# Intuition behind Cauchy Riemann equations and power series representation

The Cauchy Riemann equations in effect say that a function $f(z) = u(z)+iv(z)$ can be approximated as roughly a scaled rotation

$$f(c+h) \approx f(c) + f'(c)h = f(c) + \begin{bmatrix}u_x & -v_x\\v_x & u_x\end{bmatrix} \begin{bmatrix}h_x \\ h_y \end{bmatrix}$$

Informally, if $f$ has this kind of approximation at every point in a domain then it admits a power series representation at each point.

Intuitively why should having the scaled rotation approximation give a full power series representation? Suppose at some point $c$ the Cauchy Riemann equations are satisfied but there does not exist a power series representation at $c$, then presumably there should be a geometric point of view to see that this is ridiculous and leads to $f$ not satisfying the Cauchy Riemann equations elsewhere.

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I am not sufficiently prepared to answer you now but I can point you to a book where you will find a very convincing treatment of your question: it is Visual Complex Analysis by Tristan Needham. –  Giuseppe Negro Dec 17 '12 at 4:08