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Power series in one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742401.png" />.
+
Power series in one complex variable $  z $.
  
 
A series (representing a function) of the form
 
A series (representing a function) of the form
  
<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/p/p074/p074240/p0742402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
s(z) \  = \  \sum _ { k=0 } ^  \infty  b _ {k} (z-a) ^ {k} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742403.png" /> is the centre, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742404.png" /> are the coefficients and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742405.png" /> are the terms of the series. There exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742407.png" />, called the radius of convergence of the power series (1) and determined by the Cauchy–Hadamard formula
+
where $  a $
 +
is the centre, $  b _ {k} $
 +
are the coefficients and $  b _ {k} (z-a)  ^ {k} $
 +
are the terms of the series. There exists a number $  r $,  
 +
0 \leq  r \leq  \infty $,  
 +
called the radius of convergence of the power series (1) and determined by the Cauchy–Hadamard formula
  
<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/p/p074/p074240/p0742408.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
r \  =
 +
\frac{1}{\lim\limits _ {k \rightarrow \infty } \  \sup \  | b _ {k} |  ^ {1/k} }
 +
,
 +
$$
  
such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p0742409.png" /> the series (1) converges absolutely, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424010.png" />, it diverges (the Cauchy–Hadamard theorem). Accordingly, the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424011.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424012.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424013.png" /> is called the disc of convergence of the power series (Fig. a).
+
such that if $  | z-a | < r $
 +
the series (1) converges absolutely, while if $  | z-a | > r $,  
 +
it diverges (the Cauchy–Hadamard theorem). Accordingly, the disc $  D = \{ {z \in \mathbf C } : {| z-a | < r } \} $
 +
in the complex $  z $-
 +
plane $  \mathbf C $
 +
is called the disc of convergence of the power series (Fig. a).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074240a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074240a.gif" />
Line 17: Line 33:
 
Figure: p074240a
 
Figure: p074240a
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424014.png" />, the disc of convergence degenerates to the single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424015.png" />, for example for the power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424016.png" /> (this case is not of interest, and it is assumed from now on that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424017.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424018.png" />, the disc of convergence coincides with the entire plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424019.png" />, for example for the power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424020.png" />. The set of convergence, i.e. the set of all points of convergence of the series (1), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424021.png" />, consists of the points of the disc of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424022.png" /> plus all, some or none of the points of the circle of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424023.png" />. The disc of convergence in this case is the interior of the set of points of absolute convergence of the power series.
+
When $  r=0 $,  
 +
the disc of convergence degenerates to the single point $  z=a $,  
 +
for example for the power series $  \sum _ {k=0}  ^  \infty  k!(z-a)  ^ {k} $(
 +
this case is not of interest, and it is assumed from now on that $  r> 0 $).  
 +
When $  r = \infty $,  
 +
the disc of convergence coincides with the entire plane $  \mathbf C $,  
 +
for example for the power series $  \sum _ {k=0}  ^  \infty  (z-a)  ^ {k} /k! $.  
 +
The set of convergence, i.e. the set of all points of convergence of the series (1), when $  0 < r < \infty $,  
 +
consists of the points of the disc of convergence $  D $
 +
plus all, some or none of the points of the circle of convergence $  S = \{ {z \in \mathbf C } : {| z-a | = r } \} $.  
 +
The disc of convergence in this case is the interior of the set of points of absolute convergence of the power series.
  
Within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424024.png" />, i.e. on any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424025.png" />, the power series (1) converges absolutely and uniformly. Thus, the sum of the series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424026.png" />, is defined, and is a regular [[Analytic function|analytic function]] at least inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424027.png" />. It has at least one singular point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424028.png" /> to which the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424029.png" /> cannot be analytically continued. There exist power series with exactly one singular point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424030.png" />; there also exist power series for which the entire circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424031.png" /> consists of singular points.
+
Within $  D $,  
 +
i.e. on any compact set $  K \subset  D $,  
 +
the power series (1) converges absolutely and uniformly. Thus, the sum of the series, $  s(z) $,  
 +
is defined, and is a regular [[Analytic function|analytic function]] at least inside $  D $.  
 +
It has at least one singular point on $  S $
 +
to which the sum $  s(z) $
 +
cannot be analytically continued. There exist power series with exactly one singular point on $  S $;  
 +
there also exist power series for which the entire circle $  S $
 +
consists of singular points.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424032.png" />, the series (1) either terminates, i.e. it is a polynomial,
+
When $  r = \infty $,  
 +
the series (1) either terminates, i.e. it is a polynomial,
  
<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/p/p074/p074240/p07424033.png" /></td> </tr></table>
+
$$
 +
s(z) \  = \  \sum _ { k=0 } ^ { m }  b _ {k} (z-a)  ^ {k} ,
 +
$$
  
or its sum is an entire transcendental function, which is regular in the entire place <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424034.png" /> and which possesses an [[Essential singular point|essential singular point]] at infinity.
+
or its sum is an entire transcendental function, which is regular in the entire place $  \mathbf C $
 +
and which possesses an [[Essential singular point|essential singular point]] at infinity.
  
