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Abel's theorem on algebraic equations: Formulas expressing the solution of an arbitrary equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101801.png" /> in terms of its coefficients using radicals do not exist for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101802.png" />. The theorem was proved by N.H. Abel in 1824 . Abel's theorem may also be obtained as a corollary of [[Galois theory|Galois theory]], from which a more general theorem follows: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101803.png" /> there exist algebraic equations with integer coefficients whose roots cannot be expressed in terms of radicals of rational numbers. For a modern formulation of Abel's theorem for equations over an arbitrary field, see [[Algebraic equation|Algebraic equation]].
+
{{TEX|done}}
 +
 
 +
Abel's theorem on algebraic equations: Formulas expressing the solution of an arbitrary equation of degree $  n $
 +
in terms of its coefficients using radicals do not exist for any $  n \geq  5 $.  
 +
The theorem was proved by N.H. Abel in 1824 . Abel's theorem may also be obtained as a corollary of [[Galois theory|Galois theory]], from which a more general theorem follows: For any $  n \geq  5 $
 +
there exist algebraic equations with integer coefficients whose roots cannot be expressed in terms of radicals of rational numbers. For a modern formulation of Abel's theorem for equations over an arbitrary field, see [[Algebraic equation|Algebraic equation]].
  
 
Abel's theorem on power series: If the power series
 
Abel's theorem on power series: If 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/a/a010/a010180/a0101804.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{*}
 +
S ( z ) \  = \  \sum _ {k = 0} ^  \infty  a _ {k} ( z - b ) ^ {k} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101805.png" /> are complex numbers, converges at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101806.png" />, then it converges absolutely and uniformly within any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101807.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101808.png" /> and with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a0101809.png" />. The theorem was established by N.H. Abel . It follows from the theorem that there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018010.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018011.png" /> the series is convergent, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018012.png" /> the series is divergent. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018013.png" /> is called the radius of convergence of the series (*), while the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018014.png" /> is known as the disc of convergence of the series (*).
+
where a _ {k} ,\  b,\  z $
 +
are complex numbers, converges at $  z = z _ {0} $,  
 +
then it converges absolutely and uniformly within any disc $  | z - b | \leq  \rho $
 +
of radius $  \rho < | z _ {0} - b | $
 +
and with centre at $  b $.  
 +
The theorem was established by N.H. Abel . It follows from the theorem that there exists a number $  R \in [ 0,\  \infty ] $
 +
such that if $  | z - b | < R $
 +
the series is convergent, while if $  | z - b | > R $
 +
the series is divergent. The number $  R $
 +
is called the radius of convergence of the series (*), while the disc $  | z - b | < R $
 +
is known as the disc of convergence of the series (*).
  
 
Abel's continuity theorem: If the power series
 
Abel's continuity theorem: If the power series
  
converges at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018015.png" /> on the boundary of the disc of convergence, then it is a continuous function in any closed triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018016.png" /> with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018018.png" /> are located inside the disc of convergence. In particular
+
converges at a point $  z _ {0} $
 +
on the boundary of the disc of convergence, then it is a continuous function in any closed triangle $  T $
 +
with vertices $  z _ {0} ,\  z _ {1} ,\  z _ {2} $,  
 +
where $  z _ {1} ,\  z _ {2} $
 +
are located inside the disc of convergence. In particular
  
<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/a/a010/a010180/a01018019.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\begin{array}{c}
 +
z \rightarrow z _ {0} , \\
 +
z \in T
 +
\end{array}
 +
} \
 +
S ( z ) \  = \  S ( z _ {0} ) .
 +
$$
  
 
This limit always exists along the radius: The series
 
This limit always exists along the radius: The series
  
converges uniformly along any radius of the disc of convergence joining the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018021.png" />. This theorem is used, in particular, to calculate the sum of a power series which converges at the boundary points of the disc of convergence.
+
converges uniformly along any radius of the disc of convergence joining the points $  b $
 +
and $  z _ {0} $.  
 +
This theorem is used, in particular, to calculate the sum of a power series which converges at the boundary points of the disc of convergence.
  
