Answer & Explanation
The definition that an infinitely differentiable function f is holomorphic on an open subset
a perfectly good definition and is essentially just a restatement of the Cauchy Riemann equations. It seems in Griffiths and Harris they prove a generalized version of the Cauchy integral formula which holds for any
function (which reduces to the standard version when
If you want to learn about the equivalent definitions of holomorphic/complex analytic functions and theorems about their equivalence I'd suggest a book specifically on complex analysis.
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