How I can understand the meaning of analytical function ? Does it have any geometric representation? I know definition of analytic function and Cauchy theorem.
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3Conformal maps are one way to understand them geometrically. – timur Jan 02 '14 at 17:33
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The function can be represented by a power series. – Wintermute Jan 02 '14 at 17:35
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Consider a map $f\colon\mathbb{C}\to\mathbb{C}$ which is differentiable when considered as a map $\mathbb{R}^2\to\mathbb{R}^2$. Then its derivative at any point is a linear map $\mathbb{R}^2\to\mathbb{R}^2$, or if you wish (and I do), an $\mathbb{R}$-linear map $\mathbb{C}\to\mathbb{C}$. Analyticity requires this map to be $\mathbb{C}$-linear, i.e., multiplication by a complex number. Geometrically, this requires the map to be the composition of a uniform scaling and a rotation.
So there you have it: An analytic map needs to look like scaling and rotation around every point in its domain. This is equivalent to conformality, i.e., preservation of angles.
Harald Hanche-Olsen
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Your question is a little vague and effortless, so I will just give you two good references.
The first is "Visual Complex Analysis" by Tristan Needham. It is a fantastic book which explains complex analysis from the ground up. Perhaps the most important thing to understand is that multiplying $z$ by $e^{i\theta} = \cos\theta + i\sin\theta$ rotates $z$ by $\theta$. http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469
The other way that is very ftuiful, and the reason for this post (others may be interested), are complex phase plots. Visualizing complex functions is difficult because you would need four dimensions to see the graph of $f$. Some people plot the modulus ($|f(z)|$) and color the landscape to describe the phase (the angle $\theta$). On the other hand, this notice of the ams surveys "phase plots", which are really cool! http://www.ams.org/notices/201106/rtx110600768p.pdf
snar
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