• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Taylor series of i[x] Xorter Fellow Posts: 93 Threads: 30 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: 93 Threads: 30 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: 93 Threads: 30 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 Perhaps a new series for log^0.5(x) Gottfried 3 3,963 03/21/2020, 08:28 AM Last Post: Daniel An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 22,826 01/14/2017, 09:09 PM Last Post: Vladimir Reshetnikov Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 3,086 12/26/2016, 03:01 AM Last Post: tommy1729 2 fixpoints , 1 period --> method of iteration series tommy1729 0 3,132 12/21/2016, 01:27 PM Last Post: tommy1729 Taylor series of cheta Xorter 13 23,818 08/28/2016, 08:52 PM Last Post: sheldonison Taylor polynomial. System of equations for the coefficients. marraco 17 27,737 08/23/2016, 11:25 AM Last Post: Gottfried Tetration series for integer exponent. Can you find the pattern? marraco 20 28,362 02/21/2016, 03:27 PM Last Post: marraco [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 62,792 03/27/2015, 11:28 PM Last Post: tommy1729 [integral] How to integrate a fourier series ? tommy1729 1 4,519 05/04/2014, 03:19 PM Last Post: tommy1729 Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 24,121 02/04/2012, 05:02 AM Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s)