Derivatives of Complex Functions

Because the complex derivative is defined using a limit, and we need the limit to be the same when approaching a point from any direction, we generally only consider interior points. That is, if a function $f(z)$ is defined at points @interior to and on a curve $C$, then $f^{\prime }(z)$ is only considered at points interior to $C$. This is automatically the case when the domain of definition of $f(z)$ is an open set.

We can still talk about points!

The usual rules for real derivatives work for complex derivatives too, that is, sum, product, quotient, power and chain rules all remain in effect.

However, the real derivatives rules don't help with functions like $\mathrm{Re}(z),\mathrm{Im}(z),\Vert z\Vert$ or $\mathrm{Arg}z.$ We could use the limit definition of the derivative, but there is a better way.

Also, we might have a complex function defined in terms of its real and imaginary parts rather than in terms of $z$, i.e. $f(z)=\frac{x}{x^2+y^2}+\frac{-y}{x^2+y^2}i$. We could try to rewrite $f(z)$ in terms of $z$, but this could be hard.

Moreover, it's not yet clear how to tell when a @complex-combination $u(x,y)+v(x,y)i$ of real functions $u(x,y)$ and $v(x,y)$ define a function $f(z)$ of $z=x+yi$.

The following theorem is super important and helps clear all of that up. It may be the most important theorem in complex analysis.

We can break this theorem down into a couple of parts - that if a function is analytic, the CR equations hold, and that if the CR equations hold, the function is analytic.

See this page for the proof of the converse theorem.

Some other potentially useful theorems:

If $f^{\prime }(z)=0$ at every point of a domain $D$, then $f(z)$ must be constant in $D$.

If $f(z)=u+vi$ is analytic in a domain $D$, and if either $u(x,y)$ or $v(x,y)$ is constant in $D$, then $f(z)$ is constant in $D$.

Also, here are the Cauchy-Reimann equations in terms of the modulus $r$ and argument $\theta$ of a functon of $z$, $f(z)=u(r,\theta )+v(r,\theta )i$:

$$r\frac{\partial u}{\partial r}=\frac{\partial v}{\partial \theta },r\frac{\partial v}{\partial r}=-\frac{\partial u}{\partial \theta }$$

And here are formulas to express the derivative function in such cases:

$$f^{\prime }(z)=e^{-\theta i}(\frac{\partial u}{\partial r}+\frac{\partial v}{\partial r}i)$$

or

$$f^{\prime }(z)=e^{-\theta i}(\frac{\partial v}{\partial \theta }-\frac{\partial u}{\partial \theta }i)$$

Given a complex function $f(z)=u(x,y)+v(x,y)i$, we can use a Jacobian matrix to represent the transformation from $(x,y)$ to $(u,v)$. The Jacobian matrix of $f$ is:

and provides a linear transformation that represents the function's behavior locally around a given point.

To satisfy the Cauchy-Riemann equations, this matrix must have the form:

where $a=\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $b=\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$.

The matrix form shows that the linear transformation corresponding to the derivative of $f$ (if $f$ is differentiable and satisfies the Cauchy-Riemann equations) is a rotation combined with a scaling. In fact, $\Vert J_f\Vert =\Vert f^{\prime }(z){\Vert }^2$ and gives the area magnification/scaling factor of $f$ at $z$ and $\text{atan2}(b,a)$ gives an argument of $f^{\prime }(z)$, which is a measure of the rotational effect of $f$.