0 < | z | < \infty, - \pi < \mathop{\rm arg} z < \pi } \} $, and can be "continuously extended to -∞, 0" , i.e. for $ \zeta \in ( - \infty , 0 ) $ the limit

$$ \lim\limits _ {\begin{array}{c} z \rightarrow \zeta \\ \mathop{\rm Im} z > 0 \end{array} } \ \mathop{\rm Ln} z = \mathop{\rm Ln} _ {+} z $$

exists. (Likewise, $ \lim\limits _ {z \rightarrow \zeta , \mathop{\rm Im} z < 0 } \mathop{\rm Ln} z = \mathop{\rm Ln} _ {-z} $ exists; these limit values do not coincide (their difference is constant, equal to $ 2 \pi i $).)

Inversion of the Weierstrass elements (1) and (2) (see Inversion of a series) gives rise to elements of more general nature, correspondingly defined by Puisieux series:

$$ \tag{3 } f _ {a} = \sum _ {k= \mu } ^ \infty c _ {k} ( z- a) ^ {k/ \nu } ,\ \ f _ \infty = \sum _ {k= \mu } ^ \infty c _ {k} z ^ {- k/ \nu } , $$

where $ \mu $ is an integer and $ \nu $ is a natural number, and the discs of convergence of these series are $ U( a, R) $ and $ U( \infty , R) $. In particular, for $ \mu \geq 0 $ and $ \nu = 1 $, the series (3) coincide with (1) and (2), which define regular elements; a difference from these is that the elements defined by the series (3) are called singular for $ \mu < 0 $ or $ \nu > 1 $. For $ \nu = 1 $ and $ \nu > 1 $, the series of (3) define correspondingly unbranched and (algebraic) branched elements.

If under continuation of the initial element $ ( U( a, R), f _ {a} ) $ one allows for the occurrence of special elements, with series of the form (3), which in general are multi-valued (for $ \nu > 1 $) and have singularities of pole-type (for $ \mu < 0 $), one gets the Riemann domain of existence $ E _ {R} $( which is larger than the Weierstrass domain of existence), and the correspondingly larger set of elements which are defined by series of the form (3) is called the analytic image. An analytic image differs from a complete analytic function by the addition of all singular elements obtained under extension of a given regular element. When a corresponding topology has been introduced, the analytic image becomes the Riemann surface for the given function.

For the construction of the complete analytic function $ f _ {W} $ one can use the concept of a germ of an analytic function instead of the concept of an element. It involves localizing the concept of an element, and discarding the radius of convergence, which in this case is not important. Two elements $ ( D, f) $ and $ ( G, h) $ such that the domains $ D $ and $ G $ contain a common point $ a $ are called equivalent at the point $ a $ if there exists a neighbourhood around $ a $ at which $ f \equiv h $. This equivalence relation has the usual properties of reflexivity, symmetry and transitivity. An equivalence class of elements at a given point $ a \in \overline{\mathbf C}\; $ is called a germ $ \mathbf f _ {a} $ of the analytic function at the point $ a $. The germ characterizes the local properties of the function at the given point. Two germs $ \mathbf f _ {a} $ and $ \mathbf g _ {a} $ are equal if any two representatives of the equivalence classes coincide in some neighbourhood of $ a $. Similarly, one can define arithmetic operations and differentiation on germs by means of representatives. The complete analytic function $ f _ {W} $ is the set of all germs $ \mathbf f _ \zeta $, $ \zeta \in E _ {f} $, of the analytic function obtained from a given $ \mathbf f _ {a} $ by analytic continuation along all paths in $ \overline{\mathbf C}\; $. Equality of two complete analytic functions $ f _ {W} $ and $ g _ {W} $ and operations on complete analytic functions are defined as equality of the germs $ \mathbf f _ {a} $ and $ \mathbf g _ {a} $ at some point $ a \in E _ {f} \cap E _ {g} $ and operations on the germs.

The elements $ ( D, f) $, the Weierstrass elements $ ( U ^ {n} ( a, R), f _ {a} ) $ and the germs of analytic functions of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n \geq 1 $, are defined exactly as above, but by means of domains $ D $ in the complex space $ \mathbf C ^ {n} $ or by polydiscs of convergence,

$$ U ^ {n} ( a, R) = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} - a _ {j} | < R _ {j} ,\ j= 1 \dots n } \} ; $$

$$ R _ {j} > 0, j = 1 \dots n; \ a = ( a _ {1} \dots a _ {n} ); $$

$$ R = ( R _ {1} \dots R _ {n} ), $$

of multiple power series

$$ f _ {a} = f _ {a} ( z) = $$

$$ = \ \sum _ {k _ {1} \dots k _ {n} = 0 } ^ \infty c _ {k _ {1} \dots k _ {n} } ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} } . $$

The concept of a complete analytic function of several complex variables is constructed in complete analogy with the case of one variable.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[3] G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) MR0122987 MR1530201 MR0092855 Zbl 0501.30039
[4] B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian) MR0155003

Comments

The construct here called the analytic image is variously called the analytic configuration, the analytic entity and the analytische Gebilde in the English literature.

Additional references include H. Weyl's important monograph [a1] (translated into English as [a2]) and the more modern [a3].

References

[a1] H. Weyl, "Die Idee der Riemannschen Fläche" , Teubner (1955) MR0069903 Zbl 0068.06001
[a2] H. Weyl, "The concept of a Riemann surface" , Addison-Wesley (1955) (Translated from German) MR1440406 MR0166351
[a3] H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 MR0583745 Zbl 0475.30001
[a4] O. Foster, "Riemannsche Flächen" , Springer (1977)
How to Cite This Entry:
Complete analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_analytic_function&oldid=15898
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article