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

312 Chapter 16. The idea of inverting elliptic integrals
ABEL’S deduction. ABEL’S way of obtaining the described results was to expand the
function φ in a Taylor series
φ (x + v) =φ (x)
� �� �
=y
+
∞
∑ v
k=1
2k Q2k +
�
ψ (y)
With y = a k and α k equal to the corresponding value of x,
ABEL found
� ak
αk =
0
φ (α k + v) = a k +
f (y) dy
� ψ (y) ,
∞
∑
k=0
∞
∑ v
k=1
2k Q2k and thus in this case, φ (α k + v) was an even function of v,
By inserting v ′ = α k − v, ABEL obtained
Therefore,
φ (α k + v) = φ (α k − v) .
φ � 2α k − v ′� = φ � v ′� .
φ (2α k − 2αm + v) = φ (v) ,
and the function was therefore periodic. In general
�
�
φ
v + 2
m
∑
k,k ′ =1
n k,k ′ (α k − α k ′)
In particular, by taking for v a zero of φ, the values
were also zeros of φ.
v + 2
16.2.7 Conclusion
m
∑
k=1
n kα k for n1, . . . , nm ∈ Z with
= φ (v) .
v 2k+1 Q 2k+1.
m
∑ nk = 0
k=1
Because ABEL’S general inversion — which admittedly did not explicitly concern el-
liptic integrals — was written before he embarked on the European tour in 1825, it
cannot rely on any knowledge of the new theory of complex integration which was
presented that same year. Also admittedly, the manuscript does not contain any com-
plex integrals or complex periods but I find the suggested application of the result
plausible. I do so, because I read ABEL’S inversion of elliptic integrals in the Recherches
rather literally and see in it a formal substitution without any justification in complex
integration. This theme will surface again when the need and means of representa-
tions for elliptic functions are discussed in chapter 17.