FANDOM


Problem

Cauchy Riemann

Consider the complex number in polar form $ z = r{e}^{i \theta} $, and the complex function in polar form $ f(z) = u(r, \theta) + iv(r, \theta) $.

Show that the Cauchy-Riemann equations in polar form are

$ \frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta} $
$ \frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta} $.

Solution

It is important to know the Cartesian representation of the complex number and the Cauchy-Riemann equations first.

$ z = x+iy $
$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} $
$ \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y} $.

Now use the multivariate product rule to take the the derivatives of $ \frac{\partial u}{\partial r} $, $ \frac{\partial u}{\partial \theta} $, $ \frac{\partial v}{\partial r} $, and $ \frac{\partial v}{\partial \theta} $.

For the function $ u(r, \theta) $:

$ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial r} $
$ \frac{\partial u}{\partial \theta} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial \theta} $.

Since $ \frac{\partial x}{\partial r} = \cos{\theta} $, $ \frac{\partial y}{\partial r} = \sin{\theta} $, $ \frac{\partial x}{\partial \theta} = -r\sin{\theta} $, and $ \frac{\partial y}{\partial \theta} = r\cos{\theta} $, we get

$ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \cos{\theta} + \frac{\partial u}{\partial y} \sin{\theta} $
$ \frac{\partial u}{\partial \theta} = -\frac{\partial u}{\partial x} r\sin{\theta} + \frac{\partial u}{\partial y} r\cos{\theta} $.

We repeat the same calculations for $ v(r, \theta) $.

$ \frac{\partial v}{\partial r} = \frac{\partial v}{\partial x} \cos{\theta} + \frac{\partial v}{\partial y} \sin{\theta} $
$ \frac{\partial v}{\partial \theta} = -\frac{\partial v}{\partial x} r\sin{\theta} + \frac{\partial v}{\partial y} r\cos{\theta} $.

Now use the Cauchy-Riemann equations in Cartesian form and make the right substitutions.

$ \frac{\partial v}{\partial \theta} = \frac{\partial u}{\partial y} r\sin{\theta} + \frac{\partial u}{\partial x} r\cos{\theta} = r\frac{\partial u}{\partial r} $
$ \frac{\partial u}{\partial \theta} = -\frac{\partial v}{\partial y} r\sin{\theta} - \frac{\partial v}{\partial x} r\cos{\theta} = -r\frac{\partial v}{\partial r} $

Therefore,

$ \frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta} $
$ \frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta} $.
Community content is available under CC-BY-SA unless otherwise noted.