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October 1st, 2013, 07:46 PM   #1
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What is an analytic function

Hi,

We are constantly talking about analytic functions in complex variables.

can someone help me to wrap my head around what an analytic function actually is?
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October 1st, 2013, 10:29 PM   #2
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Re: What is an analytic function

http://mathworld.wolfram.com/AnalyticFunction.html
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October 2nd, 2013, 06:20 AM   #3
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Re: What is an analytic function

so basically it is a function that satisfies the Cauchy-Riemann equations?

If the function is in rectangular form it would have to satisfy and

and

A function in polar form would have to satisfy and

does that sound right?
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October 3rd, 2013, 03:08 PM   #4
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Math Focus: Theory of analytic functions
Re: What is an analytic function

Cauchy Riemann equations are necessary conditions but not sufficient we have to prove that all partial derivatives exists and are continuous . In general we use the Taylor expansion to verify analytic functions .
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October 14th, 2013, 04:44 PM   #5
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Re: What is an analytic function

A more fundamental definition is this:

A function, f(z), is said to be 'analytic' at if and only if there exist some neighborhood, A, of (an open set containing ) and a power series that converges to f(z) for all z in A.

Of course, that power series must be the same as the "Taylor series", so that f must be infinitely differentiable. For complex valued functions of a complex variable, one can show that if f is once differentiable (and so satisfies the "Cauchy-Riemann" equations) it is infinitely differentiable and analytic.

If we require that all numbers be real numbers that same definition gives "real analytic" functions. But for real functions of a real variable, a function can be differentiable without being analytic. It is even possible to have a (real) function which is infinitely differentiable, so that its Taylor series about a point can be constructed and that series converges, but not to the given function.
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