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Background

Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've observed a number of definite integrals of this function in various sources that indicate a connection with Riemann zeta values.

These include:

  1. On p. 632, entry 6.141.2 of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products [2], we find $$\label{grad1} \int_{0}^{1} \textbf{K}(k') dk = \frac{\pi^{2}}{4} = \frac{3}{2} \zeta(2). \tag{1} $$
  2. On p. 632, entry 6.144 of [2], we also obtain $$\int_{0}^{1} \frac{\textbf{K}(k) }{1+k}dk = \frac{\pi^{2}}{8} = \frac{3}{4} \zeta(2) . \tag{2} $$
  3. In the answer to the following question [3] by Tieto Piezas III, user Nico Bagis confirms that $$\label{bagis} \int_{0}^{1} \left( \textbf{K}(k^{1/2})^{2} \right) dk = \frac{7}{2} \zeta(3). \tag{3} $$ Other similar, conjectured definite integrals are described in the question as well.
  4. In the answer to GEdgar's question [4], pisco establishes - with help from Bagis' work - that $$\int_0^1 \frac{\textbf{K}'(k)^4}{\textbf{K}(k)^2} k\;dk = \frac{31}{8} \zeta(5) . \tag{4}$$

As pointed out by Paul Enta, the identity in \eqref{bagis} is generalised for other moments of $\textbf{K}'^{2}, \textbf{E}'^{2} $ (where $\textbf{E}(k)$ is the complete elliptic integral of the second kind), $\textbf{K}'\textbf{E}', \textbf{K}^{2},$ and $\textbf{E}^{2} $ in an article by James Wan [5], where he shows these can be expressed as numbers of the form $a + b \zeta(3)$ with $a,b \in \mathbb{Q} $. Wan's paper also shows integrals of sums of products of three or four elliptic integrals can yield closed forms, like $$ \int_{0}^{1} \left(3 \textbf{E}'(k) \textbf{K}'(k) \textbf{K}(k) - \textbf{K}(k) \textbf{K}'(k)^{2} \right) dk = \frac{\pi^{3}}{8} \tag{5} $$ on p. 12. Moreover, Kirill points out in his/her answer to this question [6] by Vladimir Reshetnikov that $$ \int_0^1 \textbf{K}(k)^3\,dk = \frac{3}{5} \textbf{K}(1/\sqrt{2})^4 = \frac{3\Gamma \big{(}\frac{1}{4}\big{)}^8}{1280\pi^2}. \tag{6} $$

Questions

  1. Is anything known about the evaluation of integrals of the form $$I_{m} := \int_{0}^{1} \left( \textbf{K}(k^{1/m}) \right)^{m} dk \tag{7} $$ or $$I^{'}_{m} := \int_{0}^{1} \left( \textbf{K}\big{(}(k')^{1/m}\big{)} \right)^{m} dk \tag{8} $$ for $m>2$ ? To me, these appear like natural generalizations of \eqref{grad1} and \eqref{bagis}.
  2. Is there a more general theory that describes the relationship(s) between integrals of complete elliptic integrals of the first (or second) kind on the one hand, and zeta values at positive integer arguments multiplied by rational coefficients on the other hand?

Sources

[1] Weisstein, Eric W. "Complete Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html

[2] I.S. Gradshteyn, I. M. Ryzhik. "Table of integrals, series, and products." http://fisica.ciens.ucv.ve/~svincenz/TISPISGIMR.pdf, 2007. Seventh edition.

[3] Piezas III, T., Bagis, N. "Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?" link MSE, 2019.

[4] GEdgar, pisco, "Elliptic integrals and $\zeta(5)$." link MSE, 2020.

[5] Wan, J. "Moments of products of Elliptic Integrals" link ArXiv, 2011.

[6] Reshetnikov, V., Kirill, "Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$" link MSE, 2013

Max Muller
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  • (1) "To me, these appear like natural generalizations of ....". No, I'm afraid not, do not confuse superficial resemblance with natural generalizations. (2) You could just link answers in MSE, there is no need to cite them. (3) There are lots of definite integrals involving $K,E$, most of them are non-trivial. – pisco Aug 25 '22 at 14:02
  • @pisco Can you support your claim (1), for example with a closed form evaluation of $\int_{0}^{1} \textbf{K} (k^{1/3})^3 dk $ ? And if indeed these are not natural generalizations of eq's $(1)$ and $(3)$, do you know what definite integrals of complete elliptic integrals actually are natural generalizations of them? – Max Muller Aug 25 '22 at 14:14
  • The integrals with nice results you cited mostly come from modular forms: when $K$ is treated as weight 1 modular forms, the integrals correspond to values of their L-functions at integer. Here https://math.stackexchange.com/questions/1811490 contains wealth of info. If you perform the same to $\int_{0}^{1} K(k^{1/3})^3 dk$, you will see it fails to be a modular function disastrously, one can still hope to attack it via hypergeometric tools, but the outlook seems grim. I personally would refrain from using words like "natural generalizations“ to things whose nature is still elusive. – pisco Aug 25 '22 at 14:42
  • @pisco Fair enough, I will phrase such things differently in subsequent questions. Curious about the question you mention, will take a good look at it. – Max Muller Aug 26 '22 at 10:57
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    Now I am aware of $\int_{0}^{1}K(k^{1/3})^3\text{d}k =\frac{9,\Gamma\left ( \frac{1}{4} \right )^8 }{6400\pi^2} +\frac{36}{5}\beta(4)$. – Setness Ramesory Mar 19 '23 at 03:22
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    A new result: $$\int_{0}^{1}K(x^{1/4})^4\text{d}x =\frac{93}{8}\zeta(5)+ \frac{3\pi^6}{128} ,_9F_8\left ( \begin{array}{c} \frac12,\frac12,\frac12,\frac12,\frac12,\frac12,\frac12,\frac12,\frac54\ \frac14,1,1,1,1,1,1,1 \end{array}\Bigg|\text{ }1 \right )$$ – Setness Ramesory Apr 30 '23 at 06:35

0 Answers0