# Limit of explicit complex function

what is the following limit

$$\lim_{y\rightarrow2}\frac{(\log(y-1))^{i\alpha}}{(\log(y-1))^{\frac{1}{2}}}$$ where $\alpha \in \mathbb R^{+}$ and $i$ is imaginary number. Thank you in advance.

-
How do you define $x^{i\alpha}$ and $x^{1/2}$ when $x\in\mathbb R$, $x\ne0$? – Did Jan 23 '13 at 13:39
At present, NARQ. – Did Jan 24 '13 at 6:23

$f:\mathbb R\rightarrow \mathbb C$ is defined by $$f(x)=x^{i\cdot\alpha}=e^{i\cdot\alpha\cdot\log(x)}=e^{i\cdot\alpha(\ln|x|+\arg(x))}.$$ and $g:\mathbb R\rightarrow \mathbb C$ is defined by $$g(x)=x^{\frac{1}{2}}=e^{\frac{1}{2}\cdot(\ln|x|+\arg(x))}.$$ where $\arg :\mathbb R\rightarrow(-\pi,\pi]$
(I guess this is supposed to answer my comment?) What is $\arg :\mathbb R\rightarrow(-\pi,\pi]$? This seems to allude to $\arg :\mathbb C\rightarrow(-\pi,\pi]$ but here one needs $\arg(x)$ for $x$ real... that is, $\arg(x)=0$ if $x\in\mathbb R_+^*$ and $\arg(x)=\pi$ if $x\in\mathbb R_-^*$? (And of course $\arg(0)$ undefined.) Sorry but this is not making any sense. – Did Jan 24 '13 at 6:22