 
Abel's theorem on Dirichlet series: If the [[Dirichlet series|Dirichlet series]]
 
Abel's theorem on Dirichlet series: If the [[Dirichlet series|Dirichlet 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/a/a010/a010180/a01018022.png" /></td> </tr></table>
+
$$
 +
\phi ( s ) \  = \  \sum _ {n = 1} ^  \infty 
 +
a _ {n} e ^ {- \lambda _ {n} s} ,
 +
\ \  s = \sigma + it ,\ \  \lambda _ {n} > 0 ,
 +
$$
  
converges at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018023.png" />, then it converges in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018024.png" /> and converges uniformly inside any angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018025.png" />. It is a generalization of Abel's theorem on power series (take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018026.png" /> and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018027.png" />). It follows from the theorem that the domain of convergence of a Dirichlet series is some half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018029.png" /> is the abscissa of convergence of the series.
+
converges at the point $  s _ {0} = \sigma _ {0} + i t _ {0} $,  
 +
then it converges in the half-plane $  \sigma > \sigma _ {0} $
 +
and converges uniformly inside any angle $  |  \mathop{\rm arg} (s - s _ {0} ) | \leq  \theta < \pi / 2 $.  
 +
It is a generalization of Abel's theorem on power series (take $  \lambda _ {n} = n $
 +
and put $  e  ^ {-s} = z $).  
 +
It follows from the theorem that the domain of convergence of a Dirichlet series is some half-plane $  \sigma > c $,  
 +
where $  c $
 +
is the abscissa of convergence of the series.
  
The following theorem is valid for an ordinary Dirichlet series (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018030.png" />) with a known asymptotic behaviour for the sum-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018031.png" /> of the coefficients of the series: If
+
The following theorem is valid for an ordinary Dirichlet series (when $  \lambda _ {n} = \mathop{\rm ln} \  n $)  
 +
with a known asymptotic behaviour for the sum-function $  A _ {n} = a _ {1} + \dots + a _ {n} $
 +
of the coefficients of the series: If
  
<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/a/a010/a010180/a01018032.png" /></td> </tr></table>
+
$$
 +
A _ {n} \  = \  B \  n ^ {s _ 1} (  \mathop{\rm ln} \  n )  ^  \alpha
 +
+ O ( n  ^  \beta  ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018033.png" /> are complex numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018034.png" /> is a real number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018036.png" />, then the Dirichlet series converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018037.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018038.png" /> can be regularly extended to the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018039.png" /> with the exception of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018040.png" />. Moreover
+
where $  B ,\  s _ {1} ,\  \alpha $
 +
are complex numbers, $  \beta $
 +
is a real number, $  \sigma _ {1} - 1 < \beta < \sigma _ {1} $,  
 +
$  \sigma _ {1} = \mathop{\rm Re} \  s _ {1} $,  
 +
then the Dirichlet series converges for $  \sigma _ {1} < \sigma $,  
 +
and the function $  \phi (s) $
 +
can be regularly extended to the half-plane $  \beta < \sigma $
 +
with the exception of the point $  s = s _ {1} $.  
 +
Moreover
  
<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/a/a010/a010180/a01018041.png" /></td> </tr></table>
+
$$
 +
\phi ( s ) \  = \  B \  \Gamma ( \alpha + 1 ) s ( s - s _ {1} ) ^ {- \alpha - 1} + g ( s )
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018042.png" /> and
+
if $  \alpha \neq -1,\  -2 \dots $
 +
and
  
<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/a/a010/a010180/a01018043.png" /></td> </tr></table>
+
$$
 +
\phi ( s ) \  = \  B
 +
\frac{( - 1 ) ^ {- \alpha}}{( - \alpha - 1 ) !}
 +
s ( s - s _ {1} ) ^ {- \alpha - 1} \
 +
\mathop{\rm ln} ( s - s _ {1} ) + g ( s )
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018044.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018045.png" /> is a regular function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018046.png" />.
+
if $  \alpha = -1,\  -2 ,\dots $.  
 +
Here $  g(s) $
 +
is a regular function for $  \sigma > \beta $.
  
E.g., the Riemann zeta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018047.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018052.png" />) is regular at least in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018053.png" />, with the exception of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018054.png" /> at which it has a first-order pole with residue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018055.png" />. This theorem can be generalized in various ways. E.g., if
+
E.g., the Riemann zeta-function $  \zeta (s) $(
 
+
$  A _ {n} = 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/a/a010/a010180/a01018056.png" /></td> </tr></table>
+
$  B = 1 $,  
 
+
$  s _ {1} = 1 $,  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018057.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018058.png" />), are arbitrary complex numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018059.png" />, then the Dirichlet series converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018061.png" /> is regular in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018062.png" /> with the exception of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018063.png" /> at which it has algebraic or logarithmic singularities. Theorems of this type yield information on the behaviour of the Dirichlet series in a given half-plane, based on the asymptotic behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018064.png" />.
+
$  \alpha = 0 $,  
 
