A complex number z has the form z = x + iy but can also be written in polar form as z = reiθ where θ is the angle of rotation counter clockwise from the x axis and r is the radius. Note that r < 0 is always a real number.
The multivalued function arg(z) is defined as arg(z) = ϕ such that the equation reiθ = reiϕ holds. Thus, arg(z) = θ + 2πn where n is and integer. Note that one input yeilds an infinite set of outputs. Now there is another function, Arg which is defined as Arg(z) = ϕ such that z = reiϕ and -π < ϕ ≤ π. This function is single-valued. Now, in complex analysis, the log function is also multivalued as it is defined as log(z) = ln(r) + i arg(z). Here we will use Log(z) which is defined as Log(z) = ln(r) + i Arg(z). That way, our function is not multivalued, but one to one and thus has an inverse. Thus, we will be able to map our problem into a different coordinate system, solve it and map it back with no ambiguities.
Now, there is an important formula in complex analysis which we will make use of later. It is the following
eiθ = cosθ + i sinθ
which is known as Euler's Formula. We will not prove this identity, but it can be done with Taylor series. Next, we'll need a few definitions.
Definition: A function f is analytic in an open set A if it has a derivative at each point in A.
Definition: A real valued function u(x,y) is called harmonic is it satisfies Laplace's equation. That is if
uxx + uyy = 0
Definition: A mapping (a mapping is just a function from the complex plane to the complex plane) f: A → B is called conformal if for every z in A, f rotates tangent vectors to curves through z by a definite angle θ and stretches them by a definite factor r. That is, the magnitude and orientation of angles between curves in A are preserved in their images in B under f.
We now discuss some useful theorems.
Theorem
If f: A → B is analytic and f '(z) ≠ 0 for all z in A, then f is conformal.
Theorem If a function is harmonic over a particular space where it satisfies certain boundary conditions, and it is transformed via a conformal map to another space, the transformation is also harmonic and satisfies corresponding boundary conditions.
Now, note that the map w = Log(z) is an analytic function with a non-zero derivative everywhere in the the complex plane except at z = 0 (it has the familiar 1/z derivative). Thus, w(F) = P(u,v) is a conformal map and P is thus a harmonic function as well and the boundary conditions are preserved across w as well. This is what allows us to solve it in the uv plane and map it back to the xy plane.
For a further development of the theory of conformal mappings and analytic functions and the proofs of these theorems see the books in the References page.
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