Characteristic Function of the von Mises Distribution
Formula
For integer $n$, the von Mises distribution $\operatorname{vM}(\mu, \kappa)$’s characteristic function $\phi_{n}$ is given by:
$$ \phi_{n} = \dfrac{I_{n}(\kappa)}{I_{0}(\kappa)} e^{\mathrm{i}\mu n} $$
Here $I_{n}$ is the modified Bessel function of the first kind of order $n$. (See modified Bessel function of the first kind.)
Derivation
The probability density function of the von Mises distribution is as follows.
$$ f(\theta) = f(\theta; \mu, \kappa) = \dfrac{1}{2 \pi I_{0}(\kappa)} \exp (\kappa \cos (\theta - \mu)) $$
To compute the characteristic function, rearrange the expression as follows.
$$ \begin{align*} \phi_{n} &= \mathbb{E} \left[ e^{\mathrm{i}n\theta} \right] \\ &= \dfrac{1}{2\pi I_{0}(\kappa)} \int_{-\pi}^{\pi} e^{\mathrm{i}n\theta} e^{\kappa \cos (\theta - \mu)} \mathrm{d}\theta \\[1em] &\quad (\text{Change of variable: $x = \theta - \mu$}) \\ &= \dfrac{1}{2\pi I_{0}(\kappa)} \int_{-\pi}^{\pi} e^{\mathrm{i}nx} e^{\mathrm{i}n\mu} e^{\kappa \cos x} \mathrm{d}x \\ &= \dfrac{e^{\mathrm{i}n\mu}}{2\pi I_{0}(\kappa)} \int_{-\pi}^{\pi} e^{\mathrm{i}nx} e^{\kappa \cos x} \mathrm{d}x \\ &= \dfrac{e^{\mathrm{i}n\mu}}{2\pi I_{0}(\kappa)} \int_{-\pi}^{\pi} (\cos(nx) + \mathrm{i}\sin(nx)) e^{\kappa \cos x} \mathrm{d}x \quad (\text{Euler’s formula})\\ &= \dfrac{e^{\mathrm{i}n\mu}}{2\pi I_{0}(\kappa)} \left[ \int_{-\pi}^{\pi} \cos(nx) e^{\kappa \cos x} \mathrm{d}x + \mathrm{i}\int_{-\pi}^{\pi} \sin(nx) e^{\kappa \cos x} \mathrm{d}x \right] \end{align*} $$
Looking at the second integral, $\sin(nx)$ is an odd function and $e^{\kappa \cos x}$ is an even function, so the integrand is odd and the integral is $0$.
$$ \begin{align*} \phi_{n} &= \dfrac{e^{\mathrm{i}n\mu}}{2\pi I_{0}(\kappa)} \int_{-\pi}^{\pi} \cos(nx) e^{\kappa \cos x} \mathrm{d}x \\ &= \dfrac{e^{\mathrm{i}n\mu}}{\pi I_{0}(\kappa)} \int_{0}^{\pi} \cos(nx) e^{\kappa \cos x} \mathrm{d}x &= \dfrac{e^{\mathrm{i}n\mu}}{I_{0}(\kappa)} \left[ \dfrac{1}{\pi} \int_{0}^{\pi} \cos(nx) e^{\kappa \cos x} \mathrm{d}x \right] \end{align*} $$
Here the value inside the parentheses, since $n$ is an integer, is expressed as follows. (See modified Bessel function of the first kind.)
$$ \phi_{n} = \dfrac{e^{\mathrm{i}n\mu}}{I_{0}(\kappa)} I_{n}(\kappa) = \dfrac{I_{n}(\kappa)}{I_{0}(\kappa)} e^{\mathrm{i}\mu n} $$
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