Conversely, the very concept of analyticity of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424035.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424036.png" /> is based on the fact that in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424038.png" /> can be expanded into a power series
+
Conversely, the very concept of analyticity of a function $  f(z) $
 +
at a point $  a $
 +
is based on the fact that in a neighbourhood of $  a $,  
 +
$  f(z) $
 +
can be expanded into 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/p/p074/p074240/p07424039.png" /></td> </tr></table>
+
$$
 +
f(z) \  = \  \sum _ { k=0 } ^  \infty  b _ {k} (z-a)  ^ {k} ,
 +
$$
  
which is the [[Taylor series|Taylor series]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424040.png" />, i.e. its coefficients are defined by the formulas
+
which is the [[Taylor series|Taylor series]] for $  f(z) $,  
 +
i.e. its coefficients are defined by the formulas
  
<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/p/p074/p074240/p07424041.png" /></td> </tr></table>
+
$$
 +
b _ {k} \  =
 +
\frac{f ^ {\  (k) } (a) }{k!}
 +
.
 +
$$
  
Consequently, the uniqueness property of a power series is important: If the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424042.png" /> of the series (1) vanishes on an infinite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424043.png" /> with a limit point inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424045.png" />, and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424047.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424048.png" /> in a neighbourhood of a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424049.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424050.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424051.png" />.
+
Consequently, the uniqueness property of a power series is important: If the sum $  s(z) $
 +
of the series (1) vanishes on an infinite set $  E \subset  D $
 +
with a limit point inside $  D $,  
 +
then $  s(z) \equiv 0 $,  
 +
and all $  b _ {k} = 0 $,  
 +
$  k = 0,\  1 ,\dots $.  
 +
In particular, if $  s(z) = 0 $
 +
in a neighbourhood of a certain point $  z _ {0} \in D $,  
 +
then $  s(z) \equiv 0 $
 +
and all $  b _ {k} = 0 $.
  
 
Thus, every power series is the Taylor series for its own sum.
 
Thus, every power series is the Taylor series for its own sum.
Line 41: Line 99:
 
Let there be another power series apart from (1):
 
Let there be another power series apart from (1):
  
<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/p/p074/p074240/p07424052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\sigma (z) \  = \  \sum _ { k=0 } ^  \infty  c _ {k} (z-a) ^ {k}
 +
$$
  
with the same centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424053.png" /> and with radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424054.png" />. Then, at least inside the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424056.png" />, the addition, subtraction and multiplication of the series (1) and (3) according to the following formulas hold:
+
with the same centre $  a $
 +
and with radius of convergence $  r _ {1} > 0 $.  
 +
Then, at least inside the disc $  \Delta = \{ {z \in \mathbf C } : {| z-a | < \rho } \} $,  
 +
where $  \rho = \mathop{\rm min} \{ r,\  r _ {1} \} $,  
 +
the addition, subtraction and multiplication of the series (1) and (3) according to the following formulas hold:
  
<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/p/p074/p074240/p07424057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\left .
 +
\begin{array}{c}
  
The laws of commutativity, associativity and distributivity hold, whereby subtraction is the inverse operation of addition. Thus, the set of power series with positive radii of convergence and a fixed centre is a [[Ring|ring]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424059.png" />, then division of power series is possible:
+
s(z) \pm \sigma (z) \  = \  \sum _ { k=0 } ^  \infty  (b _ {k} \pm c _ {k} )(z-a)  ^ {k} ,
 +
\\
  
<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/p/p074/p074240/p07424060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
s(z) \sigma (z) \  = \
 +
\sum _ { k=0 } ^  \infty  \left ( \sum _ { n=0 } ^ { k }  b _ {n} c _ {k-n} \right ) (z-a) ^ {k}
 +
\end{array}
 +
\right \} .
 +
$$
  
where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424061.png" /> are uniquely defined from the infinite system of equations
+
The laws of commutativity, associativity and distributivity hold, whereby subtraction is the inverse operation of addition. Thus, the set of power series with positive radii of convergence and a fixed centre is a [[Ring|ring]] over the field  $  \mathbf C $.  
 +
If  $  c _ {0} \neq 0 $,
 +
then division of power series is possible:
  
<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/p/p074/p074240/p07424062.png" /></td> </tr></table>
+
$$ \tag{5 }
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424065.png" />, the radius of convergence of (5) is also positive.
+
\frac{s(z)}{\sigma (z) }
 +
= \
 +
\sum _ { k=0 } ^  \infty  d _ {k} (z-a) ^ {k} ,
 +
$$
  
For the sake of simplicity, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424066.png" /> in (1) and (3); the composite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424067.png" /> will then be regular in a neighbourhood of the coordinate origin, and the process of expanding it into a power series is called substitution of a series in a series:
+
where the coefficients  $  d _ {k} $
 +
are uniquely defined from the infinite system of equations
  
<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/p/p074/p074240/p07424068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$
 +
\sum _ { n=0 } ^ { k }  c _ {n} d _ {k-n} \  = \  a _ {k} ,\ \
 +
k = 0,\  1 ,\dots.
 +
$$
  
The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424069.png" /> in (6) is obtained as the sum of the coefficients with the same index in the expansion of each of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424070.png" />, while the latter expansions are obtained by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424071.png" />-fold multiplication of the series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424072.png" /> by itself. The series (6) automatically converges when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424074.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424075.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424076.png" /> again, and, moreover, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424078.png" />. The problem of constructing a series for the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424079.png" />, which under the given conditions is regular in a neighbourhood of the origin, is called inversion of the series (3). Its solution is the Lagrange series:
+
When  $  c _ {0} \neq 0 $,  
 +
$  r > 0 $
 +
and $  r _ {1} > 0 $,  
 +
the radius of convergence of (5) is also positive.
  
