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Is it possible to non-numerically evaluate $x$ in this equation

$\frac{\sqrt\pi}{B(1/2,x)} + g = 0$

where g is constant?

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  • $\begingroup$ Once it seemed to me that B(m.n) is inverse to B(1/m,1/n) indeed numerically there is certain similarity, but it is not true. What do you mean by B(1/2, x) ? What are m, n? $\endgroup$ Apr 24, 2012 at 13:01
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    $\begingroup$ Well, $$B(1/2,x) = \int_0^1 \frac{u^{x-1}du}{\sqrt{1-u}},$$ and we want to solve $B(1/2,x) = -\sqrt{\pi}/g$. $\endgroup$ Apr 24, 2012 at 14:28

2 Answers 2

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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 $|y| > 1$.

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Vis. Gerald's response,

This depends on your constraints on x. B(1/2,x) is positive over the positive reals.

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