Verify second order Cauchy Riemann equations

Question

How do I differentiate the equations in 12? I understand the hint, but I'm not sure how to act on it.

Answer 1

Just for convenience I will rename the variables $\sigma \mapsto x$ and $\omega \mapsto y$ in the CR-equations. Then I'll show it for

$$\frac{\partial^2 u(x,y)}{\partial x^2} + \frac{\partial^2 u(x,y)}{\partial y^2} = 0.$$

Now note that

$$\frac{\partial^2 u(x,y)}{\partial x^2} = \frac{\partial}{\partial x} \frac{\partial u(x,y)}{\partial x} \overset{CR\,1}{=} \frac{\partial}{\partial x} \frac{\partial v(x,y)}{\partial y} = \frac{\partial}{\partial y} \frac{\partial v(x,y)}{\partial x} \overset{CR\,2}{=} - \frac{\partial}{\partial y} \frac{\partial u(x,y)}{\partial y} = - \frac{\partial^2 u(x,y)}{\partial y^2}$$

which states that

$$\frac{\partial^2 u(x,y)}{\partial x^2} + \frac{\partial^2 u(x,y)}{\partial y^2} = 0.$$

The case for $v(x,y)$ is handled analogously. Of course, we have implicitly used that the function $f(x,y) = u(x,y) + i v(x,y)$ is holomorphic, such that we are allowed to interchange the derivation with respect to the respective variables.

Answer 2

They are trying to confuse you by changing the variables. If in $(11)$ you replace all the $\sigma$'s with $x$ and all the $w$'s with $y$ are you able to continue? The meaning of the equations remains unchanged.

Answer 3

If you do not know how to differentiate, where did you get this problem? Or is it just the notation that is bothering you? The "Cauchy-Riemann" equations are $\frac{\partial u(x,y)}{\partial x}= \frac{\partial vx,y)}{\partial y}$ and $\frac{\partial u(x,y)}{\partial y}= -\frac{\partial v(x,y)}{\partial x}$ (I have simply changed the letters "$\sigma$" and "$\mu$" to "x" and "y".) Differentiating the first equation with respect to x, as you are told to do, you get $\frac{\partial}{\partial x}\left(\frac{\partial u(x,y)}{\partial x}\right)= \frac{\partial}{\partial x}\left(\frac{\partial v(x,y)}{\partial y}\right)$ $\frac{\partial^2 u(x,y)}{\partial x^2}= \frac{\partial v(x,y)}{\partial x\partial y}$