<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/p/p074/p074240/p07424080.png" /></td> </tr></table>
+
For the sake of simplicity, let  $  a = \sigma (0) = c _ {0} = 0 $
 +
in (1) and (3); the composite function  $  s( \sigma (z)) $
 +
will then be regular in a neighbourhood of the coordinate origin, and the process of expanding it into a power series is called substitution of a series in a series:
 +
 
 +
$$ \tag{6 }
 +
s( \sigma (z)) \  = \
 +
\sum _ { n=0 } ^  \infty  b _ {n} \left ( \sum _ { k=0 } ^  \infty  c _ {k} z  ^ {k} \right )  ^ {n} \  = \  \sum _ { m=0 } ^  \infty  g _ {m} z  ^ {m} .
 +
$$
 +
 
 +
The coefficient  $  g _ {m} $
 +
in (6) is obtained as the sum of the coefficients with the same index in the expansion of each of the functions  $  b _ {n} ( \sigma (z))  ^ {n} $,
 +
while the latter expansions are obtained by  $  n $-
 +
fold multiplication of the series for  $  \sigma (z) $
 +
by itself. The series (6) automatically converges when  $  | z | < \rho $,
 +
where  $  \rho $
 +
is such that  $  | \sigma (z) | < r $.
 +
Let  $  a = \sigma (0) = c _ {0} = 0 $
 +
again, and, moreover, let  $  c _ {1} = \sigma  ^  \prime  (0) \neq 0 $,
 +
$  w = \sigma (z) $.  
 +
The problem of constructing a series for the inverse function  $  z = \phi (w) $,
 +
which under the given conditions is regular in a neighbourhood of the origin, is called inversion of the series (3). Its solution is the Lagrange series:
 +
 
 +
$$
 +
z \  = \  \phi (w) \  = \
 +
\sum _ { n=1 } ^  \infty 
 +
\frac{1}{n!}
 +
\left (
 +
\frac \zeta {\sigma ( \zeta ) }
 +
\right ) _ {\zeta
 +
=0 }  ^ {(n)} w  ^ {n}
 +
$$
  
 
(for a more general inversion problem see [[Bürmann–Lagrange series|Bürmann–Lagrange series]]).
 
(for a more general inversion problem see [[Bürmann–Lagrange series|Bürmann–Lagrange series]]).
  
If the power series (1) converges at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424081.png" />, then it converges absolutely for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424082.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424083.png" />; this is the essence of Abel's first theorem. This theorem also makes it possible to establish the form of the domain of convergence of the series. Abel's second theorem provides a more detailed result: If the series (1) converges at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424084.png" /> on the circle of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424085.png" />, then
+
If the power series (1) converges at a point $  z _ {0} \neq a $,  
 +
then it converges absolutely for all $  z $
 +
for which $  | z-a | < | z _ {0} -a | $;  
 +
this is the essence of Abel's first theorem. This theorem also makes it possible to establish the form of the domain of convergence of the series. Abel's second theorem provides a more detailed result: If the series (1) converges at the point $  z _ {0} = a + re ^ {i \theta _ {0} } $
 +
on the circle of convergence $  S $,  
 +
then
  
<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/p/p074/p074240/p07424086.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\rho \rightarrow r } \  s(a+ \rho e ^ {i \theta _ {0} } ) \  = \  s(z _ {0} ),
 +
$$
  
i.e. the sum of the series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424087.png" />, at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424088.png" /> has radial boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424089.png" /> and, consequently, is continuous along the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424091.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424092.png" /> also has non-tangential boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424093.png" /> (cf. [[Angular boundary value|Angular boundary value]]). This theorem, dating back to 1827, can be seen as the first major result in the research into the boundary properties of power series. Inversion of Abel's second theorem without extra restrictions on the coefficients of the power series is impossible. However, if one assumes, for example, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424094.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424095.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424096.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424097.png" />. This type of partial inversions of Abel's second theorem are called [[Tauberian theorems|Tauberian theorems]].
+
i.e. the sum of the series, $  s(z) $,  
 +
at the point $  z _ {0} \in S $
 +
has radial boundary value $  s(z _ {0} ) $
 +
and, consequently, is continuous along the radius $  z = a + \rho ^ {i \theta _ {0} } $,  
 +
0 \leq  \rho \leq  r $;  
 +
moreover, $  s(z) $
 +
also has non-tangential boundary value $  s(z _ {0} ) $(
 +
cf. [[Angular boundary value|Angular boundary value]]). This theorem, dating back to 1827, can be seen as the first major result in the research into the boundary properties of power series. Inversion of Abel's second theorem without extra restrictions on the coefficients of the power series is impossible. However, if one assumes, for example, that $  b _ {k} = o(1/k) $,  
 +
and if $  \lim\limits _ {\rho \rightarrow r }  \  s(a+ \rho e ^ {i \theta _ {0} } ) = s _ {0} $
 +
exists, then $  \sum _ {k=0}  ^  \infty  b _ {k} (z _ {0} -a)  ^ {k} $
 +
converges to $  s _ {0} $.  
 +
This type of partial inversions of Abel's second theorem are called [[Tauberian theorems|Tauberian theorems]].
  
 
For other results relating to the boundary properties of power series and particularly to the location of singular points of power series, see [[Hadamard theorem|Hadamard theorem]]; [[Analytic continuation|Analytic continuation]]; [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Fatou theorem|Fatou theorem]] (see also –).
 
For other results relating to the boundary properties of power series and particularly to the location of singular points of power series, see [[Hadamard theorem|Hadamard theorem]]; [[Analytic continuation|Analytic continuation]]; [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Fatou theorem|Fatou theorem]] (see also –).
  
