Abel theorem

Abel's theorem on algebraic equations: Formulas expressing the solution of an arbitrary equation of degree in terms of its coefficients using radicals do not exist for any . 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 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

(*)

where are complex numbers, converges at , then it converges absolutely and uniformly within any disc of radius and with centre at . The theorem was established by N.H. Abel . It follows from the theorem that there exists a number such that if the series is convergent, while if the series is divergent. The number is called the radius of convergence of the series (*), while the disc is known as the disc of convergence of the series (*).

Abel's continuity theorem: If the power series

converges at a point on the boundary of the disc of convergence, then it is a continuous function in any closed triangle with vertices , where are located inside the disc of convergence. In particular

This limit always exists along the radius: The series

converges uniformly along any radius of the disc of convergence joining the points and . 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

converges at the point , then it converges in the half-plane and converges uniformly inside any angle . It is a generalization of Abel's theorem on power series (take and put ). It follows from the theorem that the domain of convergence of a Dirichlet series is some half-plane , where is the abscissa of convergence of the series.

The following theorem is valid for an ordinary Dirichlet series (when ) with a known asymptotic behaviour for the sum-function of the coefficients of the series: If

where are complex numbers, is a real number, , , then the Dirichlet series converges for , and the function can be regularly extended to the half-plane with the exception of the point . Moreover

if and

if . Here is a regular function for .

E.g., the Riemann zeta-function (, , , , ) is regular at least in the half-plane , with the exception of the point at which it has a first-order pole with residue . This theorem can be generalized in various ways. E.g., if

where (), are arbitrary complex numbers, and , then the Dirichlet series converges for , and is regular in the domain with the exception of the points 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 .

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 " 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

Comments

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

References

[a1]	G.H. Hardy, "Divergent series" , Clarendon Press (1949) MR0030620 Zbl 0032.05801