Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\Delta$, and the resolvent $R=(-\Delta - zI)^{-1}$ is well defined and bounded in all of $L^2(\mathbb{R}^2)$.

I'm trying to find the integral kernel for this operator, that is a function $K(x,y)$ such that almost everywhere in $\mathbb{R}^2$:

$$(Ru)(x)=\int_{\mathbb{R}^2} u(y)K(x,y) dy$$

Let $f := ( - \Delta - z I)^{- 1} u$ and let $\mathcal{F}$ be the Fourier transform. Now

$$ ( - \Delta - z I) f = u \Rightarrow \mathcal{F} ( ( - \Delta - z I) f) =\mathcal{F} ( u) \Rightarrow ( \xi^2 - z) \hat{f} ( \xi) = \hat{u} ( \xi) . $$

Solving for $\hat{f}$ and applying $\mathcal{F}^{- 1}$ we arrive at (modulo some constant depending on your favourite definition of $\mathcal{F}$)

$$ f ( x) =\mathcal{F}^{- 1} \left( \frac{1}{\xi^2 - z} \hat{u} ( \xi) \right) =\mathcal{F}^{- 1} \left( \frac{1}{\xi^2 - z} \right) \ast u ( x) $$

So I need to calculate

$$ \int_{\mathbb{R}^2} \frac{e^{ix \cdot \xi}}{\xi^2 - z} d \xi $$

and I'm completely stuck. Does anybody have any ideas or references? Thanks.