Geometrical Interpretation of Cauchy Riemann equations? Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation.

Do the Cauchy Riemann equations have a geometric interpretation?
Suppose $f: \mathbf{C} \rightarrow \mathbf{C}$ is analytic near $a \in \mathbf{C}$. Geometrically, this can be interpreted as saying that the effect of $f$ on a small vector $h$ emanating from $a$ is just multiplication by the complex number $f'(a)$: $$ f(a+h) \approx f(a) +f'(a)h,$$ where the error tends to zero sublinearly as $|h| \rightarrow 0$. That is, the local effect of an analytic function is a rotation plus a scaling. (There's an equivalent interpretation of the real derivative.)



In general, if we multiply $z = x + iy$ by a fixed complex number $a + ib$, we have

$$(x + iy) \mapsto (a +ib)(x+iy) =(ax-by) + i(bx+ay).$$ 

Thinking of $\mathbf{C}$ as $\mathbf{R}^2$, this transformation corresponds to the matrix multiplication
$$ \begin{bmatrix}a&amp;-b\\b&amp;a\end{bmatrix} \begin{bmatrix} x\\y\end{bmatrix} = \begin{bmatrix} ax - by\\bx+ay\end{bmatrix}.$$

Notice the form of this matrix - it's the composition of a dilation and a rotation about the origin.

Back in $\mathbf{C}$, if $u$ and $v$ are continuously differentiable mappings from $\mathbf{R}^2$ into $\mathbf{R}$, then the Jacobian $J$ of $f = u + iv$ is

$$J = \begin{bmatrix}\frac{\partial u}{\partial x}&amp;\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&amp;\frac{\partial v}{\partial y}\end{bmatrix} .$$ 

$J$ describes the local effect of $f$. Now, for $f$ to be analytic, the Cauchy-Riemann equations require that $u$ and $v$ satisfy $ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$  and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $. But if that's true, then $J$ must be a dilative rotation, like the matrix above. So the Cauchy-Riemann equations express the fact that the local effect of an analytic mapping is a dilative rotation, that is, multiplication by a complex number.
Example of action of a derivative of complex functions. If f(z)=z^2, f'(z)=2z.
For example, at z=1+i, the derivative is f'(1+i)=2(1+i). So if you apply this to something like 0.01, a differential along the horizontal axis in the complex plane, you get 
2(i+1)(0.01)=0.02(1+i)=0.02+0.02i. The "2" in 2(i+1) stretches 0.01 and the 1+i turns it 45 degrees, since 0.02/0.02=1 and arctan(1)=45 degrees.  So you can approximate (1.01+i)^2 as 
(1+i)^2+2(i+1)(0.01) as a first step. This has the form f(z)+f'(z)dz roughly speaking. Or in words, value of function+amplitwist * differential. (Amplitwist is from Visual Complex Analysis by Needham)