Let $\gamma$ be Euler constant and $W$ Lambert W function.

One can show:

$$-2/3\,{\frac {\gamma+\ln \left( \pi \right) }{W \left( -1/3\,{\frac { \left( \gamma+\ln \left( \pi \right) \right) { {\rm e}^{-1/3\,\gamma}}}{\sqrt [3]{\pi }}} \right) }} = 2 \qquad (1) $$

This means at least one of $ \left( -1/3\,{\frac { \left( \gamma+\ln \left( \pi \right) \right) { {\rm e}^{-1/3\,\gamma}}}{\sqrt [3]{\pi }}} \right)$ and $\gamma+\ln \left( \pi \right)$ is transcendental.

Q1 Is this known?

Q2 Can one solve (1) for $\gamma$ or $\pi$? (sage, maple and Wolfram Alpha couldn't).

In machine readable form:

 def eulerpi():
      import mpmath
      from mpmath import lambertw,euler,pi,log,exp
      a= -2*(euler + log(pi))/lambertw(-1/3*( (euler+log(pi))*exp(-euler/3))/pi**(1/3))/3-2
      print mpmath.chop(a)

By the definition of the Lambert function $W\left(-\frac13\alpha e^{-\alpha/3}\right)=-\frac\alpha3\,$ for any $\alpha$, so $$ -2/3\frac\alpha{W\left(-\frac13\alpha e^{-\alpha/3}\right)}=2. $$ Putting here $\alpha=\gamma+\ln\pi$ gives your formula.

  • $\begingroup$ I don't get numerical support for your second formula for alpha=4 and alpha=5, in both sage and Maple. Is it a CAS bug? (Though it works for gamma + log(pi)) $\endgroup$ – joro Dec 1 '13 at 15:35
  • $\begingroup$ @joro for some intervals the Lambert function is multiwalued en.wikipedia.org/wiki/Lambert_W_function. May be the program computes another branch. $\endgroup$ – Andrew Dec 1 '13 at 16:16
  • $\begingroup$ Btw, I tried to check if the stupid question is a trivial tautology, but branches of $W$ fooled me. $\endgroup$ – joro Dec 12 '13 at 15:20

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