Power series in several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p07424099.png" />, or multiple power series, are series (representing functions) of the form
+
Power series in several complex variables $  z = (z _ {1} \dots z _ {n} ) $,  
 +
$  n > 1 $,  
 +
or multiple power series, are series (representing functions) of the form
  
<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/p/p074/p074240/p074240100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
s(z) \  = \  \sum _ {\mid  k \vDash0 } ^  \infty  b _ {k} (z-a)  ^ {k\ } =
 +
$$
  
<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/p/p074/p074240/p074240101.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {k _ {1} = 0 } ^  \infty  \dots \sum _ {k _ {n} =0 } ^  \infty  b _ {k _ {1}  \dots k _ {n} } (z _ {1} - a _ {1} ) ^ {k _ {1} } \dots (z _ {n} - a _ {n} ) ^ {k _ {n} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240105.png" />, the centre of the series, is a point of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240106.png" />. The interior of the set of points of absolute convergence is called the domain of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240107.png" /> of the power series (7), but when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240108.png" />, it does not have such a simple form as when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240109.png" />. A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240110.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240111.png" /> is the domain of convergence of a certain power series (7) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240112.png" /> is a logarithmically-convex complete [[Reinhardt domain|Reinhardt domain]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240113.png" />. If a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240114.png" />, then the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240115.png" /> of the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240116.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240118.png" />, also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240119.png" />, and the series (7) converges absolutely and uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240120.png" /> (the analogue of Abel's first theorem). The polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240122.png" />, is called the polydisc of convergence of the series (7), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240123.png" /> and if in any larger polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240124.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240126.png" />, and at least one inequality is strict, there are points at which the series (7) diverges. The radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240127.png" /> of the polydisc of convergence are called conjugate radii of convergence, and satisfy a relation analogous to the Cauchy–Hadamard formula:
+
where $  b _ {k} = b _ {k _ {1}  \dots k _ {n} } $,  
 +
$  (z-a)  ^ {k} = (z _ {1} - a _ {1} ) ^ {k _ {1} } \dots (z _ {n} - a _ {n} ) ^ {k _ {n} } $,  
 +
$  | k | = k _ {1} + \dots + k _ {n} $,  
 +
and $  a = (a _ {1} \dots a _ {n} ) $,  
 +
the centre of the series, is a point of the complex space $  \mathbf C  ^ {n} $.  
 +
The interior of the set of points of absolute convergence is called the domain of convergence $  D $
 +
of the power series (7), but when $  n > 1 $,  
 +
it does not have such a simple form as when $  n=1 $.  
 +
A domain $  D $
 +
of $  \mathbf C  ^ {n} $
 +
is the domain of convergence of a certain power series (7) if and only if $  D $
 +
is a logarithmically-convex complete [[Reinhardt domain|Reinhardt domain]] of $  \mathbf C  ^ {n} $.  
 +
If a certain point $  z  ^ {0} \in D $,  
 +
then the closure $  \overline{U}\; (a,\  r) $
 +
of the polydisc $  U(a,\  r) = \{ {z \in \mathbf C  ^ {n} } : {| z _ {v} - a _ {v} | < r _ {v} ,\  v = 1 \dots n } \} $,  
 +
where $  r _ {v} = | z _ {v}  ^ {0} - a _ {v} | $,  
 +
$  r = (r _ {1} \dots r _ {n} ) $,  
 +
also belongs to $  D $,  
 +
and the series (7) converges absolutely and uniformly in $  \overline{U}\; (a,\  r) $(
 +
the analogue of Abel's first theorem). The polydisc $  U(a,\  r) $,  
 +
$  r=(r _ {1} \dots r _ {n} ) $,  
 +
is called the polydisc of convergence of the series (7), if $  U(a,\  r) \subset  D $
 +
and if in any larger polydisc $  \{ {z \in \mathbf C  ^ {n} } : {| z _ {v} - a _ {v} | < r _ {v}  ^  \prime  } \} $,  
 +
where $  r _ {v}  ^  \prime  \geq  r _ {v} $,  
 +
$  v = 1 \dots n $,  
 +
and at least one inequality is strict, there are points at which the series (7) diverges. The radii $  r _ {v} $
 +
of the polydisc of convergence are called conjugate radii of convergence, and satisfy a relation analogous to the Cauchy–Hadamard formula:
  
<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/p/p074/p074240/p074240128.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {| k | \rightarrow \infty } \  \sup \
 +
(| b _ {k} | r  ^ {k} ) ^ {1/ | k | } \  = \  1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240130.png" />. The domain of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240131.png" /> is exhausted by polydiscs of convergence. For example, for the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240132.png" /> the polydiscs of convergence take the form
+
where $  | b _ {k} | = | b _ {k _ {1}  } \dots b _ {k _ {n}  } | $,  
 +
$  r  ^ {k} = r _ {1} ^ {k _ {1} } \dots r _ {n} ^ {k _ {n} } $.  
 +
The domain of convergence $  D $
 +
is exhausted by polydiscs of convergence. For example, for the series $  \sum _ {k=0}  ^  \infty  (z _ {1} z _ {2} )  ^ {k} $
 +
the polydiscs of convergence take the form
  
<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/p/p074/p074240/p074240133.png" /></td> </tr></table>
+
$$
 +
U \left ( 0,\  r _ {1} ,\ 
 +
\frac{1}{r _ {1} }
 +
\right ) \  = \
 +
\left \{ {z = (z _ {1} ,\  z _ {2} ) \in \mathbf C  ^ {2} } : {| z _ {1} | < r _ {1} ,\  | z _ {2} | <  
 +
\frac{1}{r _ {1} }
 +
} \right \}
 +
,
 +
$$
  
while the domain of convergence is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240134.png" /> (in Fig. bit is represented in the quadrant of absolute values).
+
while the domain of convergence is $  D = \{ {z \in \mathbf C  ^ {2} } : {| z _ {1} | \cdot | z _ {2} | < 1 } \} $(
 +
in Fig. bit is represented in the quadrant of absolute values).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074240b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074240b.gif" />
Line 95: Line 275:
 
