I have a variable distributed Beta(a, b). I need to take expectations of an exponential function. $e^{c z}$.
Forgetting constants of proportionality throughout, the expectation is: $\int_0^1 e^{c z} z^{a - 1}(1 - z)^{b - 1} d z$
This is roughly in the form of a regularized regularized confluent hypergeometric function (e.g. try mathematica). No problem. If a = b, which is fine for me, then it can be further reduced to Bessel functions of the first kind.
However, if I want to take a conditional distribution. e.g. z > d, then I need to do something like the following (dropping the proportionality constant and leaving symmetric for simplicity)
$\int_d^{1} e^{c z}z^{a - 1}(1- z)^{a-1} d z$
Mathematica couldn't solve this, and I can't find any special form that this matches. Is this essentially intractable, or something like an incomplete regularized confluent hypergeometric function-if such things even exist?