Consider the complex number in polar form , and the complex function in polar form .
Show that the Cauchy-Riemann equations in polar form are
It is important to know the Cartesian representation of the complex number and the Cauchy-Riemann equations first.
Now use the multivariate product rule to take the the derivatives of , , , and .
For the function :
Since , , , and , we get
We repeat the same calculations for .
Now use the Cauchy-Riemann equations in Cartesian form and make the right substitutions.