# Abel theorem

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## Abel theorem

[′ä·bəl ′thir·əm]
(mathematics)
A theorem stating that if a power series in z converges for z = a, it converges absolutely for | z | < |="">a |.
A theorem stating that if a power series in z converges to f (z) for | z | < 1="" and="" to="">a for z = 1, then the limit of f (z) as z approaches 1 equals a.
A theorem stating that if the three series with n th term an, bn, and cn = a0 bn + a1 bn-1+ ⋯ + anb0, respectively, converge, then the third series equals the product of the first two series.
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References in periodicals archive ?
Mathematics: ABEGG'S RULE, ABEL'S THEOREM, ARCHIMEDES' PROBLEM, BERNOULLI'S THEOREM, DE MOIVRE'S THEOREM, DE MORGAN'S THEOREM, DESARGUES' THEOREM, DESCARTES' RULE OF SIGNS, EUCLID'S ALGORITHM, EULER'S EQUATION/FORMULA, FERMAT'S PRINCIPLE, FOURIER'S THEOREM, GAUSS'S THEOREM, GOLDBACH'S CONJECTURE, HUDDE'S RULES, LAPLACE'S EQUATIONS, NEWTON'S METHOD/PARALLELOGRAM, PASCAL'S LAW/TRIANGLE, RIEMANN'S HYPOTHESIS
 Alekseev, V.B., Abel's Theorem in Problems and Solutions, Moscow State University, Moscow (1976).
It covers classical proofs, such as Abel's theorem, and topics not included in standard textbooks like semi-direct products, polycyclic groups, Rubik's Cube-like puzzles, and Wedderburn's theorem, as well as problem sequences on depth.
We proceed to prove an analogue of Abel's theorem for boundedly convergent double series.
He continues with Abel's theorem, the gamma function, universal covering spaces, Cauchy's theorem for non-holomorphic functions and harmonic conjugates.
He then considers the work of Lagrange, Galois and Kronecker in concert, the process of computing Galois groups, solvable permutation groups, and the lemniscate, including the lemniscatic function, complex multiplication and Abel's theorem. He kindly provides information on abstract algebra as well as hints on selected exercises and a very well-organized bibliography.

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