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
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(which incidentally comes from a hyperderivative (1/k!)(d/dX)^{k}X^{z} 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) = (e^{−}^{πy} − e^{πy})/(2i) grows exponentially as the real y goes to infinity. So there are arbitrarily large ...
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