A theorem about functions with this property uses a sequence of functions E-sub-n, the nth of which equals one minus z all times e to the partial sum of z to the k over k; such functions are called elementary factors. Given a sequence with moduli tending to infinity, there exists a function with this property whose zeroes are precisely that sequence. The Taylor coefficients, a-sub-n, of functions with this property satisfy the condition that the lim-sup (“lim soup”) of the absolute value of a-sub-n to the power of one over n is (*) zero. The Hadamard factorization theorem applies to functions with this property whose Cauchy–Hadamard radius is infinity. Non-constant functions with this property can miss at most one point by Picard’s little theorem. Bounded functions with this property are constant by Liouville’s theorem. For 10 points, name this property of functions that are holomorphic everywhere on the complex plane. ■END■
ANSWER: entire [accept answers indicating that the function is everywhere differentiable, everywhere analytic, or everywhere holomorphic until “everywhere” is read; prompt on holomorphic until read; prompt on complex differentiable or analytic by asking “what is this called for complex function?”]
<Tim Morrison, Other Science - Math>
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