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 modern framework of elliptic functions and analysis, what is the fundamental point of significance in this developement?
  • $\begingroup$ I think that $Al_0(u),Al_1(u),Al_2(u),Al_3(u)$ are essentially the four Weierstrass sigma functions and they are related to Jacobi's four theta functions. The significance is that they are entire with no poles. $\endgroup$
    – Somos
    Oct 15 at 23:24
  • $\begingroup$ @Somos - can you elaborate the last point (that they are entire with no poles) more comprehensively? And why are Weierstrass sigma functions considered seminal in the development of Weierstrass's version of elliptic functions? Also, what does Gauss's derivation of the coefficients of $Al_0(u), Al_1(u)$ tell us about the level of maturity of his ideas in this direction (i.e, towards Weierstrass's theory)? I ask these questions just to describe what is the type of discussion i want to read. $\endgroup$
    – user2554
    Oct 16 at 15:31
  • $\begingroup$ About Weierstrass, read Weierstrass functions which begins with defining sigma and then deriving zeta and p. The explicit product for sigma makes clear what its zeros are and no poles. For more on this read Weierstrass factorization theorem. $\endgroup$
    – Somos
    Oct 16 at 17:15

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