08/23/2009, 02:07 PM
(This post was last modified: 08/24/2009, 04:39 PM by Base-Acid Tetration.)
I mean, what would we call "exponentiating" a function by plugging it into the taylor series, if we were to call f's superfuntion an f-exponential?
I suggest the term "f-iteration series of g" for the function/operator g plugged into f's taylor series.
So for a function/operator f, "exp-iteration series of f" would mean the series: \( \sum_{n=0}^\infty \frac{f^{\circ n}(x)}{n!} \) The taylor's theorem will say: for a smooth function f, f approximately equals f plugged into the exp-iteration series of (x-x0)D around x0.
"sin-iteration series of f" will mean: \( \sum_{n=0}^\infty \frac{-^n f^{\circ 2n+1}(x)}{(2n+1)!} \)
etc. etc.
I suggest the term "f-iteration series of g" for the function/operator g plugged into f's taylor series.
So for a function/operator f, "exp-iteration series of f" would mean the series: \( \sum_{n=0}^\infty \frac{f^{\circ n}(x)}{n!} \) The taylor's theorem will say: for a smooth function f, f approximately equals f plugged into the exp-iteration series of (x-x0)D around x0.
"sin-iteration series of f" will mean: \( \sum_{n=0}^\infty \frac{-^n f^{\circ 2n+1}(x)}{(2n+1)!} \)
etc. etc.