doom2day Posted February 21, 2007 f(x) = X to the -x power, what an unusual function. (If any enlightenment or world peace secrets come from this function, I made it) I would love to find the limit of the function but I can't really figure it out. It is about 4.3 but, is it rational? 0 Share this post Link to post

GGG Posted February 22, 2007 You should probably wait until Pi Day to post something like this. 0 Share this post Link to post

Antidote Posted February 22, 2007 F(x)= X^-x? I don't know much about limits, but as to if 4.3 is rational... well it certainly isn't irrational. 0 Share this post Link to post

Fredrik Posted February 22, 2007 Since you didn't define "X", I assume "X^x" is the same as "x^x". What do you mean by "limit of the function"? f(x) = x^-x = 1/x^x is a transcendental function. Along the positive real line, we have f(0) = 1, from where the function increases to a local maximum ~1.444667861 at x = 1/e ~= 0.3678794412. It then rapidly decays to zero as x goes to infinity. Would you like to know its structure in the complex plane too? Pari plot: 1.4443362 |'''x"""_''''''''''''''''''''''''''''''''''''''''''''''''''''''| | " "_ | | x x | |_ x | |: " | : " | _ " | | " | | "_ | | _ | | _ | | x | | x | | "_ | | x | | "_ | | x_ | | x_ | | "x_ | | "x_ | | ""xx___ | 0 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,""""xxxxxxx________ 1e-07 4 It is interesting to attempt to integrate f. The indefinite integral of x^-x is nonelementary, as can be proved with the Risch algorithm / Liouville's principle. A curious fact is that the integral from 0 to 1 of x^-x equals the sum 1 + 2^-2 + 3^-3 + 4^-4 + ... ~= 1.29128599 (search for "Sophomore's dream"). The integral from 0 to infinity is approximately 1.995455958. As far as I know, none of these constants has a closed expression in terms of familiar special functions. 0 Share this post Link to post

ducon Posted February 22, 2007 You can also write x^−x as e^(−x ln(x)), −x ln(x)→−∞ when x→+∞ so its exponential→0. Note that I don’t know where you search its limit: 0, +∞, somewhere else? 0 Share this post Link to post

printz Posted February 22, 2007 It's simple. x^-x = 1/(x^x). x^x leads to infinite, so 1/(x^x) leads to zero. The thread title looks like a dead emoticon. 0 Share this post Link to post

Little Faith Posted February 22, 2007 Fredrik said:<Pretty Graph> Do you have some sort of program that renders graphs in ASCII? 0 Share this post Link to post

Fredrik Posted February 22, 2007 Little Faith said:Do you have some sort of program that renders graphs in ASCII? http://pari.math.u-bordeaux.fr/dochtml/html.stable/Plotting_functions.html#plot 0 Share this post Link to post

exp(x) Posted February 22, 2007 Little Faith said:Do you have some sort of program that renders graphs in ASCII? Maple can do it, too. 0 Share this post Link to post

Shadow Dweller Posted February 22, 2007 damn it, i just got home from school too. curse you, algebra 2... 0 Share this post Link to post