I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT), and it's been an absolute revelation for my research. The thing is, my current work centers around complex-valued functions of a real-variable. BGT mention in their appendices that a theory of functions of regular variation can be developed for such functions, and point to Elliot (1979)'s Probabilistic Number Theory as one such example; I have that book, and there isn't anywhere near as much there as I was expecting.

My research involving asymptotics is not centered on functions of regular variation, but rather, utilizes their properties. As such, it would be most useful to know exactly which aspects of this subject (Karamata Theory) hold in the case of complex-valued functions of a real variable, and those which do not.

First, recall that for any $\alpha\in\mathbb{R}$, we write $R_{\alpha}$ to denote the set of all regularly varying functions of index $\alpha$; these are functions of the form $f\left(x\right)=x^{\alpha}\varphi\left(x\right)$, where $\varphi$ is slowly varying at $\infty$.

Of particular interest to me are the following results:

BGT - Definition: If $f$ is well-defined, real-valued, and locally bounded on $\left[X,\infty\right)$, and tends to $\infty$ as $x\rightarrow\infty$, the generalized inverse of $f$ is given by: $$f^{\leftarrow}\left(x\right)\overset{\textrm{def}}{=}\inf\left\{ y\in\left[X,\infty\right):f\left(y\right)>x\right\} $$ This function is then defined on $\left[f\left(X\right),\infty\right)$ and is monotone increasing to $\infty$.

BGT - Theorem 1.5.1.12: If $f\in R_{\alpha}$ with $\alpha>0$, there exists $g\in R_{1/\alpha}$ with: $$f\left(g\left(x\right)\right)\sim g\left(f\left(x\right)\right)\sim x\textrm{ as }x\rightarrow\infty$$ Moreover, $g$, which is called an asymptotic inverse of $f$ is determined uniquely, up to asymptotic equivalence, and one such inverse is $f^{\leftarrow}\left(x\right)$.

BGT - Theorem 1.5.13 (de Bruijn (1959): If $\varphi$ varies slowly at $\infty$, there is a slowly varying function $\varphi^{\#}$, unique up to asymptotic equivalence, so that:

$$\varphi^{\#}\left(x\varphi\left(x\right)\right)\sim\frac{1}{\varphi\left(x\right)}$$ $$\varphi\left(x\varphi^{\#}\left(x\right)\right)\sim\frac{1}{\varphi^{\#}\left(x\right)}$$

as $x\rightarrow\infty$. Moreover, $\varphi^{\#\#}\sim\varphi$. $\varphi^{\#}$ is then called the de Bruijn conjugate of $\varphi$ and $\left(\varphi,\varphi^{\#}\right)$ is called a conjugate pair.

BGT - Proposition 1.5.14: If $\left(\varphi,\varphi^{\#}\right)$ is a conjugate pair of slowly varying functions, and $A,B,\nu$ are positive real constants, then, the following are also conjugate pairs:

$$\left(\varphi\left(Ax\right), \varphi^{\#}\left(Bx\right)\right)$$ $$\left(A\cdot\varphi\left(x\right),\frac{1}{A}\cdot\varphi^{\#}\left(x\right)\right)$$ $$\left(\left(\varphi\left(x^{\nu}\right)\right)^{1/\nu},\left(\varphi^{\#}\left(x^{\nu}\right)\right)^{1/\nu}\right)$$

I can't see how the given definition of the generalized inverse could be modified to apply to complex-valued functions of a real variable. Nevertheless, BGT have an appendix in which a method of deriving the de Bruijn conjugate of a function is given, in which it is assumed that $$\varphi\left(x\right)=\exp\left(h\left(\ln x\right)\right)$$ where $h$ is a holomorphic function defined on a complex domain $D$ that contains all sufficiently large real numbers. This suggests to me that there is probably a means of extending the above results to apply to complex-valued functions of a real variable.

Finally, I am interested in generalizing the following approximation result, to the case where $g$ is complex-valued:

BGT - Theorem 2.3.11: Let $g:\left(0,\infty\right)\rightarrow\left(0,\infty\right)$ be a function ($g$ need not be continuous, nor even measurable), and suppose that: $$\alpha=\limsup_{x\rightarrow\infty}\frac{\ln g\left(x\right)}{\ln x}<\infty$$ ($\alpha$ is called the upper order of $g$). Then, there is an $f\in R_{\alpha}$ so that: $$\limsup_{x\rightarrow\infty}\frac{f\left(x\right)}{g\left(x\right)}=1$$