Figure: p074240b
 
Figure: p074240b
  
The uniqueness property of power series is preserved in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240135.png" /> in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240136.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240137.png" /> (it is sufficient even in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240138.png" />, i.e. on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240139.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240140.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240141.png" />.
+
The uniqueness property of power series is preserved in the sense that if $  s(z) = 0 $
 +
in a neighbourhood of the point $  z  ^ {0} $
 +
in $  \mathbf C  ^ {n} $(
 +
it is sufficient even in $  \mathbf R  ^ {n} $,  
 +
i.e. on a set $  \{ {z = x + iy \in \mathbf C  ^ {n} } : {| x -  \mathop{\rm Re} \  a | < r,\  y = \mathop{\rm Im} \  a } \} $),  
 +
then $  s(z) \equiv 0 $
 +
and all $  b _ {k} = 0 $.
  
Operations with multiple power series are carried out, broadly speaking, according to the same rules as when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240142.png" />. For other properties of multiple power series, see, for example, , .
+
Operations with multiple power series are carried out, broadly speaking, according to the same rules as when $  n=1 $.  
 +
For other properties of multiple power series, see, for example, , .
  
Power series in real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240144.png" />, are series of functions of the form
+
Power series in real variables $  x = (x _ {1} \dots x _ {n} ) $,  
 +
$  n \geq  1 $,  
 +
are series of functions of the form
  
<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/p/p074/p074240/p074240145.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
s(x) \  = \  \sum _ {| k | =0 } ^  \infty  b _ {k} (x-a) ^ {k} ,
 +
$$
  
where abbreviated notations are used, as in , and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240146.png" /> is the centre of the series. If the series (8) converges absolutely in a parallelepipedon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240147.png" />, then it also converges absolutely in the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240149.png" />. The sum of the series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240150.png" />, being an analytic function of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240151.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240152.png" />, is continued analytically in the form of the power series
+
where abbreviated notations are used, as in , and $  a = (a _ {1} \dots a _ {n} ) \in \mathbf R  ^ {n} $
 +
is the centre of the series. If the series (8) converges absolutely in a parallelepipedon $  \Pi = \{ {x \in \mathbf R  ^ {n} } : {| x _ {k} - a _ {k} | < r _ {k} ,\  k = 1 \dots n } \} $,  
 +
then it also converges absolutely in the polydisc $  U(a,\  r) = \{ {z \in \mathbf C  ^ {n} } : {| z -a | < r } \} $,  
 +
$  r = (r _ {1} \dots r _ {n} ) $.  
 +
The sum of the series, $  s(x) $,  
 +
being an analytic function of the real variables $  x = (x _ {1} \dots x _ {n} ) $
 +
in $  \Pi $,  
 +
is continued analytically in the form of the 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/p/p074/p074240/p074240153.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
$$ \tag{9 }
 +
s(z) \  = \  \sum _ {\mid  k \vDash0 } ^  \infty  b _ {k} (z-a) ^ {k}
 +
$$
  
to the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240154.png" /> of the complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240155.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240156.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240157.png" /> is the domain of convergence of (9) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240158.png" /> (complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240159.png" />), then its restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240160.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240161.png" /> in the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240162.png" /> is the domain of convergence of (8), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240163.png" />. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240165.png" /> is a disc of convergence and its restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240166.png" /> is an interval of convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240167.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240168.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240169.png" /> is the radius of convergence.
+
to the analytic function $  s(z) $
 +
of the complex variables $  z = x+iy = (z _ {1} = x _ {1} + iy _ {1} \dots z _ {n} = x _ {n} + iy _ {n} ) $
 +
in $  U(a,\  r) $.  
 +
If $  D $
 +
is the domain of convergence of (9) in $  \mathbf C  ^ {n} $(
 +
complex variables $  z=x+iy $),  
 +
then its restriction $  \Delta $
 +
to $  \mathbf R  ^ {n} $
 +
in the real variables $  x = (x _ {1} \dots x _ {n} ) $
 +
is the domain of convergence of (8), $  \Delta \subset  D $.  
 +
In particular, when $  n=1 $,  
 +
$  D $
 +
is a disc of convergence and its restriction $  \Delta $
 +
is an interval of convergence on $  \mathbf R $,
 +
and  $  \Delta = \{ x \in \mathbf R ,\  a-r < x < a+r \} $,  
 +
where $  r $
 +
is the radius of convergence.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of analytic functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Landau,  D. Gaier,  "Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint  (1986)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1985)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S. Bochner,  W.T. Martin,  "Several complex variables" , Princeton Univ. Press  (1948)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.I. Yanushauskas,  "Double series" , Novosibirsk  (1980)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of analytic functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Landau,  D. Gaier,  "Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint  (1986)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1985)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S. Bochner,  W.T. Martin,  "Several complex variables" , Princeton Univ. Press  (1948)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.I. Yanushauskas,  "Double series" , Novosibirsk  (1980)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The approach to analytic functions via power series is the so-called Weierstrass approach. For a somewhat more abstract setting (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240170.png" /> replaced by a suitable field) see [[#References|[a3]]], Chapts. 2–3 (cf. also [[Formal power series|Formal power series]]).
+
The approach to analytic functions via power series is the so-called Weierstrass approach. For a somewhat more abstract setting ( $  \mathbf C $
 +
replaced by a suitable field) see [[#References|[a3]]], Chapts. 2–3 (cf. also [[Formal power series|Formal power series]]).
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074240/p074240171.png" /> see [[#References|[a1]]]–[[#References|[a2]]].
+
For $  \mathbf C  ^ {n} $
 +
see [[#References|[a1]]]–[[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Narasimhan,  "Several complex variables" , Univ. Chicago Press  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Diederich,  R. Remmert,  "Funktionentheorie" , '''I''' , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Narasimhan,  "Several complex variables" , Univ. Chicago Press  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Diederich,  R. Remmert,  "Funktionentheorie" , '''I''' , Springer  (1972)</TD></TR></table>

Latest revision as of 11:57, 10 February 2020


Power series in one complex variable $ z $.

