• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Taylor series of i[x] Xorter Fellow Posts: 76 Threads: 25 Joined: Aug 2016 04/11/2017, 12:18 PM Hi, again! Maybe I have found something related to the Nth derivative of i[x] (I may or may not know how to get the derivative of a non-analytic function like y = 1 if x=rational otherwise -1). Really interesting. But first of all, I would like to check it myself. But I will need your help. Here is a produktum: $\prod_{k=0}^{N-1} k + {{1+(-1)^k} \over 2}$ I would like to interpolate this expression/function, but how? Any idea? (If I will be able to get the expansion of the Taylor series of i[x], then I promise I will share it with you.) Xorter Unizo Xorter Fellow Posts: 76 Threads: 25 Joined: Aug 2016 07/10/2017, 04:07 PM (This post was last modified: 07/10/2017, 04:43 PM by Xorter.) First of all, I am afraid, that it is necessary to get the derivatives of the logical gates. That would the 0th step to our target. Okey, it is not hard to make contiously, just like this: x xor y = lim (2^h x xor 2^h y)/2^h And we know that x xor y = (x and -y) or (-x and y) But how to derivate? Xorter Unizo Xorter Fellow Posts: 76 Threads: 25 Joined: Aug 2016 02/20/2018, 09:55 PM Okay, let us sign i[x] to just i and I[x] to just I where I[x] = int i[x] dx. D := d/dx So D I = i D i = i' D  i I = i' I + i^2 = i' I - 1 D I i = i^2 + I i' = I i' - 1 Therefore i' I = 1 + (i I)' and I i' = 1 + (I i)' D 1 I = D I 1 = 0 + i = i But D i÷I = (i' I - i^2)÷I^2 = (i' I + 1)÷I^2 = (2 + (i I)')÷I^2 D I÷i = (i^2 - I i')÷i^2 = I i' = 1 + (I i)' D I÷i = -D i I = 1 - i' I Thus D I i = - i' I = I i' - 1 furthermore i' I = 1 - I i' I do not have to talk about i' I is not I i', right? D 1÷I = (0 - i)÷I^2 = - i÷I^2 What is next? The only useful information is that 1 ÷ i = -i ... or anything else? Xorter Unizo « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Taylor series of cheta Xorter 13 4,897 08/28/2016, 08:52 PM Last Post: sheldonison Taylor polynomial. System of equations for the coefficients. marraco 17 9,051 08/23/2016, 11:25 AM Last Post: Gottfried Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 11,552 02/04/2012, 05:02 AM Last Post: Kouznetsov Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 1,992 09/04/2011, 05:59 AM Last Post: Gottfried Does anyone have taylor series approximations for tetration and slog base e^(1/e)? JmsNxn 18 13,851 06/05/2011, 08:47 PM Last Post: sheldonison paradox, accurate taylor series half iterate of eta not analytic at e sheldonison 9 8,771 06/03/2011, 05:56 PM Last Post: bo198214 weird series expansion tommy1729 2 2,946 07/05/2010, 07:59 PM Last Post: tommy1729 Something interesting about Taylor series Ztolk 3 4,608 06/29/2010, 06:32 AM Last Post: bo198214 Transseries, nest-series, and other exotic series representations for tetration mike3 26 20,453 12/15/2009, 01:00 PM Last Post: bo198214 Taylor-Dirichlet conversion andydude 1 1,948 11/20/2009, 10:22 AM Last Post: andydude

Users browsing this thread: 1 Guest(s)