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## Relating two integral functions

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### Relating two integral functions

Thu Oct 10, 2013 6:41 pm

Posts: 1

Greetings from a new-comer ...

I'm more a combinatorial geometer than an real analyst, so I'm seeking assistance with an integral-based conundrum.

Define these (special-to-me) functions for $1/3 \leq x \leq 1/2$:
\begin{align}L_3(x) &:= \frac{3}{2} \int_{1/3}^x \; \frac{1}{\sqrt{t(1-t)}} \;\operatorname{atanh}\sqrt{\frac{3t-1}{1-t}} \; dt \\[8pt] H_3(x) &:= 6 \int_{1/3}^x \;\; \frac{1}{\sqrt{1-t^2}} \;\; \operatorname{atanh}\sqrt{\frac{3t-1}{1-t}} \; dt \end{align}

I'm looking for (non-trivial!) somehow-related functions $f$ and $g$ such that $\displaystyle f(H_3) \equiv g(L_3)$ (that is, equal for all $x$).
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The context of this problem is hyperbolic 4-space, but let's start a little simpler.

In the hyperbolic plane, a right triangle with legs of length $A_1$ and $B_1$ and hypotenuse of length $H_1$ gives rise to the hyperbolic Pythagorean Theorem
$$$$\cosh H_1 = \cosh A_1 \cosh B_1$$$$
In hyperbolic 3-space, a "right-corner" tetrahedron with right-triangle leg-faces of area $A_2$, $B_2$, $C_2$ and hypotenuse-face of area $H_2$ gives rise to this relation
$$$$\cos\frac{H_2}{2} = \cos{\frac{A_2}{2}}\cos{\frac{B_2}{2}}\cos{\frac{C_2}{2}} - \sin{\frac{A_2}{2}}\sin{\frac{B_2}{2}}\sin{\frac{C_2}{2}}$$$$
In hyperbolic 4-space, the analogous simplex with right-corner leg-tetrahedra of volume $A_3$, $B_3$, $C_3$, $D_3$ and hypotenuse-tetrahedron of volume $H_3$ gives rise to ... no known relation. Ultimately, I want to find the general relation here (if there is one), but for now, I'm concentrating on the special "isosceles" case, where $A_3 = B_3 = C_3 = D_3 =: L_3$. The integrals for $L_3$ and $H_3$ reduce to the formulas given above. (The parameter $x$ is related to a dihedral angle in the configuration.)

By analogy with the lower-dimensional cases, when I say that I'm looking for $f(H_3)\equiv g(L_3)$ for "somehow-related" functions $f$ and $g$, I'm anticipating $f$ to be "cosine-like", and for $g$ to effectively be a four-parameter function $g(w, x, y, z)$ with all parameters equal (for now) and with "cosine-like" (and/or "sine-like") effects on each parameter. However, given the dramatic difference between the 2- and 3-dimensional identities, I really have no idea what I should be expecting here.

I'll close by mentioning the special-special case where $x = 1/2$, for which we get
$$H_3^\star := \operatorname{Cl}_2\left(\frac{\pi}{3}\right) = \sum_{k=1}^{\infty} \frac{1}{k^2} \sin\frac{\pi k}{3} = 1.01494\dots \qquad L_3^\star := \frac{1}{2} \operatorname{Cl}_2 \left(\frac{\pi}{2}\right) = \frac{1}{2}\; \sum_{k=1}^{\infty} \frac{1}{k^2} \sin\frac{\pi k}{2} = 0.45978\dots$$
where $\operatorname{Cl}_2$ is one of the Clausen functions, coincidentally the subject of a thread by @DreamWeaver in the Tutorials section of this forum. If there's going to be an $f$ and $g$ that work at all, then they're going to have to work for $H^\star_3$ and $L^\star_3$. I've scanned DreamWeaver's tutorials (and other listings of relations among these kinds of functions), but I don't see an obvious connection here.

Any and all insights greatly appreciated.