the maximum term and central index
An entire function is a sum of infinitely many monomial terms, f(z) = sum a_n z^n. On a circle |z| = r, each term has size |a_n| r^n, and as you increase r the balance among the terms shifts — small-n terms dominate for small r, large-n terms eventually take over. The maximum term and central index are a simple, almost combinatorial way to track this shifting balance, and they give a surprisingly powerful handle on growth that bypasses M(r) entirely.
Fix a radius r. The maximum term mu(r) is the largest of the term sizes: mu(r) = max over n of |a_n| r^n. The central index nu(r) is the value of n that achieves that maximum — the index of the currently dominant term (if several tie, take the largest such n). As r grows, mu(r) increases and nu(r) jumps up through integer values, never decreasing, because higher-power terms overtake lower ones at larger radii. Worked picture with e^z = sum z^n / n!: the n-th term is r^n / n!, and comparing consecutive terms, the (n+1)-th beats the n-th exactly when r > n+1; so at radius r the dominant index nu(r) is about r, and the maximum term mu(r) is about e^r / sqrt(2 pi r). The central index of e^z grows linearly with r — directly reflecting its order 1.
These quantities power the Wiman-Valiron theory, which shows that near the radius where it is largest, an entire function behaves locally like the single monomial a_nu z^nu — so the central index acts like a local 'effective degree.' This gives slick proofs of growth relations: log mu(r) is comparable to log M(r), so the order can be computed from mu(r) just as well, and nu(r) ties the growth to which coefficients dominate, linking back to the coefficient formula for the order. The honest limitation: the maximum-term picture is a tool for entire functions specifically (the series must converge everywhere), and the local approximation by the dominant monomial holds only away from an exceptional small set of radii, not for every r — it is a powerful heuristic made rigorous with care, not an exact identity.
For f(z) = sum z^n / n! (that is e^z), find the central index at r = 10. The term sizes r^n / n! increase as long as the ratio r / n > 1, i.e. n < 10, and decrease after; so the largest term sits at n = 10, giving nu(10) = 10. In general nu(r) is the integer part of r for e^z — the central index tracks r almost exactly, which is why log mu(r) is about r and the order reads off as 1.
For e^z the central index nu(r) is essentially r itself, so the dominant monomial at radius r is z^r / r!.
The Wiman-Valiron local approximation f(z) is about a_nu z^nu holds only outside a thin exceptional set of radii, not for every r, so it is a rigorous tool used with care, not a blanket identity. The machinery applies to entire functions, where the series converges for all z.