Exercise 2, Appendix 0

a) Show that every (H) function having a definition sequence of just one term is equivalent to a function of one of the forms $x^p (\log x)^q$, or $x^p e^{g(1)}$, where $p$ and $q$ are rational integers and $g$ is a polynomial in $x$ (method of exerc. 1).

b) Deduce from a) that every function in the Hardy field $\mathbf{R}(x, u_1, \ldots, u_p)$, where $u_1, \ldots, u_p$ are (H) functions having a single-term definition sequence, is equivalent to a function of the form $x^p (\log x)^q e^{g(1)}$ where $p$ and $q$ are rational integers and $g$ is a polynomial in $x$.