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+--- 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.