# The Beltrami equation and Neumann series

Let $$\mu: \mathbb{C} \to D(0,1).$$ A quasiconformal map is a $$W^{1}_{2,loc}-$$solution to the Beltrami equation $$\bar{\partial}f = \mu \partial f$$. In this paper, the authors remark that one can formally solve for $$f$$ in this equation by the Neumann series $$\bar{\partial}f = \mu + \mu T \mu + \mu T \mu T \mu + \cdots$$ where $$T$$ is the Beurling transform $$Tf(z) = \partial\left(\frac{\bar{\partial}f}{\partial z}\right)^{-1}(z) = -\frac{1}{\pi} p.v \int_{\mathbb{C}}\frac{f(w) \ \mathrm dw}{(z-w)^2}.$$ I cannot see how, even formally, this is true, and no source I can find shows how to do this. What is the heuristic behind this?

The story of the solution of the Betrami equation using the Beurling transformation in $$L^2$$ has an elegant and elementary explanation in:
I can't see how it becomes precisely what you have quoted, but is similar. If we let $$f=(I-\mu L)^{-1}\mu$$, where $$L$$ the Beurling transform, then $$(I-\mu L)f=\mu$$, so $$f=\mu+\mu L f$$. If $$\varphi$$ satisfies $$\bar\partial \varphi=f$$, then $$\bar\partial (z+\varphi)=f=\mu+\mu L f=\mu+\mu\bar\varphi=\mu(\bar z+\bar\varphi)$$. So $$z+\varphi$$ is the solution you are looking for, modulo various details explained clearly in that paper.