Wednesday, January 23, 2013

EM and Complex Analysis

There are an increasing number of apparent correspondences between EM this semester and our section on complex analysis in math methods last semester.  These are just notes on a few of them.

Uniqueness of the Electrostatic Potential Solution and Liouville's theorem
After stating that we would be solving Poisson's equation

to determine electrostatic potentials, our professor then launched into a proof that the solutions, once found, would be unique.  We first defined a potential psi equal to the difference between non-unique solutions, (assuming for the moment in our proof by contradiction that there could be more than one unique solution).  We placed psi back into Poisson's equation and ran through the following steps:

Ultimately we wound up proving that at best psi is a constant, but that it must be zero everywhere on the surface that defines the Dirichlet boundary conditions that the two 'non-unique' solutions both satisfy, so it's constant value must be zero and the two 'non-unique' solutions are in fact the same.

Here's the question.  At the step shown above and highlighted here:

could we have fast tracked the entire proof by invoking a result from complex analysis?  In complex analysis we learned that analytic functions satisfy the condition on psi highlighted above, (Laplace's equation).  We also learned a version of Liousville's theorem that went:

"A function which is analytic for all finite values of z and is bounded everywhere (including infnity) is a constant."

It seems this would have immediately brought us to the conclusion that psi was constant and things could have moved on from there.

Finding Potentials Actually Just Complex Analysis?
There are an accumulation of pointers in my mind that what we're doing when solving for potentials is very closely related to complex analysis and in particular to Cauchy integrals.  When solving for the potential within a volume, we're told that either the value of the potential everywhere on the surface bounding the volume, (Dirichlet boundary conditions), or the value of the normal derivative of the potential everywhere on the boundary, (Neumann boundary conditions), is sufficient information.  This looks a lot like the Cauchy integral idea where if we know the values of a function around a contour, we can calculate the value of the function at any point inside the contour.  Is there anything to this?

Please excuse the obligatory coffee stains.

Picture of the Day:
From 1/23/13

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