I am interested to do the proof of Runge's Approximation theorem by using the pompeiu lemma instead of well-known proof by using Cauchy Integral formula.
I have come up with some ideas. I am thinking to extend that holomorphic function to smooth function with compact support.But the question is can we extend that function to the desired function? If I am able to find such extension then we can express it in Pompeiu lemma in which boundary integral is not involved.After that, we can approximate that integral by rational function(but I am not sure how to do it).
If someone can give me the hint and motivation behind this.It will be appreciable.