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

308 Chapter 16. The idea of inverting elliptic integrals
Thus, the function sin lemnα produces the upper limit of the lemniscate integral
whose value is α. This is the inverse function of the lemniscate integral and the direct
counterpart to (special case of) the elliptic function φ which ABEL later, independently,
introduced in his Recherches.
Was GAUSS’ sin lemn a complex function? Clearly, GAUSS had taken the step of
considering the inverse function of the lemniscate integral. He invested extensive
effort in developing representations by infinite series, infinite products, and ratios of
infinite series. In the literature, GAUSS is universally credited with the discovery of the
doubly periodic nature of the lemniscate function. 25 This claim is generally supported
by GAUSS’ consideration of the degree of the division problem for the lemniscate.
GAUSS found, and noted in his diary, that the division problem for the lemniscate into
n parts led to an equation of degree n 2 . 26 This may well have been GAUSS’ motivation
for considering complex values of the argument. In a manuscript, in which GAUSS
wrote the lemniscate function as the ratio of two infinite products
he stated formulae such as
�
4√
2P φ + 1
2 ω
�
which amounted to
sin lemnφ =
P (φ)
Q (φ) ,
�
= pφ and p φ + 1
2 ω
�
= − 4√ 2P (φ)
P (φ + ω) = 1
�
4√ p φ +
2 1
2 ω
�
= −P (φ) .
This demonstrated the periodic nature of P and a similar result was obtained for Q.
More interestingly, GAUSS also wrote
which would indicate that
P (iψω) = ie πψ2
P (ψω) and Q (iψω) = e πψ2
Q (ψω)
sin lemn (iψ) = i sin lemn (ψ)
and therefore produce the second period of sin lemnφ. GAUSS’ manuscripts also con-
tain numerous formulae expressing the addition and multiplication of the lemniscate
function. 27
24 “Valorem huius integralis ab x = 0 usque ad x = 1 semper per 1 2 ¯ω designamus. Variabilem x
respectu integralis per signum sin lemn denotamus, respectu vero complementi integralis ad 1 2 ¯ω
per cos lemn. Ita ut
�
sin lemn
dx
√ = x,
1 − x4 � �
1
cos lemn ¯ω −
2
�
dx
√ = x.”
1 − x4 (C. F. Gauss, 1863–1933, III, 404).
25 (Schlesinger, 1922–1933) and see (J. J. Gray, 1984, 102–103).
26 (ibid., 102).
27 [Ref]