a Weierstrass elementary factor
/ VY-er-shtrahss /
Here is the problem the elementary factors solve. To build a function with a zero at a point a, the obvious factor is (1 - z/a): it vanishes at z = a and equals 1 at z = 0. But if you want zeros at infinitely many points a_1, a_2, ... marching off to infinity, the naive product of all the (1 - z/a_n) usually DIVERGES — the factors do not approach 1 fast enough. We need a way to keep the zero at a_n while gently nudging each factor back toward 1 so the product can converge. Weierstrass found exactly the right nudge.
The elementary factor of degree p is E_p(z) = (1 - z) exp(z + z^2/2 + z^3/3 + ... + z^p/p), and E_0(z) = 1 - z. The idea is surgical: (1 - z) supplies the single zero at z = 1, and the exponential factor is a non-vanishing correction (an exponential is never zero) whose exponent is exactly the first p terms of the Taylor series of -log(1 - z). Because those terms cancel the leading behaviour of log(1 - z), the factor E_p(z) is extremely close to 1 when z is small — precisely, log E_p(z) is of order z^(p+1), so |1 - E_p(z)| <= |z|^(p+1) for |z| <= 1. The higher you take p, the flatter E_p sits near 0 while still having its one zero at z = 1.
So the elementary factors are a tunable kit. To put a zero at a_n, use E_p(z/a_n), which vanishes at z = a_n. By choosing p large enough (depending on how fast the a_n run to infinity) you make each |1 - E_p(z/a_n)| as small as |z/a_n|^(p+1), and then the product of these factors converges. This single, clever device — a zero-supplying linear part times a convergence-forcing exponential part — is the whole secret that makes the Weierstrass factorization theorem work.
E_1(z) = (1 - z) e^z. Near z = 0, expand: (1 - z)(1 + z + z^2/2 + ...) = 1 - z^2/2 - ... so 1 - E_1(z) is of order z^2, much smaller than the order-z deviation of the bare factor 1 - z. The single zero is still exactly at z = 1, but the factor now hugs 1 near the origin, which is what lets a product of such factors converge.
E_1 = (1 - z)e^z keeps the zero at 1 but flattens the deviation from order z to order z^2 near the origin.
A frequent confusion: the exponential part of E_p adds no zeros and no poles — an exponential is entire and never vanishes — so the factor's only zero is the one at z = 1 from (1 - z). Its sole job is to force convergence, not to change the zero set.