 Solution

The real part of a holomorphic function is real analytic and has a local complexification in its two real coordinates. Define $f^{\ast }(w)=\overline{f(\bar{w})}$. On real $(x,y)$,and the same identity holds after analytic continuation to complex $(x,y)$. Substituting $x=(z+{\bar{z}}_0)/2$ and $y=(z-{\bar{z}}_0)/(2i)$ gives $x+iy=z$ and $x-iy={\bar{z}}_0$, henceThe conjugation on the final constant is present in the PDF but was lost in the converted TeX. Without it the formula already fails when $z=z_0$ and $f(z_0)$ is not real.

Now $z{e}^{iz}=(x+iy)e^{-y}(\cos x+i\sin x)$ has the specified real part. Thus the complete family isAny two such holomorphic functions differ by a holomorphic function with zero real part; the Cauchy-Riemann equations make its imaginary part constant on a connected neighborhood.