A series (representing a function) of the form

$$ \tag{1 } s(z) \ = \ \sum _ { k=0 } ^ \infty b _ {k} (z-a) ^ {k} , $$

where $ a $ is the centre, $ b _ {k} $ are the coefficients and $ b _ {k} (z-a) ^ {k} $ are the terms of the series. There exists a number $ r $, $ 0 \leq r \leq \infty $, called the radius of convergence of the power series (1) and determined by the Cauchy–Hadamard formula

$$ \tag{2 } r \ = \ \frac{1}{\lim\limits _ {k \rightarrow \infty } \ \sup \ | b _ {k} | ^ {1/k} } , $$

such that if $ | z-a | < r $ the series (1) converges absolutely, while if $ | z-a | > r $, it diverges (the Cauchy–Hadamard theorem). Accordingly, the disc $ D = \{ {z \in \mathbf C } : {| z-a | < r } \} $ in the complex $ z $- plane $ \mathbf C $ is called the disc of convergence of the power series (Fig. a).

Figure: p074240a

When $ r=0 $, the disc of convergence degenerates to the single point $ z=a $, for example for the power series $ \sum _ {k=0} ^ \infty k!(z-a) ^ {k} $( this case is not of interest, and it is assumed from now on that $ r> 0 $). When $ r = \infty $, the disc of convergence coincides with the entire plane $ \mathbf C $, for example for the power series $ \sum _ {k=0} ^ \infty (z-a) ^ {k} /k! $. The set of convergence, i.e. the set of all points of convergence of the series (1), when $ 0 < r < \infty $, consists of the points of the disc of convergence $ D $ plus all, some or none of the points of the circle of convergence $ S = \{ {z \in \mathbf C } : {| z-a | = r } \} $. The disc of convergence in this case is the interior of the set of points of absolute convergence of the power series.

Within $ D $, i.e. on any compact set $ K \subset D $, the power series (1) converges absolutely and uniformly. Thus, the sum of the series, $ s(z) $, is defined, and is a regular analytic function at least inside $ D $. It has at least one singular point on $ S $ to which the sum $ s(z) $ cannot be analytically continued. There exist power series with exactly one singular point on $ S $; there also exist power series for which the entire circle $ S $ consists of singular points.

When $ r = \infty $, the series (1) either terminates, i.e. it is a polynomial,

$$ s(z) \ = \ \sum _ { k=0 } ^ { m } b _ {k} (z-a) ^ {k} , $$

or its sum is an entire transcendental function, which is regular in the entire place $ \mathbf C $ and which possesses an essential singular point at infinity.

Conversely, the very concept of analyticity of a function $ f(z) $ at a point $ a $ is based on the fact that in a neighbourhood of $ a $, $ f(z) $ can be expanded into a power series

$$ f(z) \ = \ \sum _ { k=0 } ^ \infty b _ {k} (z-a) ^ {k} , $$

which is the Taylor series for $ f(z) $, i.e. its coefficients are defined by the formulas

$$ b _ {k} \ = \ \frac{f ^ {\ (k) } (a) }{k!} . $$

Consequently, the uniqueness property of a power series is important: If the sum $ s(z) $ of the series (1) vanishes on an infinite set $ E \subset D $ with a limit point inside $ D $, then $ s(z) \equiv 0 $, and all $ b _ {k} = 0 $, $ k = 0,\ 1 ,\dots $. In particular, if $ s(z) = 0 $ in a neighbourhood of a certain point $ z _ {0} \in D $, then $ s(z) \equiv 0 $ and all $ b _ {k} = 0 $.

Thus, every power series is the Taylor series for its own sum.

Let there be another power series apart from (1):

$$ \tag{3 } \sigma (z) \ = \ \sum _ { k=0 } ^ \infty c _ {k} (z-a) ^ {k} $$

with the same centre $ a $ and with radius of convergence $ r _ {1} > 0 $. Then, at least inside the disc $ \Delta = \{ {z \in \mathbf C } : {| z-a | < \rho } \} $, where $ \rho = \mathop{\rm min} \{ r,\ r _ {1} \} $, the addition, subtraction and multiplication of the series (1) and (3) according to the following formulas hold:

$$ \tag{4 } \left . \begin{array}{c} s(z) \pm \sigma (z) \ = \ \sum _ { k=0 } ^ \infty (b _ {k} \pm c _ {k} )(z-a) ^ {k} , \\ s(z) \sigma (z) \ = \ \sum _ { k=0 } ^ \infty \left ( \sum _ { n=0 } ^ { k } b _ {n} c _ {k-n} \right ) (z-a) ^ {k} \end{array} \right \} . $$

The laws of commutativity, associativity and distributivity hold, whereby subtraction is the inverse operation of addition. Thus, the set of power series with positive radii of convergence and a fixed centre is a ring over the field $ \mathbf C $. If $ c _ {0} \neq 0 $, then division of power series is possible:

$$ \tag{5 } \frac{s(z)}{\sigma (z) } \ = \ \sum _ { k=0 } ^ \infty d _ {k} (z-a) ^ {k} , $$

where the coefficients $ d _ {k} $ are uniquely defined from the infinite system of equations

$$ \sum _ { n=0 } ^ { k } c _ {n} d _ {k-n} \ = \ a _ {k} ,\ \ k = 0,\ 1 ,\dots. $$

When $ c _ {0} \neq 0 $, $ r > 0 $ and $ r _ {1} > 0 $, the radius of convergence of (5) is also positive.

