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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

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

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|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} $.

====References====
<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>

====Comments====
More on Abel's theorems 2)–4) can be found in [[#References|[a1]]].

====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>