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

304 Chapter 16. The idea of inverting elliptic integrals
Periods of φ. With the addition formulae in place, ABEL inserted β = ± ω 2 and β = ¯ω 2
to find by direct computation
�
φ α ± ω
�
= ±φ
2
Similarly, for the auxiliary functions
�
f α ± ω
�
2
�
F α ± ω
�
2
�
ω
� �
f (α)
and φ α ±
2 F (α) ¯ω
2 i
� �
¯ω
= ±φ
2 i
�
F (α)
f (α) .
�
φ � φ (α)
�
ω and f
F (α) 2 � �
and F α ±
F (α) ¯ω
2 i
�
= ∓ F � ω 2
= F � ω 2
�
α ± ¯ω
2 i
�
= f � ¯ω 2 i �
f (α) ;
= ∓ f � ¯ω
2 i �
φ � φ (α)
�
¯ω
2 i f (α) .
When he combined these and inserted e.g. α = α + ω 2 and β = ω 2 , ABEL found
�
φ (α + ω) = φ α + ω
2
= φ
�
ω
� −
2
F( ω 2 )
φ( ω 2 )
ω
�
+ = φ
2
F( ω 2 )
F(α)
φ(α)
F(α)
�
ω
�
� �
ω f α + 2
2 F � α + ω �
2
= −φ (α) .
In other words, φ (α + 2ω) = φ (α), and ABEL had discovered that 2ω was a period of
φ. Similarly, 2 ¯ωi was also found to be a period of φ.
The value of φ for any complex value α + βi of its argument could thus be found,
ABEL emphasized, from the values φ (α) , f (α) , F (α) and φ (iβ) , f (iβ) , F (iβ). Fur-
thermore, if
α + βi = � mω ± α ′� + � n ¯ω ± β ′� i
such that α ′ ∈ � 0, ω � � �
2 and β ′ ¯ω ∈ 0, 2 , the values of these six functions could be obtained
from the values of φ (α ′ ) , f (α ′ ) , F (α ′ ) and φ (β ′ i) , f (β ′ i) , F (β ′ i) by formulae
such as
φ (α) = φ � mω ± α ′� = ± (−1) m φ � α ′� .
Consequently, the value of φ (and of f and F) at any complex argument was determined
by the values of φ (α) (and f and F) in which α ∈ � 0, ω � � �
¯ω
2 or α ∈ 0, 2 i.
ABEL’S extension of the elliptic function φ to the entire complex plane may thus be
summarized in the following steps (see figure 16.2):
1. The elliptic function φ (α) was obtained by inversion of the elliptic integral on a
�
. Because the function was odd, it was simultane-
segment of the real axis � 0, ω 2
ously found for α ∈ � − ω 2 , 0� .
2. By a formal, imaginary substitution the function φ (iβ) was found for β ∈ � 0, ¯ω �
2 i
and consequently for β ∈ � − ¯ω 2 , 0� i. The value of φ (c,e) (iβ) was obtained from
the inversion of a related elliptic function φ (e,c) (β) on a segment of the real axis.