\[e^{z\sin\theta}=I_{0}\left(z\right)+2\sum_{k=0}^{\infty}(-1)^{k}I_{2k+1}\left(%
z\right)\sin\left((2k+1)\theta\right)+2\sum_{k=1}^{\infty}(-1)^{k}I_{2k}\left(%
z\right)\cos\left(2k\theta\right).\]