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I am an experimental physicist and have a math question. There are two functions, one being

$-1/\log(x)$

the other

$\sqrt{x}/(1-x)$

which are remarkably similar in the domain ]0,1[.

Does anybody have an idea why?

J.G.
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Daniel R
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  • In what sense are they similar? The first one tends to $0$ slower than the second one as $x \to 0$. – Kavi Rama Murthy Jan 11 '20 at 13:30
  • Did you intentionally put a $-1$ in your $log$ equation? They seem to be more similar is it was $1$ instead of $-1$ – Sina Babaei Zadeh Jan 11 '20 at 13:31
  • Ah, I am sorry. The values are very similar! See here: https://www.wolframalpha.com/input/?i=plot+-1%2Flog%28x%29+and+sqrt%28x%29%2F%281-x%29+from+0+to+1 – Daniel R Jan 11 '20 at 13:36

3 Answers3

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I just figured it out myself. Develop $\log(x)$ around 1, then write out $-\frac{1}{\log(x)}$, factor out (1-x) and then write $\sqrt(x)$ as $1/1/\sqrt(x)$ and develop $1/\sqrt(x)$ around 1, then compare.

Daniel R
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The approximation $\ln x\approx x-1$ shows these functions are similar when $x$ is close to $1$. When $x$ is small and positive, so are both functions. But they look less similar if you zoom to say $0\le x\le 0.1$, and even less similar if you zoom to $0\le x\le 0.001$. Indeed,$$\lim_{x\to0^+}\frac{\sqrt{x}/(1-x)}{-1/\ln x}=\lim_{x\to0^+}\frac{-2\sqrt{x}\ln\sqrt{x}}{1-x}=-2\lim_{y\to^+}y\ln y=0.$$

J.G.
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  • True, I did not notice that! But for my application, x is 1-losses, and I want to decrease losses, so I expect x to be closer to one. It is however strange that the functions are not be exactly the same, since both ways of calculating them should be equivalent! – Daniel R Jan 12 '20 at 14:35
  • @DanielR. Are you saying you solved some problem in two ways, and they gave those two functions, and you want to make sense of the answers differing? If so, maybe you should ask another question about that. – J.G. Jan 12 '20 at 15:12
  • Originally, I wanted a formal way to show they are the same, or at least explain the similarity for that large part of the interval I am interested in. Now, I am thinking about what you said last: Work out why they differ. – Daniel R Jan 12 '20 at 15:29
  • @DanielR. Well, as I said, they're both roughly $1/(1-x)$ if $x\approx1$, but if $x\ll1$ one function is much smaller than the other. How did you come across them, exactly? – J.G. Jan 12 '20 at 15:46
  • I was calculating the finesse of a ring resonator using the phase response function. – Daniel R Jan 12 '20 at 19:02
  • @DanielR. Ah, well, as I said, you may need to ask a question about why calculations differed. – J.G. Jan 12 '20 at 19:07
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The series development around $x=1$ gives: $$ \eqalign{ & {{\sqrt {\left( {1 + x} \right)} } \over {\left( {1 - \left( {1 + x} \right)} \right)}} = - {{\sqrt {\left( {1 + x} \right)} } \over x} = - \left( {{1 \over x} + {1 \over 2} - {x \over 8} + {{x^{\,2} } \over {16}} + O\left( {x^{\,3} } \right)} \right) \cr & - {1 \over {\ln \left( {1 + x} \right)}} = - \left( {{1 \over x} + {1 \over 2} - {x \over {12}} + {{x^{\,2} } \over {24}} + O\left( {x^{\,3} } \right)} \right) \cr} $$

G Cab
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