Functions that fill the whole plane
An entire function is one that is holomorphic on the entire complex plane — no poles, no branch cuts, no edge anywhere. You have met the headliners already: polynomials, e^z, cos z and sin z, and any convergent power series sum a_n z^n that never stops converging. These are the cleanest objects in all of complex analysis, and that very cleanliness is the problem we tackle on this rung. Strip away the boundary, the poles, the singularities, and what is left to distinguish one entire function from another? The answer is how big it gets. Growth becomes the fingerprint.
Recall a fact from the Liouville rung that sets the stage. Liouville's theorem says a bounded entire function must be constant — there is no nonconstant entire function that stays small everywhere. So every interesting entire function is unbounded; it must run off to infinity in some direction. That immediately raises the only sensible question left: how fast? A polynomial of degree n grows like |z|^n. But e^z grows far faster, and e^(z^2) faster still. We need a ruler graded finely enough to separate all of these. Two numbers will do it.
The maximum modulus: turning a function into a curve
To measure growth we first need a single number that captures 'how big f is out near radius r'. The honest choice is the maximum modulus function, written M(r) = max of |f(z)| over the circle |z| = r. As you let the radius r expand, M(r) records the largest value the modulus reaches on each circle. By the maximum modulus principle this maximum is always attained on the boundary circle, never strictly inside, so M(r) is also the biggest |f| over the whole closed disk of radius r — and it climbs steadily as r grows. We have traded a function of a complex variable for a single increasing real-valued curve M(r). Now we just have to read its growth.
Let us look at our three examples through this lens. For a polynomial of degree n, M(r) grows like r^n — polynomial growth, the gentlest kind. For f(z) = e^z, the modulus is |e^(x + i y)| = e^x, which is biggest when x = r, so M(r) = e^r — exponential growth, a wholly different league. For f(z) = e^(z^2), a short calculation gives M(r) = e^(r^2). And cos z and sin z? Recall from the elementary-functions rung that they are unbounded in the plane — cos z is not trapped in [-1, 1] once z leaves the real axis. In fact cos z = (e^(i z) + e^(-i z))/2, so along the imaginary axis it grows like e^r, the same exponential rate as e^z.
Order: which power of r sits in the exponent
Look back at the three exponential cases: M(r) = e^r for e^z, M(r) = e^(r^2) for e^(z^2), and you can check M(r) = e^(r^3) for e^(z^3). The growth is exponential in all of them, but the power of r inside the exponent differs: 1, then 2, then 3. That power is exactly what the order measures. Loosely, if M(r) behaves like e^(r^rho) for large r, then rho is the order. The order sorts entire functions by which exponential-of-a-power they keep pace with — it is a far coarser, far more robust label than the function's formula.
To pin it down precisely we take a double logarithm, because the exponent is buried two layers deep. The clean definition of the order rho is the limit superior, as r goes to infinity, of log log M(r) over log r. Walking through why: M(r) ~ e^(r^rho) means log M(r) ~ r^rho, so log log M(r) ~ rho log r, and dividing by log r leaves rho. The limit superior (rather than an ordinary limit) is there for honesty — M(r) need not be a perfect e^(r^rho), it only has to be comparable, and the lim sup extracts the right exponent even when the growth wobbles.
rho = limsup as r->inf of log log M(r) / log r polynomial M(r) ~ r^n -> log M(r) ~ n log r -> rho = 0 e^z M(r) = e^r -> log M(r) = r -> rho = 1 e^(z^2) M(r) = e^(r^2) -> log M(r) = r^2 -> rho = 2 cos z, sin z M(r) ~ e^r -> rho = 1
Type: the fine tuning within an order
Order alone is too coarse to separate e^z from e^(2z): both have M(r) = e^r and M(r) = e^(2r), both order 1, yet the second clearly grows twice as fast in the exponent. To split them we read the constant in front of r^rho. If M(r) behaves like e^(tau r^rho), that leading constant tau is the type. For e^z the exponent is exactly r, so tau = 1; for e^(2z) it is 2r, so tau = 2; for e^((1/2) z) it is r/2, so tau = 1/2. Type is the second dial, the fine tuning you reach for once order has already chosen the league.
The precise definition of the type tau is the limit superior, as r goes to infinity, of log M(r) divided by r^rho — once the order rho is known, you peel off one logarithm instead of two and read the constant directly. Functions of order rho get a three-way nickname: minimal type when tau = 0, mean (or normal) type when tau is a finite positive number, and maximal type when tau = infinity. The order says how aggressively a function grows; the type says exactly how hard it leans on that aggression. Together they give a remarkably complete growth profile from just two real numbers.
- Form the maximum modulus M(r) = max of |f(z)| over |z| = r, and find how it grows for large r — for these examples just compute |f| along the direction where it is largest.
- Take the order: rho = limsup of log log M(r) / log r. This tells you which exponential league the function is in (rho = 0 for polynomial-gentle, rho = 1 for plain exponential, and so on).
- With rho in hand, take the type: tau = limsup of log M(r) / r^rho. This reads the leading constant and pins the function down within its order.
- Sanity-check against the known menu: e^(c z) has order 1 and type |c|; cos z and sin z have order 1, type 1; e^(z^2) has order 2, type 1; any polynomial has order 0.
Reading order from the Taylor coefficients
Here is the bridge that makes order and type computable without ever drawing M(r). An entire function is a globally convergent power series f(z) = sum a_n z^n, and its growth is encoded in how fast the coefficients a_n shrink. The intuition is a duality you have felt before with the radius-of-convergence formula: for an entire function the series converges everywhere, so the a_n must die off — and the faster they die, the slower the function grows, while coefficients that die only sluggishly let the function explode. There is an exact formula turning the decay rate of |a_n| into the order rho, with no maximum modulus computation at all.
Where does such a bridge come from? It is built from a tool you already own: the Cauchy estimates bound each coefficient by |a_n| <= M(r) / r^n, valid for every radius r. Optimizing that bound over r — choosing the radius that makes the right side smallest — ties the size of M(r) directly to the smallness of |a_n|, and running the inequality both directions yields the coefficient formula for the order. So the growth of an entire function in the far field and the decay of its Taylor coefficients at the origin are two readings of one underlying fact. That duality is one of the quietly beautiful themes of the whole subject.
Why these two numbers run the whole rung
Order and type are not just bookkeeping — they are the load-bearing parameters for everything ahead. The next guide proves Jensen's formula, which links the growth of M(r) to the number and location of the zeros: a function cannot grow slowly and still carry a flood of zeros, so the order caps how densely the zeros may pack. The guide after that, the Hadamard factorization theorem, turns this into a structure theorem — an entire function of finite order is its zeros times an exponential of a polynomial whose degree is bounded by the order. Finite order buys you a factorization; the order tells you how complicated it is allowed to be.
Then the rung turns to value distribution — not how big f gets, but which values it takes. The little Picard theorem says a nonconstant entire function omits at most one complex value (e^z omits exactly 0, recall that e^z is never zero), and the great Picard theorem makes the wildness near an essential singularity precise. Finally Nevanlinna theory generalizes the maximum modulus M(r) into a subtler growth gauge — the Nevanlinna characteristic — that counts value-taking and unifies Jensen, Picard, and the whole circle of ideas. Every one of those results is phrased in the language we built today. Order and type are the coordinates of the entire world.