# Abel theorem

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 $A _ {n}$.