Let us begin with some (standard, I think) definitions.

Def: An elliptic function is a doubly periodic meromorphic function on $\mathbb{C}$.

Def: An elliptic integral is an integral of the form $$f(x) = \int_{a}^x R\left(t,\sqrt{P(t)}\right)\ dt,$$ where $R$ is a rational fucntion of its arguments and where $P(t)$ is a third or fourth degree polynomial with simple roots.

I have often heard the claim that an elliptic function is (or can be) defined as the inverse of an elliptic integral. However, I have never seen a proof of this statement. As someone who is largely unfamiliar with the subject, most of the references I could dig up seem to refer to the special case of the Jacobi elliptic functions, which appear as inverse functions of the elliptic integrals of the first kind. Maybe the claim I'm referring to is simply talking about the special case of Jacobi elliptic functions, but I believe the statement holds in generality (I could be wrong).

So, can anyone provide a proof or reference (or counter-example) to something akin to the following?

Claim: The elliptic functions are precisely the inverses of the elliptic integrals, as I've defined them above. That is, every elliptic function arises as the inverse of some elliptic integral, and conversely every elliptic integral arises as the inverse of some elliptic function.

  • $\begingroup$ Maybe interesting: mhtlab.uwaterloo.ca/courses/me755/web_chap3.pdf. $\endgroup$ – Martín-Blas Pérez Pinilla Dec 15 '15 at 9:09
  • $\begingroup$ @Martín-BlasPérezPinilla I had actually seen this reference already, and it treats the elliptic integrals of the first and second kinds along with the Jacobi and Weirstrass elliptic functions. The general claim doesn't seem to be there. $\endgroup$ – EuYu Dec 15 '15 at 9:18
  • $\begingroup$ Short note: much of the theory of elliptic functions have inverses that can be expressed in terms of the elliptic integral of the first kind, and compositions of it with the inverse trigonometric functions. I don't believe I've seen anybody make a sizable theory of inverses of the second- or third-kind integrals. $\endgroup$ – J. M. is a poor mathematician Jan 24 '17 at 15:21

The claim as stated is not true. (E.g., if $R$ has only even powers of the second variable, the resulting function $f$ is the integral of a rational function.) What is true is that every general elliptic integral of this form can be expressed as a linear combination of integrals of rational functions and the three Legendre canonical forms (elliptic integrals of the first, second, and third kind). This is a classical result, and there are several different algorithms to reduce a general elliptic integral to this form, some of them implemented in common computer algebra systems.

A modern (freely available) reference with a list of classical references is here: B.C. Carlson, Toward Symbolic Integration of Elliptic Integrals, Journal of Symbolic Computation, 28 (6), 1999, 739–753

  • $\begingroup$ Thank you for the answer. The counter-example provided seems to be rather trivial, which is perhaps due to my poor definitions. What if we insist that the integrand must be of the form $\frac{A(x)+B(x)\sqrt{P(x)}}{C(x)+D(x)\sqrt{P(x)}}$, with $B$ or $D$ non-zero? I know of the decomposition in terms of the three elliptic integrals, but I am not interested in evaluating the elliptic integrals. I am simply curious whether the modern definition of elliptic functions as doubly periodic meromorphic functions can be realized as the inverses of elliptic integrals, as they were considered classically. $\endgroup$ – EuYu Dec 23 '15 at 6:56
  • $\begingroup$ Yes, that counterexample is trivial, but even with your additional assumption the integrand might be a sum of a standard elliptic integral and an integral of a rational function (say, a linear function, to make it easy), in which case it is not the inverse of an elliptic function. $\endgroup$ – Lukas Geyer Dec 23 '15 at 18:00

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