potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...

, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic

Newtonian potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...

is small.

Statement of the lemma

The following statement can be found in Levin's book.B.Ya. Levin, ''Lectures on Entire Functions'' Let ''μ'' be a finite positive Borel measure on the complex plane C with ''μ''(C) = ''n''. Let ''u''(''z'') be the logarithmic potential of ''μ'': :

u(z) = \frac\int_\mathbf \log, z-\zeta, \,d\mu(\zeta)

Given ''H'' ∈ (0, 1), there exist discs of radii ''r''_''i'' such that :

\sum_i r_i < 5H

and :

u(z) \ge \frac\log \frac

for all ''z'' outside the union of these discs.

Notes

Complex analysis {{mathanalysis-stub