Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 7 / a

Table of contents
- Solution a

Solution

For a holomorphic function with

z = r e^{i θ}

, the chain rule gives

f_{r} = e^{i θ} f^{'} (z), f_{θ} = i r e^{i θ} f^{'} (z) = i r f_{r} .

f = u + i v

and equating real parts and imaginary parts,

u_{θ} = - r v_{r}, v_{θ} = r u_{r} .

For

r > 0

r

gives the Cauchy-Riemann equations in polar coordinates:

\frac{1}{r} u_{θ} = - v_{r}, u_{r} = \frac{1}{r} v_{θ} .

The restriction

r > 0

reflects the coordinate singularity at the origin; the Cartesian equations have no such restriction.

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version