Analytic iff Holomorphic on open domainss of C

Margie Marx

Margie Marx

Answered

2022-01-14

Analytic iff Holomorphic on open domainss of C

Answer & Explanation

scoollato7o

scoollato7o

Expert

2022-01-15Added 26 answers

Step 1
The definition that an infinitely differentiable function f is holomorphic on an open subset UC if
fz=0
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 C
function (which reduces to the standard version when fz¯=0) and use this to prove that such functions are analytic.
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.

MoxboasteBots5h

MoxboasteBots5h

Expert

2022-01-16Added 35 answers

Step 1
I do not have a copy of that text book at hande, but the standard definition of holomorphic function is that it is a differentiable function, that is, that, for each z0 in its domain, the limit
limzz0f(z)f(z0)zz0
exists.Quite often, but not always, it is added to the definition the condition that the domain of f is an open subset of C

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