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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.

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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
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1 Answer 1

$f:\mathbb R\rightarrow \mathbb C$ 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 $\arg :\mathbb R\rightarrow(-\pi,\pi]$

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(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
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