Timeline for A variant of Lambert function
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2020 at 9:13 | comment | added | lchen | Thank you both. | |
| Mar 22, 2020 at 4:53 | history | edited | Alexey Ustinov | CC BY-SA 4.0 |
added 114 characters in body
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| Mar 22, 2020 at 4:49 | comment | added | Alexey Ustinov | @SimplyBeautifulArt You are write, my answer is wrong. | |
| Mar 22, 2020 at 0:43 | comment | added | Simply Beautiful Art | Assuming I haven't made a mistake, using this approach but modified with $\ln(x)=W(z)$ will give us $$W'(z)=\frac1{x(\ln(x)+1)}$$which is reduces to$$\frac e{ex\ln(ex)}$$which is invertible using the Lambert W function. It does go to show, however, that the inverse of $W'$ is solvable in terms of $W$ and is given by$$(W')^{-1}(z)=x\ln(x),~x=\frac e{W^{-1}(\ln(z)+1)}$$ | |
| Mar 22, 2020 at 0:19 | comment | added | Simply Beautiful Art | Correct me if I'm wrong, but if $z=\ln(t)$ and $x=W(z)$, then $z=xe^x$ and $t=e^z=e^{xe^x}\ne x^x$. | |
| Mar 21, 2020 at 23:37 | comment | added | Simply Beautiful Art | Huh I've never thought to use the derivative of the Lambert W function, interesting approach. | |
| Mar 21, 2020 at 15:48 | comment | added | Alexey Ustinov | @lchen If you'll replace $x$ by $-x$ and $a$ by $-a$, then you'll get $\left(\frac{x}{a}\right)^{x-1}=b.$ | |
| Mar 21, 2020 at 15:44 | comment | added | lchen | Thank you all the same. I thought that it was related to Lambert function but could not figure out the solution. | |
| Mar 21, 2020 at 15:36 | comment | added | Alexey Ustinov | @lchen I don't know. | |
| Mar 21, 2020 at 15:32 | comment | added | lchen | Ｉmean to solve the equation $\left(\frac{a}{x}\right)^{x+1}=b$. | |
| Mar 21, 2020 at 15:16 | comment | added | Alexey Ustinov | @lchen It is not clear. | |
| Mar 21, 2020 at 14:40 | comment | added | lchen | What about if $x$ is in the denominator, i.e., how to solve $\left(\frac{a}{x}\right)^{x+1}=b$? Thank you | |
| Mar 21, 2020 at 12:37 | history | answered | Alexey Ustinov | CC BY-SA 4.0 |