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real analysis

Source: miklos schweitzer 2011 q8

August 29, 2021
real analysis

Problem Statement

Given a nonzero real number a≤1/ea\leq 1/e, let z1,...,zn∈Cz_1, ..., z_n \in C be non-real numbers for which zez+a=0ze^z + a = 0 holds, and let c1,...,cn∈Cc_1, ..., c_n \in C be arbitrary. Show that the function f(x)=Re(∑j=1ncjezjx)f(x)=Re(\sum_{j=1}^n c_j e^{z_j x}) (x∈Rx \in R) has a zero in every closed interval of length 1.
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