Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Taylor series of i[x]
#13
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
Reply


Messages In This Thread
Taylor series of i[x] - by Xorter - 01/12/2017, 04:50 PM
RE: Taylor series of i[x] - by sheldonison - 01/12/2017, 08:50 PM
RE: Taylor series of i[x] - by Xorter - 01/13/2017, 05:26 PM
RE: Taylor series of i[x] - by Xorter - 01/13/2017, 07:13 PM
RE: Taylor series of i[x] - by Xorter - 01/14/2017, 10:14 AM
RE: Taylor series of i[x] - by mike3 - 01/23/2017, 07:38 AM
RE: Taylor series of i[x] - by Xorter - 02/26/2017, 11:10 AM
RE: Taylor series of i[x] - by Xorter - 03/01/2017, 03:06 PM
RE: Taylor series of i[x] - by Xorter - 03/04/2017, 09:40 AM
RE: Taylor series of i[x] - by Xorter - 04/06/2017, 03:42 PM
RE: Taylor series of i[x] - by Xorter - 04/11/2017, 12:18 PM
RE: Taylor series of i[x] - by Xorter - 07/10/2017, 04:07 PM
RE: Taylor series of i[x] - by Xorter - 02/20/2018, 09:55 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Perhaps a new series for log^0.5(x) Gottfried 3 4,105 03/21/2020, 08:28 AM
Last Post: Daniel
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 23,273 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 3,153 12/26/2016, 03:01 AM
Last Post: tommy1729
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 3,212 12/21/2016, 01:27 PM
Last Post: tommy1729
  Taylor series of cheta Xorter 13 24,262 08/28/2016, 08:52 PM
Last Post: sheldonison
  Taylor polynomial. System of equations for the coefficients. marraco 17 28,245 08/23/2016, 11:25 AM
Last Post: Gottfried
  Tetration series for integer exponent. Can you find the pattern? marraco 20 28,877 02/21/2016, 03:27 PM
Last Post: marraco
  [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 63,820 03/27/2015, 11:28 PM
Last Post: tommy1729
  [integral] How to integrate a fourier series ? tommy1729 1 4,605 05/04/2014, 03:19 PM
Last Post: tommy1729
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 24,472 02/04/2012, 05:02 AM
Last Post: Kouznetsov



Users browsing this thread: 1 Guest(s)