Akhiezer's theorem

Akhiezer's theorem In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.[1] Contents 1 Statement 2 Related results 3 Notes 4 References Statement Let f(z) be an entire function of exponential type τ, with f(x) ≥ 0 for real x. Then the following are equivalent: There exists an entire function F, of exponential type τ/2, having all its zeros in the (closed) upper half plane, such that {displaystyle f(z)=F(z){overline {F({overline {z}})}}} One has: {displaystyle sum |operatorname {Im} (1/z_{n})|

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