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


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
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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


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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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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