Wiman–Valiron theory

Summary

Wiman–Valiron theory studies a transcendental entire function through the largest term of its Taylor series on each circle $\abs{z} = r$. The index of this maximal term, the central index $\nu(r)$, grows to infinity with $r$. Near points where $\abs{f}$ attains its maximum on the circle, $f$ behaves like the single monomial $a_{\nu(r)} z^{\nu(r)}$, outside a small exceptional set of radii. A main application is to the growth of entire solutions of linear differential equations with polynomial coefficients.