The Cauchy integral formula and distributional integration

ABSTRACT

We study the Cauchy representation formula for analytic functions on the unit disc whose pointwise boundary value function is distributionally integrable. We prove that the formula holds when the distributional boundary values exist, and give examples that show that it may not be true when that is not the case. We also prove a maximum principle for pointwise boundary values valid for functions with distributional boundary values.

KEYWORDS:

• Cauchy integral formula
• distributional boundary values of analytic functions
• distributional integral

AMS SUBJECT CLASSIFICATIONS:

• 30E20
• 46F10
• 26A39

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Example 4.3 shows that this may not hold if F does not have distributional boundary values.

2. One could also see that E must be empty by using Lemma 2.3, since if not empty it would have an isolated point and it is easy to see that that is not possible.

3. This also holds in several variables [31].

4. Another proof of the Zielézny result, for a general distribution, not necessarily the distributional boundary value of an analytic function, can be obtained from the fact that if a harmonic function and its conjugate have radial limits and the boundary distribution is bounded at a point then both have point values there [32].