We show that on the space of smooth functions of compact support then the Cauchy transform inverts the operation of taking the ant-conformal part of the gradient (note that this is does not show that the operation of taking the ant-conformal part of the gradient is invertible and the kernel of this operator is all holomorphic mappings). We use this to show that we can take gradients "through" the Cauchy transform. We then define the Beurling transform and using a version of the "divergence theorem in the complex plane" we show that the Beurling transform is the Cauchy transform acting on on the conformal part of a gradient of a function. This shows that the limit in the definition of the Beurling transform is well defined. And also as a corollary we see that the we can take gradients "through" the Beurling transform.
