Cauchy-Riemann equations for $z=x+iy$ and $f(z)=R(x,y)e^{itheta(x,y)}$
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I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake): $frac{partial u}{partial x} = frac{partial v}{partial y}$ , $frac{partial u}{partial y} = -frac{partial v}{partial x}$ , where $z=x+iy$ , $f(z)=u(x,y)+iv(x,y)$ , $rfrac{partial u}{partial r} = frac{partial v}{partial theta}$ , $frac{partial u}{partial theta} = -rfrac{partial v}{partial r}$ , where $z=re^{itheta}$ , $f(z)=u(r,theta)+iv(r,theta)$ , $frac{partial ln R}{partial ln r} = frac{partial varphi}{partial theta}$ , $frac{partial ln R}{partial theta} = -frac{partial varphi}{partial ln r}$ , where $z=re^{itheta}$ , $f(z)=R(r,theta)e^{ivarphi(r,theta)}$ . Can someone show me the fourth form, i.e. with $z=x+iy$ and $f(z)=R(x,y)e^{ivarphi(x,y)}$ , preferably with a proof if it is not too bothersome?
