Thread: When exactly do we use the Cauchy-Riemann equations?

1. When exactly do we use the Cauchy-Riemann equations?

I'm having a really hard time trying to figure out when to use the Cauchy-Riemann equations, u_x = v_y and u_y = -v_x. My textbook says that they are necessary, but not sufficient conditions for differentiability. My textbook also says that they can be used to show that a function is not differentiable at a point, by showing that the CR equations do not hold. So is that all they can be used for, to show non-differentiability?

2. Originally Posted by Alexrey
So is that all they can be used for, to show non-differentiability?
We can also use the definition of differentiability. For example $f(z)=\bar{z}$

$\dfrac{\overline{z+h}-\bar{z}}{h}=\dfrac{\bar{h}}{h}=e^{-2i\theta}$

so, the limit as $h\to 0$ depends on $\theta$ as a consequence , f is not differentiable at z.

3. I know that we can use the above to prove that a function is not differentiable, but lets say that I get a question that says, "Prove that the following function is NOT differentiable", then instead of using the above method that you mentioned, could I straight away use the Cauchy-Riemann equations to prove the statement?

Also my textbook says that if a function has continuous first order partial derivatives that satisfy the Cauchy-Riemann equations, then it is differentiable, so if I get a question that says, "The following function has continuous partial derivatives, prove that it is differentiable", then again can I use the CR equations instead of the usual method for proving differentiability that you used?

4. Originally Posted by Alexrey
I know that we can use the above to prove that a function is not differentiable, but lets say that I get a question that says, "Prove that the following function is NOT differentiable", then instead of using the above method that you mentioned, could I straight away use the Cauchy-Riemann equations to prove the statement?

Of course you can, $f(z)=\bar{z}=x-iy\Rightarrow u_x=1\neq -1=v_y$ so, f is not differentiable at $z=x+iy$

Also my textbook says that if a function has continuous first order partial derivatives that satisfy the Cauchy-Riemann equations, then it is differentiable, so if I get a question that says, "The following function has continuous partial derivatives, prove that it is differentiable", then again can I use the CR equations instead of the usual method for proving differentiability that you used?
Yes, you can.

5. Awesome, thanks very much!