The task is to evalute $ \int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}, \;\;\; \theta \in \mathbb{C}^2 \setminus ( \mathbb{S}^1 \cup \left\{ 0 \right\} ), \;\;\; \theta_1^2 + \theta_2^2 = 1,\;\;\; x\in \mathbb{R}^2 $ Obviously it is the Fourrier transform of $f(\xi) = \langle \xi,\theta \rangle ^ {-1}$. I tried to make the change $y_{1} = \langle\xi,\Re \theta\rangle, \;\; y_2 = \langle \xi, \Im \theta \rangle$, but i faced with difficult calculations.
Update: let $ f(x) = \frac{1}{2\pi i}\int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle} $ Then, using inverse Fourrier transform we get $ i\langle\theta,\xi\rangle \hat{f}(\xi) = \frac{1}{\sqrt{2\pi}} $ Taking Fourier transform now we recieve $ \langle \theta, \nabla f(x) \rangle = \delta(x) $ In other words, f(x) is a Green function for operator $L = \langle \theta, \nabla \rangle$.