+
$  \beta > 0 $)  
====References====
+
is regular at least in the half-plane $  \sigma > 0 $,  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.H. Abel, , ''Oeuvres complètes, nouvelle éd.'' , '''1''' , Grondahl &amp; Son , Christiania (1881) (Edition de Holmboe)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.H. Abel, "Untersuchungen über die Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010180/a01018065.png" />" ''J. Reine Angew. Math.'' , '''1''' (1826) pp. 311–339 {{MR|}} {{ZBL|26.0277.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR></table>
+
with the exception of the point $  s = 1 $
 +
at which it has a first-order pole with residue $  1 $.  
 +
This theorem can be generalized in various ways. E.g., if
  
 +
$$
 +
A _ {n} \  = \  \sum _ {j = 1} ^ { k }  B _ {j} n ^ {s _ j}
 +
(  \mathop{\rm ln} \  n ) ^ {\alpha _ j} + O ( n  ^  \beta  ) ,
 +
$$
  
 +
where  $  B _ {j} ,\  s _ {j} ,\  \alpha _ {j} $(
 +
$  1 \leq  j \leq  k $),
 +
are arbitrary complex numbers, and  $  \sigma _ {k} - 1 < \beta < \sigma _ {k} < \dots < \sigma _ {1} $,
 +
then the Dirichlet series converges for  $  \sigma > \sigma _ {1} $,
 +
and  $  \phi (s) $
 +
is regular in the domain  $  \sigma > \beta $
 +
with the exception of the points  $  s _ {1} \dots s _ {k} $
 +
at which it has algebraic or logarithmic singularities. Theorems of this type yield information on the behaviour of the Dirichlet series in a given half-plane, based on the asymptotic behaviour of  $  A _ {n} $.
  
 
====Comments====
 
====Comments====
Line 52: Line 130:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Clarendon Press (1949) {{MR|0030620}} {{ZBL|0032.05801}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> N.H. Abel, , ''Oeuvres complètes, nouvelle éd.'' , '''1''' , Grondahl &amp; Son , Christiania (1881) (Edition de Holmboe)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> N.H. Abel, "Untersuchungen über die Reihe $1+mx/2+m(m-1)x^2/(2\cdot 1)+\cdots$" ''J. Reine Angew. Math.'' , '''1''' (1826) pp. 311–339 {{MR|}} {{ZBL|26.0277.02}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Clarendon Press (1949) {{MR|0030620}} {{ZBL|0032.05801}} </TD></TR></table>

Latest revision as of 06:14, 26 March 2023


Abel's theorem on algebraic equations: Formulas expressing the solution of an arbitrary equation of degree $ n $ in terms of its coefficients using radicals do not exist for any $ n \geq 5 $. The theorem was proved by N.H. Abel in 1824 . Abel's theorem may also be obtained as a corollary of Galois theory, from which a more general theorem follows: For any $ n \geq 5 $ there exist algebraic equations with integer coefficients whose roots cannot be expressed in terms of radicals of rational numbers. For a modern formulation of Abel's theorem for equations over an arbitrary field, see Algebraic equation.

Abel's theorem on power series: If the power series

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

where $ a _ {k} ,\ b,\ z $ are complex numbers, converges at $ z = z _ {0} $, then it converges absolutely and uniformly within any disc $ | z - b | \leq \rho $ of radius $ \rho < | z _ {0} - b | $ and with centre at $ b $. The theorem was established by N.H. Abel . It follows from the theorem that there exists a number $ R \in [ 0,\ \infty ] $ such that if $ | z - b | < R $ the series is convergent, while if $ | z - b | > R $ the series is divergent. The number $ R $ is called the radius of convergence of the series (*), while the disc $ | z - b | < R $ is known as the disc of convergence of the series (*).

Abel's continuity theorem: If the power series

converges at a point $ z _ {0} $ on the boundary of the disc of convergence, then it is a continuous function in any closed triangle $ T $ with vertices $ z _ {0} ,\ z _ {1} ,\ z _ {2} $, where $ z _ {1} ,\ z _ {2} $ are located inside the disc of convergence. In particular

$$ \lim\limits _ {\begin{array}{c} z \rightarrow z _ {0} , \\ z \in T \end{array} } \ S ( z ) \ = \ S ( z _ {0} ) . $$

This limit always exists along the radius: The series

converges uniformly along any radius of the disc of convergence joining the points $ b $ and $ z _ {0} $. This theorem is used, in particular, to calculate the sum of a power series which converges at the boundary points of the disc of convergence.

