• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 iterated derivation Xorter Fellow Posts: 93 Threads: 30 Joined: Aug 2016 06/09/2019, 09:43 PM Hi, everyone! To be honest, iterated derivation as shifting may have been better, because as you will see, it is connected to shifting, but not known if it is just luck or what. As you can see in Dr. Peyam's vlog the exponential derivation is simply shifting the variable of function by the power of the derivation. So for example: (exp D f)(x) = f(x+1) ((exp D)^N f)(x) = f(x+N) And the so-called normal derivative of a function can also be represented as shifting if you draw the Taylor series of the function as in a "number" system: each coefficients are shifted to right (the radix is the variable x). What if all the function of the operator derivation is a kind of shifting of the function? Well, I have checked the (natural) logarithm of the operator, I do not want to make this post long and boring, so I am writing only the result. (If needed, it is okay for me to share the whies and hows.) So: (log D)(c x^n) = c Li2 x^n Which you can apply to all the coefficients of the Taylor series of a function. Would it be a kind of shifting? Furthermore I have checked also the double exponentiation of derivation, I have no so many results, but I know that it is related to numbers of stirling, but e. g. (exp exp D)(c x) = e(1 + c x). What is the rule of (exp^oN D f)(x) or more generally the rule of ((e [M] x)^oN o D f)(x)? Of course, (f^oN)(x) = (f o f o ... o f)(x) and D = d/dx, sorry, I should learn to use LaTex, again. Xorter Unizo « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Generalized Kneser superfunction trick (the iterated limit definition) MphLee 25 4,007 05/26/2021, 11:55 PM Last Post: MphLee 1st iterated derivatives and the tetration of 0 Xorter 0 2,510 05/12/2018, 12:34 PM Last Post: Xorter Iterated nand Xorter 2 6,064 03/27/2017, 06:51 PM Last Post: Xorter Iterated compositions Xorter 0 2,646 08/20/2016, 01:19 PM Last Post: Xorter Grzegorczyk hierarchy vs Iterated differential equations? MphLee 0 3,210 01/03/2015, 11:02 PM Last Post: MphLee Iterated polynomials JmsNxn 4 10,129 12/16/2010, 09:00 PM Last Post: JmsNxn The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] jaydfox 16 26,390 09/09/2007, 01:21 AM Last Post: jaydfox Bell formula for iterated exponentiation bo198214 1 5,692 08/26/2007, 12:27 PM Last Post: Gottfried computing the iterated exp(x)-1 Daniel 14 26,187 08/17/2007, 08:44 PM Last Post: andydude

Users browsing this thread: 1 Guest(s)