1. ## Complex analysis question...

In...

Cauchy-Riemann Equations -- from Wolfram MathWorld

... is written that, given the function of x and y...

$\displaystyle f(x,y) = u(x,y) + i\ v(x,y)$ (1)

... and setting...

$\displaystyle z= x + i\ y \implies dz= dx + i\ dy$ (2)

... is...

$\displaystyle \displaystyle \frac{df}{dz} = \frac{\partial {f}}{\partial {x}}\ \frac{\partial {x}}{\partial {z}} + \frac{\partial {f}}{\partial {y}}\ \frac{\partial {y}}{\partial {x}} = \frac{1}{2}\ (\frac{\partial {f}}{\partial {x}} - i\ \frac{\partial {f}}{\partial {y}})$ (3)

It would for me very interesting to have more detail about the steps to arrive at the result (3). Any help will be greatly appreciated!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

2. As always with partial derivatives, you have to ask what the "other variable(s)" are. When you see the notation $\displaystyle \frac{\partial {x}}{\partial {z}}$, it is implicitly telling you that x is a function of z and some other (unspecified) variable. The usual convention when dealing with functions of a complex variable is that the "other variable" is the complex conjugate $\displaystyle \overline{z}$, so that $\displaystyle x = \frac12(z+\overline{z})$ and $\displaystyle y = \frac1{2i}(z-\overline{z})$. It's then clear that $\displaystyle \frac{\partial {x}}{\partial {z}} = \frac12$ and $\displaystyle \frac{\partial {y}}{\partial {z}} = \frac1{2i}$, and equation (3) follows by using the chain rule. Notice that f is meant to be an analytic function, so it is not dependent on $\displaystyle \overline{z}$. Thus the derivative $\displaystyle \frac{df}{dz}$ can be written with straight d's rather than as a partial derivative.

In books on complex function theory, the condition for f to be analytic is sometimes written $\displaystyle \frac{\partial f}{\partial\overline{z}} = 0.$ In fact, the condition $\displaystyle \frac{\partial f}{\partial\overline{z}} = 0$ is exactly equivalent to the Cauchy–Riemann equations.

3. ThankYou very much!... sincerly I never suspected of this 'trap'!... it's a very powerful 'key'!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$