Geometrical Meaning of derivative of complex function

What's the geometrical meaning of $f'(z)$ in complex analysis, as we know in real analysis $f'(x)$ has meaning ie. Slope of curve or gives max/ min. But what does derivative $f'(z)$ has geometrical meaning in complex analysis

Solution 1:

Multiplying by a complex number other than $0$ consists of rotating and dilating. To multiply by $i$ is to rotate $90^\circ$ counterclockwise; to multiply by $4+3i$ is to rotate counterclockwise through the angle $\arctan(3/4)$ and dilate by $5=\sqrt{3^2+4^2}$, etc.

So say we have $$ \left.\frac{dw}{dz}\right|_{z=z_0} = f'(z_0). $$ Then at $z=z_0$ we have $dw=f'(z_0)\,dz$, i.e. if $dz$ is an infinitely small change in $z$, from $z_0$ to $z_0+dz$, then the corresponding infinitely small change in $w$ from $f(z_0)$ to $f(z_0) + dw$, is what you get from rotating and dilating $dz$ by the amounts corresonding to $f'(z_0)$.

This explains why holomorphic functions are conformal except at points where the derivative is $0$: where two curves intersect, the process of rotating does not change the angle between them since they're both rotated by the same amount, and of course dilating does not change the angle.

Solution 2:

The first-order approximation of a differentiable function $f$ around $z_0$ is $$ f(z_0+\Delta z) = f(z_0) + f'(z_0)\cdot \Delta z \text{.} $$

Now, complex multiplication geometrically corresponds to a uniformly scaled rotation (if $z = re^{i\varphi}$, then multiplying with $z$ means scaling with $r \in \mathbb{R}$ and rotating by angle $\varphi$). Thus, if a complex function is differentiable, it locally acts as a scaled rotation. In other words, if you pick some points close to some $z_0$, their image under $f$ will have (approximately) the same geometric shape, but will now be close to $f(z_0)$, will be rotated by angle $\varphi$ and scaled by factor $r$.