e-mail: george@math.nwu.edu

**Using squares of real-valued functions to prove that
the $\Xi^*(z)$ function and certain other entire functions have only real
zeros**

**Abstract**

It is shown how squares of real-valued functions can be used to give new proofs of the reality of the zeros of

$$ \Xi^*(z) = 4\pi^2 \int_0^\infty \cosh{9\over 2}u \ e^{-2\pi \cosh 2u} \cos zu \ du,$$

$$ K_{iz}(a) = \int_0^\infty e^{-a \cosh u} \cos zu \ du, \ \ a > 0,$$

and of some other entire functions.