Elementary approach to the Hartogs extension theorem

In this paper we present a proof of Hartogs’ extension theorem, following T. Sobieszek’s paper from 2003. Hartogs’ theorem provides a large class of domains where holomorphic functions have analytic continuation to larger domains, and is “a several complex variables theorem” in nature because its conclusion is false in the complex plane. Sobieszek’s proof is quite remarkable because he uses, stated in his paper without proofs, only higher-dimensional identity principle for holomorphic functions and Cauchy’s integral formula for compact sets. We proved this two theorems here, making this exposition self-contained. The only background required is an undergraduate course in real and complex analysis and in point-set topology.

Elementaren pristop k Hartogsovemu razširitvenemu izreku

V članku predstavimo dokaz Hartogsovega razširitvenega izreka, kjer sledimo članku T. Sobieszeka iz leta 2003. Hartogsov izrek podaja veliko primerov domen, kjer imajo holomorfne funkcije analitično nadaljevanje na večje domene, in je izrazito “izrek analize več kompleksnih spremenljivk”, kajti na kompleksni ravnini ni resničen. Sobieszekov dokaz je precej izreden, saj v članku uporablja le večdimenzionalni princip identitete za holomorfne funkcije in Cauchyjevo integralsko formulo za kompaktne množice, oboje zapisano brez dokazov. S poudarkom na enostavnosti tega članka oba izreka tudi dokažemo. Edino potrebno predznanje je dodiplomski tečaj iz realne in kompleksne analize ter splošne topologije.