Borel's theorem
/ Borel -> boh-REL /
Picard's theorem tells you an entire function misses at most one value, but it is silent on how OFTEN each value is taken. Borel's theorem refines this into a quantitative law about the abundance of solutions. It says that for an entire function of finite order rho, the solutions of f(z) = w are plentiful for essentially every w: the zeros of f(z) - w have counting function growing at the full rate r^rho, for every value w with at most one exception. The one stingy value — the value for which solutions are unusually sparse — is the exceptional (Borel) value, and there can be at most one.
Precisely: let f be entire of finite positive order rho. For a value w, the order of growth of the zeros of f(z) - w is its 'convergence exponent.' Borel's theorem asserts that this exponent equals rho for every w, with at most one exceptional w for which it is strictly smaller. In other words, almost every value is attained densely — at the maximal rate the order permits — and at most one value is attained more sparingly. The exponential again models the extreme: e^z has order 1, and for every w not 0 the solutions of e^z = w have counting function growing like r (exponent 1), while w = 0 is the Borel exceptional value, attained never at all (exponent 0). So Picard's omitted value and Borel's sparse value coincide here.
Borel's theorem is the bridge from the qualitative Picard statements to the quantitative theory of value distribution that Nevanlinna later completed. It refines 'misses at most one value' into 'attains every value with the maximal density, save at most one,' connecting the order rho not just to the zeros of f but to the w-points of f for every target w. The careful caveat: the exceptional value is defined by a deficiency in the rate of the counting function, not necessarily by outright omission — a Borel exceptional value might still be taken, just too rarely to keep pace with the order; outright omission (a Picard exceptional value) is the most extreme form of being Borel exceptional.
Take f(z) = e^z + z, which is entire of order 1. The equation f(z) = w has solutions whose counting function grows like r for every value w — including w = 0 now, since e^z + z does hit 0. Here f has NO exceptional value: the allowed single exception is permitted but not forced. Borel's theorem guarantees at most one exceptional value, not that one must exist.
Borel allows at most one exceptional value but does not require one — e^z + z has none, e^z has exactly one.
Borel exceptional means the counting function for that value grows strictly slower than the order, which is more permissive than Picard's outright omission; a value can be Borel-exceptional yet still attained. And the theorem needs finite positive order — it says nothing for order 0 or infinity.