## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

## Integer-valued entire functions I – Pólya

Are there entire functions f such that f (0), f (1), f (2), . . . are rational integers? Of course: any polynomial with integer coefficients. But also f (z) = z(z−1)/2, which for any z in Z is a binomial coefficient. And more generally

 (9.1)

(which incidentally comes from a hyperderivative (1/k!)(d/dX)kXz at X = 1). Are there any f not polynomials? Of course: for example f (z) = sin(πz). This is harmlessly bounded on R but on its rightful domain C it grows quite violently; for example f (iy) = (eπyeπy)/(2i) grows exponentially as the real y goes to infinity. So there are arbitrarily large ...

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required