For the sake of simplicity, let $ a = \sigma (0) = c _ {0} = 0 $ in (1) and (3); the composite function $ s( \sigma (z)) $ will then be regular in a neighbourhood of the coordinate origin, and the process of expanding it into a power series is called substitution of a series in a series:

$$ \tag{6 } s( \sigma (z)) \ = \ \sum _ { n=0 } ^ \infty b _ {n} \left ( \sum _ { k=0 } ^ \infty c _ {k} z ^ {k} \right ) ^ {n} \ = \ \sum _ { m=0 } ^ \infty g _ {m} z ^ {m} . $$

The coefficient $ g _ {m} $ in (6) is obtained as the sum of the coefficients with the same index in the expansion of each of the functions $ b _ {n} ( \sigma (z)) ^ {n} $, while the latter expansions are obtained by $ n $- fold multiplication of the series for $ \sigma (z) $ by itself. The series (6) automatically converges when $ | z | < \rho $, where $ \rho $ is such that $ | \sigma (z) | < r $. Let $ a = \sigma (0) = c _ {0} = 0 $ again, and, moreover, let $ c _ {1} = \sigma ^ \prime (0) \neq 0 $, $ w = \sigma (z) $. The problem of constructing a series for the inverse function $ z = \phi (w) $, which under the given conditions is regular in a neighbourhood of the origin, is called inversion of the series (3). Its solution is the Lagrange series:

$$ z \ = \ \phi (w) \ = \ \sum _ { n=1 } ^ \infty \frac{1}{n!} \left ( \frac \zeta {\sigma ( \zeta ) } \right ) _ {\zeta =0 } ^ {(n)} w ^ {n} $$

(for a more general inversion problem see Bürmann–Lagrange series).

If the power series (1) converges at a point $ z _ {0} \neq a $, then it converges absolutely for all $ z $ for which $ | z-a | < | z _ {0} -a | $; this is the essence of Abel's first theorem. This theorem also makes it possible to establish the form of the domain of convergence of the series. Abel's second theorem provides a more detailed result: If the series (1) converges at the point $ z _ {0} = a + re ^ {i \theta _ {0} } $ on the circle of convergence $ S $, then

$$ \lim\limits _ {\rho \rightarrow r } \ s(a+ \rho e ^ {i \theta _ {0} } ) \ = \ s(z _ {0} ), $$

i.e. the sum of the series, $ s(z) $, at the point $ z _ {0} \in S $ has radial boundary value $ s(z _ {0} ) $ and, consequently, is continuous along the radius $ z = a + \rho ^ {i \theta _ {0} } $, $ 0 \leq \rho \leq r $; moreover, $ s(z) $ also has non-tangential boundary value $ s(z _ {0} ) $( cf. Angular boundary value). This theorem, dating back to 1827, can be seen as the first major result in the research into the boundary properties of power series. Inversion of Abel's second theorem without extra restrictions on the coefficients of the power series is impossible. However, if one assumes, for example, that $ b _ {k} = o(1/k) $, and if $ \lim\limits _ {\rho \rightarrow r } \ s(a+ \rho e ^ {i \theta _ {0} } ) = s _ {0} $ exists, then $ \sum _ {k=0} ^ \infty b _ {k} (z _ {0} -a) ^ {k} $ converges to $ s _ {0} $. This type of partial inversions of Abel's second theorem are called Tauberian theorems.

For other results relating to the boundary properties of power series and particularly to the location of singular points of power series, see Hadamard theorem; Analytic continuation; Boundary properties of analytic functions; Fatou theorem (see also –).

Power series in several complex variables $ z = (z _ {1} \dots z _ {n} ) $, $ n > 1 $, or multiple power series, are series (representing functions) of the form

$$ \tag{7 } s(z) \ = \ \sum _ {\mid k \vDash0 } ^ \infty b _ {k} (z-a) ^ {k\ } = $$

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

where $ b _ {k} = b _ {k _ {1} \dots k _ {n} } $, $ (z-a) ^ {k} = (z _ {1} - a _ {1} ) ^ {k _ {1} } \dots (z _ {n} - a _ {n} ) ^ {k _ {n} } $, $ | k | = k _ {1} + \dots + k _ {n} $, and $ a = (a _ {1} \dots a _ {n} ) $, the centre of the series, is a point of the complex space $ \mathbf C ^ {n} $. The interior of the set of points of absolute convergence is called the domain of convergence $ D $ of the power series (7), but when $ n > 1 $, it does not have such a simple form as when $ n=1 $. A domain $ D $ of $ \mathbf C ^ {n} $ is the domain of convergence of a certain power series (7) if and only if $ D $ is a logarithmically-convex complete Reinhardt domain of $ \mathbf C ^ {n} $. If a certain point $ z ^ {0} \in D $, then the closure $ \overline{U}\; (a,\ r) $ of the polydisc $ U(a,\ r) = \{ {z \in \mathbf C ^ {n} } : {| z _ {v} - a _ {v} | < r _ {v} ,\ v = 1 \dots n } \} $, where $ r _ {v} = | z _ {v} ^ {0} - a _ {v} | $, $ r = (r _ {1} \dots r _ {n} ) $, also belongs to $ D $, and the series (7) converges absolutely and uniformly in $ \overline{U}\; (a,\ r) $( the analogue of Abel's first theorem). The polydisc $ U(a,\ r) $, $ r=(r _ {1} \dots r _ {n} ) $, is called the polydisc of convergence of the series (7), if $ U(a,\ r) \subset D $ and if in any larger polydisc $ \{ {z \in \mathbf C ^ {n} } : {| z _ {v} - a _ {v} | < r _ {v} ^ \prime } \} $, where $ r _ {v} ^ \prime \geq r _ {v} $, $ v = 1 \dots n $, and at least one inequality is strict, there are points at which the series (7) diverges. The radii $ r _ {v} $ of the polydisc of convergence are called conjugate radii of convergence, and satisfy a relation analogous to the Cauchy–Hadamard formula:

