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Apr
22
comment Approximation of a quotient that involves the Lambert function.
I understood that there are two solutions, and, in fact, that disturbed me a little. Thanks again for your answer :-)
Apr
22
accepted Approximation of a quotient that involves the Lambert function.
Apr
21
comment Approximation of a quotient that involves the Lambert function.
I want to solve the equation $xn^{1/x}=(\ln n)^{1+c}$ with an asymptotic approximation. I had the intuition that $x=O(\ln n/(\ln \ln n))$.
Apr
21
comment Approximation of a quotient that involves the Lambert function.
Thanks very much for your detailed answer @JJacquelin. So, what branch should I use ? $W_{-1}$ or $W_0$ ?
Apr
21
comment Approximation of a quotient that involves the Lambert function.
Thanks again :-)
Apr
21
comment Approximation of a quotient that involves the Lambert function.
I don't understand very well. You say that I have to choose what I want ?
Apr
21
comment Approximation of a quotient that involves the Lambert function.
Thanks very much @user90369. If have to use $W_{-1}$, then we have $\frac{-\ln n}{W_{-1}(- \ln^{-c}n)}=O\left(\frac{\ln n}{\ln \ln n}\right)$. What is the argument for using $W_{-1}$ ? Thanks again.
Apr
21
comment Approximation of a quotient that involves the Lambert function.
Thanks you @user90369. Your proof is interesting. Is $k^{1/k}$ always lower than $e^{1/e}$ ? I don't know this property. Using the Lambert function allows to obtain an accurate solution (with a denominator which is function of $n$). It is true that I'm interested in a solution $x < \ln n$; but I would prefer a tight asymptotic upper bound. To do that, I think we have to use the branch $W_{-1}$, but I don't know if we can. Thanks again.
Apr
20
comment Approximation of a quotient that involves the Lambert function.
In fact I'm looking for a slightly tighter bound. For information: I've obtained $x=\frac{-\ln n}{W(- \ln^{-c}n)}$ by solving the equation $xn^{1/x}= (\ln n)^{1+c}$. I know that $x$ should lower than $\ln n$, but I don't know in what extent...
Apr
20
comment Approximation of a quotient that involves the Lambert function.
Thanks for your answer. What about the second branch $W_{-1}$ ? I'm reading on Wiki that when $x$ approaches $0$, we have $W_{-1}(x)$ close to $\ln(-x) - \ln(-\ln(-x))$.
Apr
20
asked Approximation of a quotient that involves the Lambert function.
Apr
20
comment Problem of simplification
Thanks for your comments.
Apr
20
comment Problem of simplification
Thanks very much.
Apr
16
comment Problem of simplification
Thanks for your comment @martycohen. Do you want to say that I should be satisfied with such a form ?
Apr
16
comment Problem of simplification
@ClaudeLeibovici In fact, I don't like the negative signs.
Apr
16
comment Problem of simplification
@ClaudeLeibovici I've added it.
Apr
16
revised Problem of simplification
added 34 characters in body
Apr
16
asked Problem of simplification
Apr
8
comment How to solve the equation $x \log \log x = n$
Thanks for this comment.
Apr
8
accepted How to solve the equation $x \log \log x = n$