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New definition of a singular integral operator

Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA.

* Corresponding Author
Annals of Communications in Mathematics 2023
, 6 (4),
Received: 13 September 2023 |
Accepted: 15 November 2023 |
Published: 31 December 2023


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


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New definition of a singular integral operator.

Annals of Communications in Mathematics,

6 (4):
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  • Copyright (c) 2023 by the Author(s). Licensee Techno Sky Publications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (


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