In a fragment entitled "inversion of the elliptic integral of the first genus" (Gauss's werke, volume 8, p. 96-97), Gauss inverts the general elliptic integral of the first kind: he writes $\int\frac{dx}{\sqrt{(1-x^2)(1-\mu x^2)}} = \varphi$, and by a certain algebraic developement derives the inverse function
$$x = f(\varphi) = \frac{P(\varphi)}{Q(\varphi)} = \frac{\varphi+A_1\varphi^3+A_2\varphi^5+A_3\varphi^7+...}{1+B_2\varphi^4+B_3\varphi^6+B_4\varphi^8+...}$$
where the coefficients $A_1,A_2,A_3...$ and $B_2,B_3,B_4...$ are polynomials in $\mu$.
Initially, I thought this result is only one of many others that Gauss obtained on inversion of elliptic integrals, but than i saw that in his remarks on Gauss's fragment, Fricke writes:
This note, the date of which is presumed to be based on a diary from May 6, 1806..., is one of the most interesting Gauss made in the field of elliptical functions... Objectively, the present developement is illuminated in a very interesting way by the letter from Gauss to Bessel printed above. There Gauss repeatedly emphasizes the importance of entire transcendent functions. Here the function $f\varphi$... is represented in an elegant developement as the quotient of two functions $P\varphi,Q\varphi$, which Gauss undoubtedly knew its property as complete transcendent functions. It may also be remarked that the functions $P,Q$, which Gauss introduces here, are none other than those which Weierstrass later called $Al(\varphi)_1,Al(\varphi)_0$, in connection with Abel's work.
I also checked in volume 1 of Weierstrass's werke, and in a paper "on the developement of modular functions", Weierstrass also derives $A_1,A_2,A_3,B_2,B_3,B_4$ (and he adds more coefficients). The coefficients he writes down are exactly those which Gauss wrote, so i didn't misunderstand Fricke's remark.
I tried to search for information on the so-called "Weierstrass's Al functions" and i didn't find much material in the internet on those functions (except the lecture notes "Karl Weierstraß and the theory of Abelian and elliptic functions", and two other sources). In a synopsis of volume 8 of Gauss's werke, the author describes Gauss's derivation as "nothing more nor less than Weierstrass's celebrated expression of $x$ as the quotient of two entire transcendental functions, viz, $Al_1(u),Al_0(u)$", so it seems that there is really an important point here in the work of both Gauss and Weierstrass.
Therefore, my question is:
- From the perspective of the modern framework of elliptic functions and analysis, what is the fundamental point of significance in this developement?