How can you prove the analyticity of this function using Cauchy-Riemann Equations

Question

Let $$f_{\varepsilon,+} = \frac{i}{\pi}\oint_{L_\varepsilon,+}\frac{h(t)}{t-z}dz$$

Assuming there's enough regularity and all hypothesis are satisfied for the derivatives to go through the integral. How can you prove its analytic?

Answer 1

Under your hypotheses all is noise but $\frac1{t-z} $. You need the real and imaginary parts $u$ and $v$: $$ 2u=2\operatorname{Re} \frac1{t-z} =\frac1{t-z} +\frac1{t-\overline z} =\frac{z+\overline z} {t^2+|z|^2-2\operatorname{Re}tz}=\frac{2x}{t^2+x^2+y^2-2tx}, $$ $$ 2iv=2i\operatorname{Im} \frac1{t-z} =\frac1{t-z} - \frac1{t-\overline z} =\frac{z-\overline z} {t^2+|z|^2-2\operatorname{Re}tz}=\frac{2iy}{t^2+x^2+y^2-2tx}. $$ Now you can check directly that $u$ and $v$ satisfy the Cauchy-Riemnan equations.