LogarithmicPotential - crowlogic/arb4j GitHub Wiki

Bôcher's Theorem and Logarithmic Potential

Context

In the realm of complex analysis, Bôcher's theorem offers a way to understand rational functions by associating their zeros and poles with a "potential field." In this metaphor, zeros are points of positive mass, while poles are points of negative mass.

Logarithmic Potential

The term "logarithmic potential" has a particular definition in potential theory, given by:

[ u(x) = \int \ln \frac{1}{|x - y|} d\mu(y) ]

This definition has specific mathematical properties and applications, often related to problems in potential theory.

Bôcher-style "Potential"

In the context of Bôcher's theorem, the idea of a potential is more of a metaphorical or heuristic concept. For a rational function ( r(z) ) with zeros at ( z_i ) of multiplicity ( m_i ) and poles at ( p_j ) of multiplicity ( n_j ), the potential ( U(z) ) is:

[ U(z) = \sum_{i} m_{i} \log|z - z_{i}| - \sum_{j} n_{j} \log|z - p_{j}| ]

Example: ( f(t) = \tanh(\ln(1 + t^2)) )

For ( f(t) = \tanh(\ln(1 + t^2)) ), we find a zero at ( t = 0 ) with multiplicity 2, and poles at ( t = \pm \sqrt{-1 \pm i} ). The associated Bôcher-style "potential" would account for these zeros and poles.

What's Lost?

When simplifying ( f(t) = \tanh(\ln(1 + t^2)) ) to a form that focuses only on zeros and poles, some properties of ( f(t) ) are lost. These properties could be important in problems involving Laplace or Poisson equations, as would be the case with a proper logarithmic potential in potential theory.