Self-root function and reciprocal self-power function have same integrals - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Self-root function and reciprocal self-power function have same integrals (/showthread.php?tid=436) Self-root function and reciprocal self-power function have same integrals - Ztolk - 04/03/2010 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. RE: Self-root function and reciprocal self-power function have same integrals - tommy1729 - 04/07/2010 (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 ? RE: Self-root function and reciprocal self-power function have same integrals - Ztolk - 04/07/2010 What is that? RE: Self-root function and reciprocal self-power function have same integrals - tommy1729 - 04/07/2010 (04/07/2010, 04:04 PM)Ztolk Wrote: What is that? http://en.wikipedia.org/wiki/Integration_by_substitution 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 RE: Self-root function and reciprocal self-power function have same integrals - tommy1729 - 04/10/2010 (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 RE: Self-root function and reciprocal self-power function have same integrals - Ztolk - 04/11/2010 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.