A friend of mine is reading this article (eq.2,pg-5) about Plasma Physics and trying to find what the s function $Z_r(x)$ is. . He's been trying to find out for half an hour. I'm typing this question on his behalf.
In plasma physics, one common special function is $Z(x)$, called the plasma dispersion function, but this article uses $Z_r(x)$, i.e., with the subscript $r$. It's surely not the plasma dispersion function, because $Z_r(x)$ is real (as there are no complex numbers involved in the article) while the plasma dispersion function is a complex function. The function $Z_r(x)$ appears in the article as a result of 3D integration of some exponents and logarithms and then making a Taylor expansion. The article author seems to assume it's a commonly known function as he doesn't define it anywhere in his article.
What is the special function $Z_r(x)$? What is its name, and how is this function defined?
We've found the answer thanks to the hint by @ClaudeLeibovic in the comments below the original post. The hint was to look at https://www.researchgate.net/publication/234928947_Kinetic_theory_of_solitary_waves_on_coasting_beams_in_synchrotrons
And that article defines $Z_r(x)$. It's the real part of the plasma dispersion function $Z(x)$ mentioned in my original post.