Expectations of cosine under von Mises distribution

Question

I'm trying to work out the expectations of a few functions under the von Mises distribution:

$
p(theta mid mu, kappa) = frac{1}{2pi I_0(kappa)}
expleft{
 kappa cos left( theta - mu right)
right}
$

Specifically, for known $mu$ and $kappa$,

$
mathbb{E} left( costfrac{theta}{2} right) =
 frac{1}{2pi I_0(kappa)} int_0^{2pi}
 costfrac{theta}{2} expleft{
 kappa cos left( theta - mu right)
right} dtheta = ?
$

However, my bag of tricks for integrals is fairly small, and I've hit
a kind of brick wall.

Ultimately, I'm looking for expressions for $mathbb{E}(costheta/2)$,
$mathbb{E}(sintheta/2)$, $mathbb{E}(cos^2theta/2)$, and
$mathbb{E}(sin^2theta/2)$ for a given $mu$,$kappa$.

Is there a good lit. source (or trick) for integrals of this type?

Thanks!

Brad · Answer

Ok, after a weekend of wine, cheese and brain-bashing, I solved the integrals. For the sake of posterity:

$
leftlangle cos(theta/2) rightrangle =
 frac{1}{sqrt{2 pi kappa} I_0(kappa)} left[
 e^{ kappa} cos(mu/2) ;
  mathrm{erf} left( sqrt{2 kappa} sin(mu/2) right) -
 e^{-kappa} sin(mu/2) ;
  mathrm{erfi} left( sqrt{2 kappa} cos(mu/2) right)
 right]
$

$
leftlangle sin(theta/2) rightrangle =
 frac{1}{sqrt{2 pi kappa} I_0(kappa)} left[
 e^{ kappa} sin(mu/2) ;
  mathrm{erf} left( sqrt{2 kappa} sin(mu/2) right) +
 e^{-kappa} cos(mu/2) ;
  mathrm{erfi} left( sqrt{2 kappa} cos(mu/2) right)
 right]
$

$
leftlangle cos^2(theta/2) rightrangle =
frac{1}{2} left[
 1 + frac{I_1(kappa)}{I_0(kappa)} cos mu right]
$

$
leftlangle sin^2(theta/2) rightrangle =
frac{1}{2} left[
 1 - frac{I_1(kappa)}{I_0(kappa)} cos mu right]
$

And one more, because why not...

$
leftlangle cos(theta/2) sin(theta/2) rightrangle =
 frac{1}{2} frac{I_1(kappa)}{I_0(kappa)} sin mu
$

The one trick I hadn't tried prior to posting was the ticket: let $x = theta - mu$ and then use trig identities to simplify the expressions into known integrals.

Expectations of cosine under von Mises distribution

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