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 October 1st, 2013, 07:46 PM #1 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 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?
 October 1st, 2013, 10:29 PM #2 Senior Member   Joined: Aug 2011 Posts: 334 Thanks: 8 Re: What is an analytic function
 October 2nd, 2013, 06:20 AM #3 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 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 $f(z)=u(x,y)+iv(x,y)$ it would have to satisfy $u_x= v_y$ and $u_y= - v_x$ and A function in polar form $f(z)= u(r, \theta) + i v(r,\theta)$ would have to satisfy $ru_r=v_\theta$ and $u_\theta= -rv_r$ does that sound right?
 October 3rd, 2013, 03:08 PM #4 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 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 .
 October 14th, 2013, 04:44 PM #5 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: What is an analytic function A more fundamental definition is this: A function, f(z), is said to be 'analytic' at $z= z_0$ if and only if there exist some neighborhood, A, of $z_0$ (an open set containing $z_0$) and a power series $\sum_{n=0}^\infty a_n(z- z_0)^n$ that converges to f(z) for all z in A. Of course, that power series must be the same as the "Taylor series", $\sum f^{(n)}(z_0)(z- z_0)^n$ 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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