If we have a Beta distribution with $\alpha = 1/2$ and $\beta =1$ then it has moment generating function given by$$M(t) = \frac {\sqrt {\pi}\text{ }\text{erfi}\left (\sqrt {t} \right)} {2\sqrt {t}}.$$Is there a known formula and reference for the Legendre transform of $M(t)$, defined as $I(a) := \displaystyle\sup_{t} \left( t a - \log M(t) \right)$?
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