I'm studying J. Duoandikoetxea's book Fourier Analysis. To provide an example of standard kernels he considers the operator $C_\Gamma$ on the Schwartz space $\mathcal{S} (\mathbb{R})$ given by the Cauchy integral $$ C_\Gamma f(z)=\frac{1}{2\pi i}\int_{-\infty}^\infty\frac{f(t)(1+iA'(t))}{t+iA(t)-z}\,dt\,, $$ where $\Gamma$ is the graph of the Lipschitz function $A(x)$. It is easy to see that this defines an analytic function in $\Omega_+=\{z=x+iy:y>A(x)\}$. However, without any intermediate step, then he claims that the boundary values $\lim_{\varepsilon\to0}C_\Gamma f(x+i(A(x)+\varepsilon))$ are given by $$ \frac12\left[\,f(x)+\frac i\pi\lim_{\varepsilon\to0}\int_{|x-t|>\varepsilon}\frac{f(t)(1+iA'(t))}{x-t+i(A(x)-A(t))}\,dt\right]. $$
I've done some searching on the web and found this article by Calderón which does exactly the same without any further explanation, so I'm wondering if I am missing something obvious here.
I would appreciate any help in trying to understand how they arrive at this expression.