Remmert's Theory of Complex Functions book is great for plenty of commentary on the history behind the mathematics introduced in each section.
The picture below is of the section which talks about Riemann's own derivation of the familiar Cauchy-Riemann conditions.
I have two questions:
- Where the items in the brackets of the numerator for equation (2) come from? That is, $\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial x}i\right)$ and $\left(\frac{\partial v}{\partial y} - \frac{\partial u}{\partial y}i\right)$
- Is (1) Riemann's way of writing $\frac{df(z)}{dz}$ ?
I am not advanced, and don't have a university degree in mathematics, but I do understand the basic idea that asserting the derivative is the same, no matter which direction the limit is taken, leads to the Cauchy-Riemann relations.
We start with
$$ \begin{align} du & = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy \\ \\ dv & = \frac{\partial v}{\partial x} dx + \frac{\partial v}{\partial y} dy \end{align}$$
We construct $du + dv i$
$$ \begin{align} du + dvi & = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy + \frac{\partial v}{\partial x} dx i + \frac{\partial v}{\partial y} dy i \\ \\ & = \left ( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial x} i \right)dx + \left (\frac{\partial u}{\partial y}\frac{1}{i} + \frac{\partial v}{\partial y} \right) dyi \\ \\ & = \left ( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial x} i \right )dx + \left (\frac{\partial v}{\partial y} - \frac{\partial u}{\partial y}i \right) dyi \end{align}$$
However I am not sure about the basis of the original "start with" equations.