I'm looking to compute an exact integral formula for the reciprocal of the double factorial function, $(2n-1)!!$, or just as easily for the reciprocal gamma function, $\Gamma\left(n+\frac{1}{2}\right)$. I found the post located here and that formula works well for me when, for example, I take $c := 1$.
However, there is another known formula that I'm looking to replicate, or at least find a suitable analog to. Namely, that for integers $n \geq 0$ we have that (this formula is found in the appendices of the Concrete Mathematics book, for example): \begin{align*} \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-n\imath t} e^{e^{\imath t}} dt & = \frac{1}{n!}. \end{align*} I have had a look around google and found page 3 of this article and the Hankel loop contour described here, though I am still struggling to find the analog to this formula for the double factorial function case. I believe that the integral formula above is derived from the contour integral for the reciprocal gamma function, but when I perform a change of variable in the loop contour formula and plugin $z \mapsto n + \frac{1}{2}$, Mathematica computes the following result when $n = 3$ (the expected result is acceptably $\frac{8}{105}$, or ideally $\frac{1}{105}$): \begin{align*} \frac{1}{4\sqrt{\pi}} \int_{-\pi}^{\pi} e^{-(n_3+\frac{1}{2})\imath t} e^{e^{\imath t}} dt & = -\frac{1}{21 e \sqrt{\pi}} + \frac{8}{105} \operatorname{erf}(1). \end{align*} The result is obviously close to the intended formula, so I'm thinking that perhaps it's an issue with the bounds on the integral. I would like to keep the bounds of integration finite as in the factorial function formula if possible. Does anyone have any thoughts, advice, or solutions for this problem?