Conformal and non-conformal transformations

[JulieK]

It is well known that from a two-dimensional solution of Laplace equation for a particular geometry, other solutions for other geometries can be obtained by making conformal transformations.

Now, I have a function defined on a disc centered at the origin and is given by

f(r) = a r

where a is constant and r is the radial distance from the origin. My function is obviously not a solution of the Laplace equation. However, I want to see if it is possible to find a transformation (conformal or non-conformal) that maps this to a rectangle (centered on the origin with length L in the x-direction and width W in the y-direction) so that I obtain the corresponding solution on the rectangular geometry similar to what is done with Laplace solutions.

 

[marcusl]

Only analytic functions have conformal maps, so your first step is to determine whether your function is analytic over the disk.