$$ \lim\limits _ {| k | \rightarrow \infty } \ \sup \ (| b _ {k} | r ^ {k} ) ^ {1/ | k | } \ = \ 1, $$

where $ | b _ {k} | = | b _ {k _ {1} } \dots b _ {k _ {n} } | $, $ r ^ {k} = r _ {1} ^ {k _ {1} } \dots r _ {n} ^ {k _ {n} } $. The domain of convergence $ D $ is exhausted by polydiscs of convergence. For example, for the series $ \sum _ {k=0} ^ \infty (z _ {1} z _ {2} ) ^ {k} $ the polydiscs of convergence take the form

$$ U \left ( 0,\ r _ {1} ,\ \frac{1}{r _ {1} } \right ) \ = \ \left \{ {z = (z _ {1} ,\ z _ {2} ) \in \mathbf C ^ {2} } : {| z _ {1} | < r _ {1} ,\ | z _ {2} | < \frac{1}{r _ {1} } } \right \} , $$

while the domain of convergence is $ D = \{ {z \in \mathbf C ^ {2} } : {| z _ {1} | \cdot | z _ {2} | < 1 } \} $( in Fig. bit is represented in the quadrant of absolute values).

Figure: p074240b

The uniqueness property of power series is preserved in the sense that if $ s(z) = 0 $ in a neighbourhood of the point $ z ^ {0} $ in $ \mathbf C ^ {n} $( it is sufficient even in $ \mathbf R ^ {n} $, i.e. on a set $ \{ {z = x + iy \in \mathbf C ^ {n} } : {| x - \mathop{\rm Re} \ a | < r,\ y = \mathop{\rm Im} \ a } \} $), then $ s(z) \equiv 0 $ and all $ b _ {k} = 0 $.

Operations with multiple power series are carried out, broadly speaking, according to the same rules as when $ n=1 $. For other properties of multiple power series, see, for example, , .

Power series in real variables $ x = (x _ {1} \dots x _ {n} ) $, $ n \geq 1 $, are series of functions of the form

$$ \tag{8 } s(x) \ = \ \sum _ {| k | =0 } ^ \infty b _ {k} (x-a) ^ {k} , $$

where abbreviated notations are used, as in , and $ a = (a _ {1} \dots a _ {n} ) \in \mathbf R ^ {n} $ is the centre of the series. If the series (8) converges absolutely in a parallelepipedon $ \Pi = \{ {x \in \mathbf R ^ {n} } : {| x _ {k} - a _ {k} | < r _ {k} ,\ k = 1 \dots n } \} $, then it also converges absolutely in the polydisc $ U(a,\ r) = \{ {z \in \mathbf C ^ {n} } : {| z -a | < r } \} $, $ r = (r _ {1} \dots r _ {n} ) $. The sum of the series, $ s(x) $, being an analytic function of the real variables $ x = (x _ {1} \dots x _ {n} ) $ in $ \Pi $, is continued analytically in the form of the power series

$$ \tag{9 } s(z) \ = \ \sum _ {\mid k \vDash0 } ^ \infty b _ {k} (z-a) ^ {k} $$

to the analytic function $ s(z) $ of the complex variables $ z = x+iy = (z _ {1} = x _ {1} + iy _ {1} \dots z _ {n} = x _ {n} + iy _ {n} ) $ in $ U(a,\ r) $. If $ D $ is the domain of convergence of (9) in $ \mathbf C ^ {n} $( complex variables $ z=x+iy $), then its restriction $ \Delta $ to $ \mathbf R ^ {n} $ in the real variables $ x = (x _ {1} \dots x _ {n} ) $ is the domain of convergence of (8), $ \Delta \subset D $. In particular, when $ n=1 $, $ D $ is a disc of convergence and its restriction $ \Delta $ is an interval of convergence on $ \mathbf R $, and $ \Delta = \{ x \in \mathbf R ,\ a-r < x < a+r \} $, where $ r $ is the radius of convergence.

References

[1] A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)
[2] A.I. Markushevich, "Theory of analytic functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[3] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)
[4] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955)
[5] E. Landau, D. Gaier, "Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint (1986)
[6] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[7] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian)
[8] S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948)
[9] A.I. Yanushauskas, "Double series" , Novosibirsk (1980) (In Russian)

Comments

The approach to analytic functions via power series is the so-called Weierstrass approach. For a somewhat more abstract setting ( $ \mathbf C $ replaced by a suitable field) see [a3], Chapts. 2–3 (cf. also Formal power series).

For $ \mathbf C ^ {n} $ see [a1][a2].

References

[a1] W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
[a2] R. Narasimhan, "Several complex variables" , Univ. Chicago Press (1971)
[a3] K. Diederich, R. Remmert, "Funktionentheorie" , I , Springer (1972)
How to Cite This Entry:
Power series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_series&oldid=44404
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article