Questions About the Proof of Cauchy–Pompeiu Integral Formula.
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I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am reading chapter 4, where he introduces the $bar{partial}$ -problem. However, he begins by proving a generalized form of the Cauchy Integral formula and I have some questions about the proof. Theorem: Let $Usubsetmathbb{C}$ be a bounded domain with piecewise $C^1$ -smooth boundary $partial U$ oriented positively, and let $f:bar{U}tomathbb{C}$ be a $ C^1$ -smooth function. Then for $ zin U$ : $ f(z)=frac{1}{2pi i}int_{partial U}frac{f(zeta)}{zeta-z}dzeta+frac{1}{2pi i}int_Udfrac{frac{partial f}{partialbar{z}}(zeta)}{zeta-z}dzetawedge dbar{zeta}$ . "Proof" : This is how he does it. He begin...