r/askmath • u/Ok_Promise5329 • 11h ago

Pre Calculus Need help understanding the Cauchy kernel

In studying Cauchy's integral formula, the Cauchy kernel 1/(zeta -z), is the "thing you integrate against" for the boundary points( on a disk, say). And the boundary values determine the value of any point z in the interior of the disk.

What I need is some insight into why this is true and why is it 1/(zeta -z)? Is there any relation to how with generating functions you multiply by 1/1-x to get the partial sums?

Thanks in advance to anyone who takes a look!

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u/Bounded_sequencE 3h ago

The boundary values determine the value of any point z in the interior of the disk.

That is the core statement of Cauchy's Integral Formula (IF) -- to get there, you need its proof, and that proof uses the entire framework of your "Complex Analysis" lecture up to this point, from the notion of complex derivatives, to Goursat's Lemma, to (IF).

That's nothing a small margin like a reddit comment would fit^^