RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

340 Chapter 18. Tools in ABEL’s research on elliptic transcendentals
of J. LIOUVILLE (1809–1882), the theory of integration in finite terms established itself
as an independent theory investigated for its own results. In this context, the theory
and ABEL’S contribution to it have been well described in J. LÜTZEN’S biography of
LIOUVILLE. 21 Referring to LÜTZEN’S description, a presentation of ABEL’S argument
and a brief discussion of relevant points of ABEL’S contribution are included below.
18.2.1 Characterization by continued fractions
In the paper from 1826, 22 ABEL investigated conditions under which the integral
�
ρ dx
√R
(18.5)
could be reduced to the logarithmic expression
log p + q√R p − q √ . (18.6)
R
The article first dealt with this question of reduction, but as ABEL ultimately noticed,
the answer obtained was actually the answer to a more general question. ABEL noted
that in case the integral (18.5) could be represented by logarithmic functions in any
way, it would always have a representation of the form (18.6). ABEL promised a proof
of this assertion but never published one; it was eventually given by P. L. CHEBYSHEV
(1821–1894). 23
A non-empty class. ABEL found by direct differentiation that for
he would have
Writing dz in the form
dz =
z = log p + q√ R
p − q √ R ,
pq dR + 2 (p dq − q dp) R
.
(p 2 − q 2 R) √ R
M dx
dz =
N √ R with
M = pq dR
�
+ 2 p
dx dq
�
− qdp R and (18.7)
dx dx
N = p 2 − q 2 R, (18.8)
he had thus found that for such values of M and N,
� M dx
N √ R = log p + q√ R
p − q √ R .
ABEL concluded:
21 (Lützen, 1990, chapter IX).
22 (N. H. Abel, 1826d).
23 (Chebyshev, 1853).