1. Van Der Pauw -completing the missing part-
B 許恭銓

2. The Problem
Van Der Pauw, in his paper, derived the equation for the special case where the four probes are in a line. He later claimed that this equation is suitable for almost all smooth lamellae, and it could be proved by using complex analysis. Unfortunately, in the paper he didn’t prove this, and so a question arises.

3. Field Representations on complex plane
The electric field in the complex plane
The complex theorem guarantees that

4. A useful derivation
Now, on the complex plane we observe the current traversing through the paths
between arbitrary points P1->P2 P2 P1

5. Applying the Derivation 1/2
Back to our main problem, on the half complex plane, z-plane, we
traverse along the edge,
the x-axis, and observe
Except near the points A and B, there are no currents flowing perpendicular to the traversal.
So by our previous derivation, the value of won’t change. A B C D

6. Applying the Derivation 2/2
When we get near points A/B, we encounter the currents flowing in/out of the points.
When jumping over point A, by our previous derivation,
Similarly, for point B,
So we have

7. Applying to arbitrary shape 1/3
Now, we consider a lamella of arbitrary shape
lying on the complex plane. We
should call it the t-plane.
Suppose there exists a conformal mapping from z-plane to t-plane, say
The corresponding conformal transform maps A , B , C , D  , , ,

8. Applying to arbitrary shape 2/3
Now, let the function
Traversing along , we still have
If we make , then we can assure that
for the corresponding points.
And by the characteristics of the conformal mapping, is guaranteed the potential function for t-plane.

9. Applying to arbitrary shape 3/3
Therefore,