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 Self-root function and reciprocal self-power function have same integrals Ztolk Junior Fellow Posts: 21 Threads: 4 Joined: Mar 2010 04/03/2010, 04:25 AM I was playing around with Maple and I noticed that. $\int_{0}^{\infty}x^{\frac{1}{x}-2}dx=\int_{0}^{\infty}x^{-x}dx$=1.995455958 I then added some parameters and came up with the following: $\int_{0}^{\infty}x^{a/x^{b}-c}dx=\int_{0}^{\infty}x^{-ax^{b}+(c-2)}dx$ For positive a and b, and c>2. I do not know why this is, but I find it very interesting. The self-root function is the inverse of an infinite order tetration. tommy1729 Ultimate Fellow Posts: 1,675 Threads: 368 Joined: Feb 2009 04/07/2010, 03:44 PM (04/03/2010, 04:25 AM)Ztolk Wrote: I was playing around with Maple and I noticed that. $\int_{0}^{\infty}x^{\frac{1}{x}-2}dx=\int_{0}^{\infty}x^{-x}dx$=1.995455958 I then added some parameters and came up with the following: $\int_{0}^{\infty}x^{a/x^{b}-c}dx=\int_{0}^{\infty}x^{-ax^{b}+(c-2)}dx$ For positive a and b, and c>2. I do not know why this is, but I find it very interesting. The self-root function is the inverse of an infinite order tetration. have you tried substitution ? a moebius substitution ? Ztolk Junior Fellow Posts: 21 Threads: 4 Joined: Mar 2010 04/07/2010, 04:04 PM What is that? tommy1729 Ultimate Fellow Posts: 1,675 Threads: 368 Joined: Feb 2009 04/07/2010, 04:17 PM (04/07/2010, 04:04 PM)Ztolk Wrote: What is that? http://en.wikipedia.org/wiki/Integration...bstitution http://en.wikipedia.org/wiki/M%C3%B6bius_transformation combining the above two and leaving out the restriction ad -bc = 0 if needed. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,675 Threads: 368 Joined: Feb 2009 04/10/2010, 10:18 PM (04/03/2010, 04:25 AM)Ztolk Wrote: I was playing around with Maple and I noticed that. $\int_{0}^{\infty}x^{\frac{1}{x}-2}dx=\int_{0}^{\infty}x^{-x}dx$=1.995455958 the substitution y = 1/x proves it. regards tommy1729 Ztolk Junior Fellow Posts: 21 Threads: 4 Joined: Mar 2010 04/11/2010, 12:52 AM (This post was last modified: 04/11/2010, 01:45 AM by Ztolk.) So it does. Neat. Thanks. I should try this with the full three parameter version. edit: 1/x substitution checks out for the full thing. « Next Oldest | Next Newest »

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