The Cauchy Integral Formula Beyond Analyticity
What survives of the Cauchy integral formula when the function is no longer analytical? A smooth version exists, and a single volume term measures exactly how far holomorphicity fails.
The main result
Dear Reader, I can finally return to my briefs in mathematics. Today I would like to revisit a thoroughly classical topic: the Cauchy integral formula, which belongs to the programme of any respectable university. The interesting question is what becomes of it once the function is no longer analytical.
There is a smooth version of the formula, and the hypothesis it needs is remarkably modest — smoothness suffices, in contrast to full analyticity. The classical statement reappears in it unchanged, joined by a single volume term built from . That term is the whole story: on a holomorphic function it vanishes, and in general it measures precisely how far the function fails to be holomorphic.
It can be found in books such as the celebrated Principles of Algebraic Geometry by Griffiths and Harris. Both statements are below, side by side.
Dr. Denys Dutykh
Associate Professor of Mathematics
Khalifa University of Science and Technology, Abu Dhabi, UAE
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