Abel's theorem on Dirichlet series: If the Dirichlet series

$$ \phi ( s ) \ = \ \sum _ {n = 1} ^ \infty a _ {n} e ^ {- \lambda _ {n} s} , \ \ s = \sigma + it ,\ \ \lambda _ {n} > 0 , $$

converges at the point $ s _ {0} = \sigma _ {0} + i t _ {0} $, then it converges in the half-plane $ \sigma > \sigma _ {0} $ and converges uniformly inside any angle $ | \mathop{\rm arg} (s - s _ {0} ) | \leq \theta < \pi / 2 $. It is a generalization of Abel's theorem on power series (take $ \lambda _ {n} = n $ and put $ e ^ {-s} = z $). It follows from the theorem that the domain of convergence of a Dirichlet series is some half-plane $ \sigma > c $, where $ c $ is the abscissa of convergence of the series.

The following theorem is valid for an ordinary Dirichlet series (when $ \lambda _ {n} = \mathop{\rm ln} \ n $) with a known asymptotic behaviour for the sum-function $ A _ {n} = a _ {1} + \dots + a _ {n} $ of the coefficients of the series: If

$$ A _ {n} \ = \ B \ n ^ {s _ 1} ( \mathop{\rm ln} \ n ) ^ \alpha + O ( n ^ \beta ) , $$

where $ B ,\ s _ {1} ,\ \alpha $ are complex numbers, $ \beta $ is a real number, $ \sigma _ {1} - 1 < \beta < \sigma _ {1} $, $ \sigma _ {1} = \mathop{\rm Re} \ s _ {1} $, then the Dirichlet series converges for $ \sigma _ {1} < \sigma $, and the function $ \phi (s) $ can be regularly extended to the half-plane $ \beta < \sigma $ with the exception of the point $ s = s _ {1} $. Moreover

$$ \phi ( s ) \ = \ B \ \Gamma ( \alpha + 1 ) s ( s - s _ {1} ) ^ {- \alpha - 1} + g ( s ) $$

if $ \alpha \neq -1,\ -2 \dots $ and

$$ \phi ( s ) \ = \ B \frac{( - 1 ) ^ {- \alpha}}{( - \alpha - 1 ) !} s ( s - s _ {1} ) ^ {- \alpha - 1} \ \mathop{\rm ln} ( s - s _ {1} ) + g ( s ) $$

if $ \alpha = -1,\ -2 ,\dots $. Here $ g(s) $ is a regular function for $ \sigma > \beta $.

E.g., the Riemann zeta-function $ \zeta (s) $( $ A _ {n} = n $, $ B = 1 $, $ s _ {1} = 1 $, $ \alpha = 0 $, $ \beta > 0 $) is regular at least in the half-plane $ \sigma > 0 $, with the exception of the point $ s = 1 $ at which it has a first-order pole with residue $ 1 $. This theorem can be generalized in various ways. E.g., if

$$ A _ {n} \ = \ \sum _ {j = 1} ^ { k } B _ {j} n ^ {s _ j} ( \mathop{\rm ln} \ n ) ^ {\alpha _ j} + O ( n ^ \beta ) , $$

where $ B _ {j} ,\ s _ {j} ,\ \alpha _ {j} $( $ 1 \leq j \leq k $), are arbitrary complex numbers, and $ \sigma _ {k} - 1 < \beta < \sigma _ {k} < \dots < \sigma _ {1} $, then the Dirichlet series converges for $ \sigma > \sigma _ {1} $, and $ \phi (s) $ is regular in the domain $ \sigma > \beta $ with the exception of the points $ s _ {1} \dots s _ {k} $ at which it has algebraic or logarithmic singularities. Theorems of this type yield information on the behaviour of the Dirichlet series in a given half-plane, based on the asymptotic behaviour of $ A _ {n} $.

Comments

More on Abel's theorems 2)–4) can be found in [a1].

References

[1] N.H. Abel, , Oeuvres complètes, nouvelle éd. , 1 , Grondahl & Son , Christiania (1881) (Edition de Holmboe)
[2] N.H. Abel, "Untersuchungen über die Reihe $1+mx/2+m(m-1)x^2/(2\cdot 1)+\cdots$" J. Reine Angew. Math. , 1 (1826) pp. 311–339 Zbl 26.0277.02
[3] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[a1] G.H. Hardy, "Divergent series" , Clarendon Press (1949) MR0030620 Zbl 0032.05801
How to Cite This Entry:
Abel theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_theorem&oldid=24359
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article