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.
\( \int_{0}^{\infty}x^{\frac{1}{x}-2}dx=\int_{0}^{\infty}x^{-x}dx \)=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.