the type of an entire function

The order of an entire function is a coarse ruler: it tells you the exponent rho in the benchmark e^(r^rho), but it cannot distinguish e^z from e^(5z), both of which have order 1. Type is the finer ruler that steps in once the order is fixed. It measures the multiplier sitting in front of r^rho — whether the function rides the benchmark closely or pulls ahead of it by a large constant factor. Order tells you the shape of the growth; type tells you the scale.

Once the order rho (with 0 < rho < infinity) is known, the type sigma is defined by sigma = limsup as r tends to infinity of log M(r) / r^rho. In other words M(r) is roughly e^(sigma r^rho) for large r. A function is of minimal type if sigma = 0, normal (mean) type if 0 < sigma < infinity, and maximal type if sigma = infinity. Worked example: e^(a z) for a constant a has order 1 and M(r) = e^(|a| r), so log M(r) / r = |a| and the type is sigma = |a|. Thus e^z has type 1 while e^(5z) has type 5 — same order, different type, exactly the distinction we wanted. The function sin z has order 1 and type 1.

Type matters in real analysis and signal processing, where entire functions of order 1 and finite type — called functions of exponential type — are central: the Paley-Wiener theorem identifies them with Fourier transforms of compactly supported functions, and the band-limited signals of sampling theory are exactly the exponential-type functions bounded on the real line. A caveat to keep straight: type is only meaningful after the order is pinned down and finite, and it too is a limsup, so it captures the largest constant the growth ever reaches, not a steady one.