Abstract
Let D be a connected bounded domain in R2 , S be its boundary which is closed, connected and smooth or S = (−∞, ∞). Let Φ(z) = 1 2πi R S f(s)ds s−z , f ∈ L1 (S), z = x+iy. The singular integral operator Af := 1 iπ R S f(s)ds s−t , t ∈ S, is defined in a new way. This definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f ∈ L1 (S) to be boundary value of an analytic in D function. The Sokhotsky-Plemelj formulas are derived for f ∈ L1 (S). Our new definition allows one to treat singular boundary values of analytic functions.
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