Is it possible to non-numerically evaluate $x$ in this equation
$\frac{\sqrt\pi}{B(1/2,x)} + g = 0$
where g is constant?
Is it possible to non-numerically evaluate $x$ in this equation
$\frac{\sqrt\pi}{B(1/2,x)} + g = 0$
where g is constant?
According to Maple, for large $|y|$, $B(1/2,x) = y$ where
$$ \eqalign{x &= {y}^{-1}+{\frac {2 \ln \left( 2 \right) }{{y}^{2}}}+{\frac {36
\, \left( \ln \left( 2 \right) \right) ^{2}-{\pi }^{2}}{6 {y}^{3}}}+{\frac {64\, \left( \ln \left( 2 \right) \right) ^{3}-4\,{\pi }^{
2}\ln \left( 2 \right) +6\,\zeta \left( 3 \right) }{3 {y}^{4}}}\cr&+{
\frac {11\,{\pi }^{4}+30000\, \left( \ln \left( 2
\right) \right) ^{4}-3000\,{\pi }^{2} \left( \ln \left( 2 \right)
\right) ^{2}+7200\,\zeta \left( 3 \right) \ln \left( 2 \right) }{360 {y
}^{5}}}+O \left( {y}^{-6} \right)\cr}$$
It appears the series converges for approximately $|y| > 1$.
Vis. Gerald's response,
This depends on your constraints on x. B(1/2,x) is positive over the positive reals.